gmp-doc-5.1.2/0002755000175000000620000000000012147565653012073 5ustar stevestaffgmp-doc-5.1.2/doc/0002755000175000000620000000000012147565653012640 5ustar stevestaffgmp-doc-5.1.2/doc/gmp.info-20000644000175000000620000057056312146435202014436 0ustar stevestaffThis is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from ../../gmp/doc/gmp.texi. This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover Texts being "You have freedom to copy and modify this GNU Manual, like GNU software". A copy of the license is included in *note GNU Free Documentation License::. INFO-DIR-SECTION GNU libraries START-INFO-DIR-ENTRY * gmp: (gmp). GNU Multiple Precision Arithmetic Library. END-INFO-DIR-ENTRY  File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms 15.3.5 Jacobi Symbol -------------------- [This section is obsolete. The current Jacobi code actually uses a very efficient algorithm.] `mpz_jacobi' and `mpz_kronecker' are currently implemented with a simple binary algorithm similar to that described for the GCDs (*note Binary GCD::). They're not very fast when both inputs are large. Lehmer's multi-step improvement or a binary based multi-step algorithm is likely to be better. When one operand fits a single limb, and that includes `mpz_kronecker_ui' and friends, an initial reduction is done with either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient. In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps.  File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms 15.4 Powering Algorithms ======================== * Menu: * Normal Powering Algorithm:: * Modular Powering Algorithm::  File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms 15.4.1 Normal Powering ---------------------- Normal `mpz' or `mpf' powering uses a simple binary algorithm, successively squaring and then multiplying by the base when a 1 bit is seen in the exponent, as per Knuth section 4.6.3. The "left to right" variant described there is used rather than algorithm A, since it's just as easy and can be done with somewhat less temporary memory.  File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms 15.4.2 Modular Powering ----------------------- Modular powering is implemented using a 2^k-ary sliding window algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 (*note References::). k is chosen according to the size of the exponent. Larger exponents use larger values of k, the choice being made to minimize the average number of multiplications that must supplement the squaring. The modular multiplies and squarings use either a simple division or the REDC method by Montgomery (*note References::). REDC is a little faster, essentially saving N single limb divisions in a fashion similar to an exact remainder (*note Exact Remainder::).  File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms 15.5 Root Extraction Algorithms =============================== * Menu: * Square Root Algorithm:: * Nth Root Algorithm:: * Perfect Square Algorithm:: * Perfect Power Algorithm::  File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms 15.5.1 Square Root ------------------ Square roots are taken using the "Karatsuba Square Root" algorithm by Paul Zimmermann (*note References::). An input n is split into four parts of k bits each, so with b=2^k we have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so that either the high or second highest bit is set. In GMP, k is kept on a limb boundary and the input is left shifted (by an even number of bits) to normalize. The square root of the high two parts is taken, by recursive application of the algorithm (bottoming out in a one-limb Newton's method), s1,r1 = sqrtrem (a3*b + a2) This is an approximation to the desired root and is extended by a division to give s,r, q,u = divrem (r1*b + a1, 2*s1) s = s1*b + q r = u*b + a0 - q^2 The normalization requirement on a3 means at this point s is either correct or 1 too big. r is negative in the latter case, so if r < 0 then r = r + 2*s - 1 s = s - 1 The algorithm is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method, or as a variation on the schoolboy method (no longer taught) for square roots two digits at a time. If the remainder r is not required then usually only a few high limbs of r and u need to be calculated to determine whether an adjustment to s is required. This optimization is not currently implemented. In the Karatsuba multiplication range this algorithm is O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n limbs. In the FFT multiplication range this grows to a bound of O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. The algorithm does all its calculations in integers and the resulting `mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.  File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms 15.5.2 Nth Root --------------- Integer Nth roots are taken using Newton's method with the following iteration, where A is the input and n is the root to be taken. 1 A a[i+1] = - * ( --------- + (n-1)*a[i] ) n a[i]^(n-1) The initial approximation a[1] is generated bitwise by successively powering a trial root with or without new 1 bits, aiming to be just above the true root. The iteration converges quadratically when started from a good approximation. When n is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized.  File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms 15.5.3 Perfect Square --------------------- A significant fraction of non-squares can be quickly identified by checking whether the input is a quadratic residue modulo small integers. `mpz_perfect_square_p' first tests the input mod 256, which means just examining the low byte. Only 44 different values occur for squares mod 256, so 82.8% of inputs can be immediately identified as non-squares. On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for a total 99.25% of inputs identified as non-squares. On a 64-bit system 97 is tested too, for a total 99.62%. These moduli are chosen because they're factors of 2^24-1 (or 2^48-1 for 64-bits), and such a remainder can be quickly taken just using additions (see `mpn_mod_34lsub1'). When nails are in use moduli are instead selected by the `gen-psqr.c' program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1 could be done with nails using some extra bit shifts, but this is not currently implemented. In any case each modulus is applied to the `mpn_mod_34lsub1' or `mpn_mod_1' remainder and a table lookup identifies non-squares. By using a "modexact" style calculation, and suitably permuted tables, just one multiply each is required, see the code for details. Moduli are also combined to save operations, so long as the lookup tables don't become too big. `gen-psqr.c' does all the pre-calculations. A square root must still be taken for any value that passes these tests, to verify it's really a square and not one of the small fraction of non-squares that get through (i.e. a pseudo-square to all the tested bases). Clearly more residue tests could be done, `mpz_perfect_square_p' only uses a compact and efficient set. Big inputs would probably benefit from more residue testing, small inputs might be better off with less. The assumed distribution of squares versus non-squares in the input would affect such considerations.  File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms 15.5.4 Perfect Power -------------------- Detecting perfect powers is required by some factorization algorithms. Currently `mpz_perfect_power_p' is implemented using repeated Nth root extractions, though naturally only prime roots need to be considered. (*Note Nth Root Algorithm::.) If a prime divisor p with multiplicity e can be found, then only roots which are divisors of e need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked.  File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms 15.6 Radix Conversion ===================== Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation. * Menu: * Binary to Radix:: * Radix to Binary::  File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms 15.6.1 Binary to Radix ---------------------- Conversions from binary to a power-of-2 radix use a simple and fast O(N) bit extraction algorithm. Conversions from binary to other radices use one of two algorithms. Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Repeated divisions by b^n are made, where b is the radix and n is the biggest power that fits in a limb. But instead of simply using the remainder r from such divisions, an extra divide step is done to give a fractional limb representing r/b^n. The digits of r can then be extracted using multiplications by b rather than divisions. Special case code is provided for decimal, allowing multiplications by 10 to optimize to shifts and adds. Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is used. For an input t, powers b^(n*2^i) of the radix are calculated, until a power between t and sqrt(t) is reached. t is then divided by that largest power, giving a quotient which is the digits above that power, and a remainder which is those below. These two parts are in turn divided by the second highest power, and so on recursively. When a piece has been divided down to less than `GET_STR_DC_THRESHOLD' limbs, the basecase algorithm described above is used. The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (*note Divide and Conquer Division::), and big divisions tend to have less overheads than lots of separate single limb divisions anyway. But in any case the cost of calculating the powers b^(n*2^i) must first be overcome. `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent the same basic thing, the point where it becomes worth doing a big division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD' includes the cost of calculating the radix power required, whereas `GET_STR_DC_THRESHOLD' assumes that's already available, which is the case when recursing. Since the base case produces digits from least to most significant but they want to be stored from most to least, it's necessary to calculate in advance how many digits there will be, or at least be sure not to underestimate that. For GMP the number of input bits is multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up. The result is either correct or one too big. Examining some of the high bits of the input could increase the chance of getting the exact number of digits, but an exact result every time would not be practical, since in general the difference between numbers 100... and 99... is only in the last few bits and the work to identify 99... might well be almost as much as a full conversion. `mpf_get_str' doesn't currently use the algorithm described here, it multiplies or divides by a power of b to move the radix point to the just above the highest non-zero digit (or at worst one above that location), then multiplies by b^n to bring out digits. This is O(N^2) and is certainly not optimal. The r/b^n scheme described above for using multiplications to bring out digits might be useful for more than a single limb. Some brief experiments with it on the base case when recursing didn't give a noticeable improvement, but perhaps that was only due to the implementation. Something similar would work for the sub-quadratic divisions too, though there would be the cost of calculating a bigger radix power. Another possible improvement for the sub-quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced, i.e. the highest power would be an b^(n*k) approximately equal to sqrt(t), not restricted to a 2^i factor. That ought to smooth out a graph of times against sizes, but may or may not be a net speedup.  File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms 15.6.2 Radix to Binary ---------------------- *This section needs to be rewritten, it currently describes the algorithms used before GMP 4.3.* Conversions from a power-of-2 radix into binary use a simple and fast O(N) bitwise concatenation algorithm. Conversions from other radices use one of two algorithms. Sizes below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups of n digits are converted to limbs, where n is the biggest power of the base b which will fit in a limb, then those groups are accumulated into the result by multiplying by b^n and adding. This saves multi-precision operations, as per Knuth section 4.4 part E (*note References::). Some special case code is provided for decimal, giving the compiler a chance to optimize multiplications by 10. Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is used. First groups of n digits are converted into limbs. Then adjacent limbs are combined into limb pairs with x*b^n+y, where x and y are the limbs. Adjacent limb pairs are combined into quads similarly with x*b^(2n)+y. This continues until a single block remains, that being the result. The advantage of this method is that the multiplications for each x are big blocks, allowing Karatsuba and higher algorithms to be used. But the cost of calculating the powers b^(n*2^i) must be overcome. `SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000 digits, and on some processors much bigger still. `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and tuned for decimal), though it might be better based on a limb count, so as to be independent of the base. But that sort of count isn't used by the base case and so would need some sort of initial calculation or estimate. The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that `mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5, or more).  File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms 15.7 Other Algorithms ===================== * Menu: * Prime Testing Algorithm:: * Factorial Algorithm:: * Binomial Coefficients Algorithm:: * Fibonacci Numbers Algorithm:: * Lucas Numbers Algorithm:: * Random Number Algorithms::  File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms 15.7.1 Prime Testing -------------------- The primality testing in `mpz_probab_prime_p' (*note Number Theoretic Functions::) first does some trial division by small factors and then uses the Miller-Rabin probabilistic primality testing algorithm, as described in Knuth section 4.5.4 algorithm P (*note References::). For an odd input n, and with n = q*2^k+1 where q is odd, this algorithm selects a random base x and tests whether x^q mod n is 1 or -1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably prime, if not then n is definitely composite. Any prime n will pass the test, but some composites do too. Such composites are known as strong pseudoprimes to base x. No n is a strong pseudoprime to more than 1/4 of all bases (see Knuth exercise 22), hence with x chosen at random there's no more than a 1/4 chance a "probable prime" will in fact be composite. In fact strong pseudoprimes are quite rare, making the test much more powerful than this analysis would suggest, but 1/4 is all that's proven for an arbitrary n.  File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms 15.7.2 Factorial ---------------- Factorials are calculated by a combination of two algorithms. An idea is shared among them: to compute the odd part of the factorial; a final step takes account of the power of 2 term, by shifting. For small n, the odd factor of n! is computed with the simple observation that it is equal to the product of all positive odd numbers smaller than n times the odd factor of [n/2]!, where [x] is the integer part of x, and so on recursively. The procedure can be best illustrated with an example, 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^19 Current code collects all the factors in a single list, with a loop and no recursion, and compute the product, with no special care for repeated chunks. When n is larger, computation pass trough prime sieving. An helper function is used, as suggested by Peter Luschny: n ----- n! | | L(p,n) msf(n) = -------------- = | | p [n/2]!^2.2^k p=3 Where p ranges on odd prime numbers. The exponent k is chosen to obtain an odd integer number: k is the number of 1 bits in the binary representation of [n/2]. The function L(p,n) can be defined as zero when p is composite, and, for any prime p, it is computed with: --- \ n L(p,n) = / [---] mod 2 <= log (n) . --- p^i p i>0 With this helper function, we are able to compute the odd part of n! using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion stops using the small-n algorithm on some [n/2^i]. Both the above algorithms use binary splitting to compute the product of many small factors. At first as many products as possible are accumulated in a single register, generating a list of factors that fit in a machine word. This list is then split into halves, and the product is computed recursively. Such splitting is more efficient than repeated Nx1 multiplies since it forms big multiplies, allowing Karatsuba and higher algorithms to be used. And even below the Karatsuba threshold a big block of work can be more efficient for the basecase algorithm.  File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms 15.7.3 Binomial Coefficients ---------------------------- Binomial coefficients C(n,k) are calculated by first arranging k <= n/2 using C(n,k) = C(n,n-k) if necessary, and then evaluating the following product simply from i=2 to i=k. k (n-k+i) C(n,k) = (n-k+1) * prod ------- i=2 i It's easy to show that each denominator i will divide the product so far, so the exact division algorithm is used (*note Exact Division::). The numerators n-k+i and denominators i are first accumulated into as many fit a limb, to save multi-precision operations, though for `mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t' and n-k+i in general won't fit in a limb at all.  File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms 15.7.4 Fibonacci Numbers ------------------------ The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for calculating isolated F[n] or F[n],F[n-1] values efficiently. For small n, a table of single limb values in `__gmp_fib_table' is used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up to F[93]. For convenience the table starts at F[-1]. Beyond the table, values are generated with a binary powering algorithm, calculating a pair F[n] and F[n-1] working from high to low across the bits of n. The formulas used are F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k F[2k-1] = F[k]^2 + F[k-1]^2 F[2k] = F[2k+1] - F[2k-1] At each step, k is the high b bits of n. If the next bit of n is 0 then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used, and the process repeated until all bits of n are incorporated. Notice these formulas require just two squares per bit of n. It'd be possible to handle the first few n above the single limb table with simple additions, using the defining Fibonacci recurrence F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to be faster for only about 10 or 20 values of n, and including a block of code for just those doesn't seem worthwhile. If they really mattered it'd be better to extend the data table. Using a table avoids lots of calculations on small numbers, and makes small n go fast. A bigger table would make more small n go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm. `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only interested in F[n]. In this case the last step of the algorithm can become one multiply instead of two squares. One of the following two formulas is used, according as n is odd or even. F[2k] = F[k]*(F[k]+2F[k-1]) F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k F[2k+1] here is the same as above, just rearranged to be a multiply. For interest, the 2*(-1)^k term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into further limbs, which saves some code size. See comments with `mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.  File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms 15.7.5 Lucas Numbers -------------------- `mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas. L[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1] `mpz_lucnum_ui' is only interested in L[n], and some work can be saved. Trailing zero bits on n can be handled with a single square each. L[2k] = L[k]^2 - 2*(-1)^k And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers, similar to what `mpz_fib_ui' does. L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k  File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms 15.7.6 Random Numbers --------------------- For the `urandomb' functions, random numbers are generated simply by concatenating bits produced by the generator. As long as the generator has good randomness properties this will produce well-distributed N bit numbers. For the `urandomm' functions, random numbers in a range 0<=R48 bit pieces is convenient. With some care though six 21x32->53 bit products can be used, if one of the lower two 21-bit pieces also uses the sign bit. For the `mpn_mul_1' family of functions on a 64-bit machine, the invariant single limb is split at the start, into 3 or 4 pieces. Inside the loop, the bignum operand is split into 32-bit pieces. Fast conversion of these unsigned 32-bit pieces to floating point is highly machine-dependent. In some cases, reading the data into the integer unit, zero-extending to 64-bits, then transferring to the floating point unit back via memory is the only option. Converting partial products back to 64-bit limbs is usually best done as a signed conversion. Since all values are smaller than 2^53, signed and unsigned are the same, but most processors lack unsigned conversions. Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or `mpn_addmul_1' with a 64-bit limb. The single limb operand V is split into four 16-bit parts. The multi-limb operand U is split in the loop into two 32-bit parts. +---+---+---+---+ |v48|v32|v16|v00| V operand +---+---+---+---+ +-------+---+---+ x | u32 | u00 | U operand (one limb) +---------------+ --------------------------------- +-----------+ | u00 x v00 | p00 48-bit products +-----------+ +-----------+ | u00 x v16 | p16 +-----------+ +-----------+ | u00 x v32 | p32 +-----------+ +-----------+ | u00 x v48 | p48 +-----------+ +-----------+ | u32 x v00 | r32 +-----------+ +-----------+ | u32 x v16 | r48 +-----------+ +-----------+ | u32 x v32 | r64 +-----------+ +-----------+ | u32 x v48 | r80 +-----------+ p32 and r32 can be summed using floating-point addition, and likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from the previous iteration. For each loop then, four 49-bit quantities are transferred to the integer unit, aligned as follows, |-----64bits----|-----64bits----| +------------+ | p00 + r64' | i00 +------------+ +------------+ | p16 + r80' | i16 +------------+ +------------+ | p32 + r32 | i32 +------------+ +------------+ | p48 + r48 | i48 +------------+ The challenge then is to sum these efficiently and add in a carry limb, generating a low 64-bit result limb and a high 33-bit carry limb (i48 extends 33 bits into the high half).  File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding 15.8.7 SIMD Instructions ------------------------ The single-instruction multiple-data support in current microprocessors is aimed at signal processing algorithms where each data point can be treated more or less independently. There's generally not much support for propagating the sort of carries that arise in GMP. SIMD multiplications of say four 16x16 bit multiplies only do as much work as one 32x32 from GMP's point of view, and need some shifts and adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile. On the x86 chips, MMX has so far found a use in `mpn_rshift' and `mpn_lshift', and is used in a special case for 16-bit multipliers in the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1', `mpn_addmul_1', and `mpn_submul_1'.  File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding 15.8.8 Software Pipelining -------------------------- Software pipelining consists of scheduling instructions around the branch point in a loop. For example a loop might issue a load not for use in the present iteration but the next, thereby allowing extra cycles for the data to arrive from memory. Naturally this is wanted only when doing things like loads or multiplies that take several cycles to complete, and only where a CPU has multiple functional units so that other work can be done in the meantime. A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage. This is like juggling. If the latency of some instruction is greater than the loop time then it will be necessary to unroll, so one register has a result ready to use while another (or multiple others) are still in progress. (*note Assembly Loop Unrolling::).  File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding 15.8.9 Loop Unrolling --------------------- Loop unrolling consists of replicating code so that several limbs are processed in each loop. At a minimum this reduces loop overheads by a corresponding factor, but it can also allow better register usage, for example alternately using one register combination and then another. Judicious use of `m4' macros can help avoid lots of duplication in the source code. Any amount of unrolling can be handled with a loop counter that's decremented by N each time, stopping when the remaining count is less than the further N the loop will process. Or by subtracting N at the start, the termination condition becomes when the counter C is less than 0 (and the count of remaining limbs is C+N). Alternately for a power of 2 unroll the loop count and remainder can be established with a shift and mask. This is convenient if also making a computed jump into the middle of a large loop. The limbs not a multiple of the unrolling can be handled in various ways, for example * A simple loop at the end (or the start) to process the excess. Care will be wanted that it isn't too much slower than the unrolled part. * A set of binary tests, for example after an 8-limb unrolling, test for 4 more limbs to process, then a further 2 more or not, and finally 1 more or not. This will probably take more code space than a simple loop. * A `switch' statement, providing separate code for each possible excess, for example an 8-limb unrolling would have separate code for 0 remaining, 1 remaining, etc, up to 7 remaining. This might take a lot of code, but may be the best way to optimize all cases in combination with a deep pipelined loop. * A computed jump into the middle of the loop, thus making the first iteration handle the excess. This should make times smoothly increase with size, which is attractive, but setups for the jump and adjustments for pointers can be tricky and could become quite difficult in combination with deep pipelining.  File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding 15.8.10 Writing Guide --------------------- This is a guide to writing software pipelined loops for processing limb vectors in assembly. First determine the algorithm and which instructions are needed. Code it without unrolling or scheduling, to make sure it works. On a 3-operand CPU try to write each new value to a new register, this will greatly simplify later steps. Then note for each instruction the functional unit and/or issue port requirements. If an instruction can use either of two units, like U0 or U1 then make a category "U0/U1". Count the total using each unit (or combined unit), and count all instructions. Figure out from those counts the best possible loop time. The goal will be to find a perfect schedule where instruction latencies are completely hidden. The total instruction count might be the limiting factor, or perhaps a particular functional unit. It might be possible to tweak the instructions to help the limiting factor. Suppose the loop time is N, then make N issue buckets, with the final loop branch at the end of the last. Now fill the buckets with dummy instructions using the functional units desired. Run this to make sure the intended speed is reached. Now replace the dummy instructions with the real instructions from the slow but correct loop you started with. The first will typically be a load instruction. Then the instruction using that value is placed in a bucket an appropriate distance down. Run the loop again, to check it still runs at target speed. Keep placing instructions, frequently measuring the loop. After a few you will need to wrap around from the last bucket back to the top of the loop. If you used the new-register for new-value strategy above then there will be no register conflicts. If not then take care not to clobber something already in use. Changing registers at this time is very error prone. The loop will overlap two or more of the original loop iterations, and the computation of one vector element result will be started in one iteration of the new loop, and completed one or several iterations later. The final step is to create feed-in and wind-down code for the loop. A good way to do this is to make a copy (or copies) of the loop at the start and delete those instructions which don't have valid antecedents, and at the end replicate and delete those whose results are unwanted (including any further loads). The loop will have a minimum number of limbs loaded and processed, so the feed-in code must test if the request size is smaller and skip either to a suitable part of the wind-down or to special code for small sizes.  File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top 16 Internals ************ *This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases. Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters.* * Menu: * Integer Internals:: * Rational Internals:: * Float Internals:: * Raw Output Internals:: * C++ Interface Internals::  File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals 16.1 Integer Internals ====================== `mpz_t' variables represent integers using sign and magnitude, in space dynamically allocated and reallocated. The fields are as follows. `_mp_size' The number of limbs, or the negative of that when representing a negative integer. Zero is represented by `_mp_size' set to zero, in which case the `_mp_d' data is unused. `_mp_d' A pointer to an array of limbs which is the magnitude. These are stored "little endian" as per the `mpn' functions, so `_mp_d[0]' is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the most significant. Whenever `_mp_size' is non-zero, the most significant limb is non-zero. Currently there's always at least one limb allocated, so for instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui' can fetch `_mp_d[0]' unconditionally (though its value is then only wanted if `_mp_size' is non-zero). `_mp_alloc' `_mp_alloc' is the number of limbs currently allocated at `_mp_d', and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine is about to (or might be about to) increase `_mp_size', it checks `_mp_alloc' to see whether there's enough space, and reallocates if not. `MPZ_REALLOC' is generally used for this. The various bitwise logical functions like `mpz_and' behave as if negative values were twos complement. But sign and magnitude is always used internally, and necessary adjustments are made during the calculations. Sometimes this isn't pretty, but sign and magnitude are best for other routines. Some internal temporary variables are setup with `MPZ_TMP_INIT' and these have `_mp_d' space obtained from `TMP_ALLOC' rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have unpredictable consequences). `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is usually a `long'. This is done to make the fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of range.  File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals 16.2 Rational Internals ======================= `mpq_t' variables represent rationals using an `mpz_t' numerator and denominator (*note Integer Internals::). The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1. It's believed that casting out common factors at each stage of a calculation is best in general. A GCD is an O(N^2) operation so it's better to do a few small ones immediately than to delay and have to do a big one later. Knowing the numerator and denominator have no common factors can be used for example in `mpq_mul' to make only two cross GCDs necessary, not four. This general approach to common factors is badly sub-optimal in the presence of simple factorizations or little prospect for cancellation, but GMP has no way to know when this will occur. As per *note Efficiency::, that's left to applications. The `mpq_t' framework might still suit, with `mpq_numref' and `mpq_denref' for direct access to the numerator and denominator, or of course `mpz_t' variables can be used directly.  File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals 16.3 Float Internals ==================== Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this. `mpf_t' floats have a variable precision mantissa and a single machine word signed exponent. The mantissa is represented using sign and magnitude. most least significant significant limb limb _mp_d |---- _mp_exp ---> | _____ _____ _____ _____ _____ |_____|_____|_____|_____|_____| . <------------ radix point <-------- _mp_size ---------> The fields are as follows. `_mp_size' The number of limbs currently in use, or the negative of that when representing a negative value. Zero is represented by `_mp_size' and `_mp_exp' both set to zero, and in that case the `_mp_d' data is unused. (In the future `_mp_exp' might be undefined when representing zero.) `_mp_prec' The precision of the mantissa, in limbs. In any calculation the aim is to produce `_mp_prec' limbs of result (the most significant being non-zero). `_mp_d' A pointer to the array of limbs which is the absolute value of the mantissa. These are stored "little endian" as per the `mpn' functions, so `_mp_d[0]' is the least significant limb and `_mp_d[ABS(_mp_size)-1]' the most significant. The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb. `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being for convenience (see below). There are no reallocations during a calculation, only in a change of precision with `mpf_set_prec'. `_mp_exp' The exponent, in limbs, determining the location of the implied radix point. Zero means the radix point is just above the most significant limb. Positive values mean a radix point offset towards the lower limbs and hence a value >= 1, as for example in the diagram above. Negative exponents mean a radix point further above the highest limb. Naturally the exponent can be any value, it doesn't have to fall within the limbs as the diagram shows, it can be a long way above or a long way below. Limbs other than those included in the `{_mp_d,_mp_size}' data are treated as zero. The `_mp_size' and `_mp_prec' fields are `int', although the `mp_size_t' type is usually a `long'. The `_mp_exp' field is usually `long'. This is done to make some fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of precision and a very large range. The following various points should be noted. Low Zeros The least significant limbs `_mp_d[0]' etc can be zero, though such low zeros can always be ignored. Routines likely to produce low zeros check and avoid them to save time in subsequent calculations, but for most routines they're quite unlikely and aren't checked. Mantissa Size Range The `_mp_size' count of limbs in use can be less than `_mp_prec' if the value can be represented in less. This means low precision values or small integers stored in a high precision `mpf_t' can still be operated on efficiently. `_mp_size' can also be greater than `_mp_prec'. Firstly a value is allowed to use all of the `_mp_prec+1' limbs available at `_mp_d', and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves `_mp_size' unchanged and so the size can be arbitrarily bigger than `_mp_prec'. Rounding All rounding is done on limb boundaries. Calculating `_mp_prec' limbs with the high non-zero will ensure the application requested minimum precision is obtained. The use of simple "trunc" rounding towards zero is efficient, since there's no need to examine extra limbs and increment or decrement. Bit Shifts Since the exponent is in limbs, there are no bit shifts in basic operations like `mpf_add' and `mpf_mul'. When differing exponents are encountered all that's needed is to adjust pointers to line up the relevant limbs. Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit shifts, but the choice is between an exponent in limbs which requires shifts there, or one in bits which requires them almost everywhere else. Use of `_mp_prec+1' Limbs The extra limb on `_mp_d' (`_mp_prec+1' rather than just `_mp_prec') helps when an `mpf' routine might get a carry from its operation. `mpf_add' for instance will do an `mpn_add' of `_mp_prec' limbs. If there's no carry then that's the result, but if there is a carry then it's stored in the extra limb of space and `_mp_size' becomes `_mp_prec+1'. Whenever `_mp_prec+1' limbs are held in a variable, the low limb is not needed for the intended precision, only the `_mp_prec' high limbs. But zeroing it out or moving the rest down is unnecessary. Subsequent routines reading the value will simply take the high limbs they need, and this will be `_mp_prec' if their target has that same precision. This is no more than a pointer adjustment, and must be checked anyway since the destination precision can be different from the sources. Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs if available. This ensures that a variable which has `_mp_size' equal to `_mp_prec+1' will get its full exact value copied. Strictly speaking this is unnecessary since only `_mp_prec' limbs are needed for the application's requested precision, but it's considered that an `mpf_set' from one variable into another of the same precision ought to produce an exact copy. Application Precisions `__GMPF_BITS_TO_PREC' converts an application requested precision to an `_mp_prec'. The value in bits is rounded up to a whole limb then an extra limb is added since the most significant limb of `_mp_d' is only non-zero and therefore might contain only one bit. `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the extra limb from `_mp_prec' before converting to bits. The net effect of reading back with `mpf_get_prec' is simply the precision rounded up to a multiple of `mp_bits_per_limb'. Note that the extra limb added here for the high only being non-zero is in addition to the extra limb allocated to `_mp_d'. For example with a 32-bit limb, an application request for 250 bits will be rounded up to 8 limbs, then an extra added for the high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then gets 10 limbs allocated. Reading back with `mpf_get_prec' will take `_mp_prec' subtract 1 limb and multiply by 32, giving 256 bits. Strictly speaking, the fact the high limb has at least one bit means that a float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but for the purposes of `mpf_t' it's considered simply to be 64 bits, a nice multiple of the limb size.  File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals 16.4 Raw Output Internals ========================= `mpz_out_raw' uses the following format. +------+------------------------+ | size | data bytes | +------+------------------------+ The size is 4 bytes written most significant byte first, being the number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first. The most significant data byte is always non-zero, so the output is the same on all systems, irrespective of limb size. In GMP 1, leading zero bytes were written to pad the data bytes to a multiple of the limb size. `mpz_inp_raw' will still accept this, for compatibility. The use of "big endian" for both the size and data fields is deliberate, it makes the data easy to read in a hex dump of a file. Unfortunately it also means that the limb data must be reversed when reading or writing, so neither a big endian nor little endian system can just read and write `_mp_d'.  File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals 16.5 C++ Interface Internals ============================ A system of expression templates is used to ensure something like `a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the scheme also ensures the precision of the final destination is used for any temporaries within a statement like `f=w*x+y*z'. These are important features which a naive implementation cannot provide. A simplified description of the scheme follows. The true scheme is complicated by the fact that expressions have different return types. For detailed information, refer to the source code. To perform an operation, say, addition, we first define a "function object" evaluating it, struct __gmp_binary_plus { static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); } }; And an "additive expression" object, __gmp_expr<__gmp_binary_expr > operator+(const mpf_class &f, const mpf_class &g) { return __gmp_expr <__gmp_binary_expr >(f, g); } The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used to encapsulate any possible kind of expression into a single template type. In fact even `mpf_class' etc are `typedef' specializations of `__gmp_expr'. Next we define assignment of `__gmp_expr' to `mpf_class'. template mpf_class & mpf_class::operator=(const __gmp_expr &expr) { expr.eval(this->get_mpf_t(), this->precision()); return *this; } template void __gmp_expr<__gmp_binary_expr >::eval (mpf_t f, mp_bitcnt_t precision) { Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); } where `expr.val1' and `expr.val2' are references to the expression's operands (here `expr' is the `__gmp_binary_expr' stored within the `__gmp_expr'). This way, the expression is actually evaluated only at the time of assignment, when the required precision (that of `f') is known. Furthermore the target `mpf_t' is now available, thus we can call `mpf_add' directly with `f' as the output argument. Compound expressions are handled by defining operators taking subexpressions as their arguments, like this: template __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > operator+(const __gmp_expr &expr1, const __gmp_expr &expr2) { return __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > (expr1, expr2); } And the corresponding specializations of `__gmp_expr::eval': template void __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, Op> >::eval (mpf_t f, mp_bitcnt_t precision) { // declare two temporaries mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); } The expression is thus recursively evaluated to any level of complexity and all subexpressions are evaluated to the precision of `f'.  File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top Appendix A Contributors *********************** Torbjörn Granlund wrote the original GMP library and is still the main developer. Code not explicitly attributed to others, was contributed by Torbjörn. Several other individuals and organizations have contributed GMP. Here is a list in chronological order on first contribution: Gunnar Sjödin and Hans Riesel helped with mathematical problems in early versions of the library. Richard Stallman helped with the interface design and revised the first version of this manual. Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions. John Amanatides of York University in Canada contributed the function `mpz_probab_prime_p'. Paul Zimmermann wrote the REDC-based mpz_powm code, the Schönhage-Strassen FFT multiply code, and the Karatsuba square root code. He also improved the Toom3 code for GMP 4.2. Paul sparked the development of GMP 2, with his comparisons between bignum packages. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code (with Torbjörn). Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact', `mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2. Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure. He has also made valuable suggestions and tested numerous intermediary releases. Joachim Hollman was involved in the design of the `mpf' interface, and in the `mpz' design revisions for version 2. Bennet Yee contributed the initial versions of `mpz_jacobi' and `mpz_legendre'. Andreas Schwab contributed the files `mpn/m68k/lshift.S' and `mpn/m68k/rshift.S' (now in `.asm' form). Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count. Robert also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3, and contributed the ARM assembly code. Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several algorithms. Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions. Kevin Ryde worked on a large number of things: optimized x86 code, m4 asm macros, parameter tuning, speed measuring, the configure system, function inlining, divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number functions, printf and scanf functions, perl interface, demo expression parser, the algorithms chapter in the manual, `gmpasm-mode.el', and various miscellaneous improvements elsewhere. Kent Boortz made the Mac OS 9 port. Steve Root helped write the optimized alpha 21264 assembly code. Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++ `istream' input routines. Jason Moxham rewrote `mpz_fac_ui'. Pedro Gimeno implemented the Mersenne Twister and made other random number improvements. Niels Möller wrote the sub-quadratic GCD, extended GCD and jacobi code, the quadratic Hensel division code, and (with Torbjörn) the new divide and conquer division code for GMP 4.3. Niels also helped implement the new Toom multiply code for GMP 4.3 and implemented helper functions to simplify Toom evaluations for GMP 5.0. He wrote the original version of mpn_mulmod_bnm1, and he is the main author of the mini-gmp package used for gmp bootstrapping. Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply strategy, and found the optimal strategies for evaluation and interpolation in Toom multiplication. Marco Bodrato helped implement the new Toom multiply code for GMP 4.3 and implemented most of the new Toom multiply and squaring code for 5.0. He is the main author of the current mpn_mulmod_bnm1 and mpn_mullo_n. Marco also wrote the functions mpn_invert and mpn_invertappr. He is the author of the current combinatorial functions: binomial, factorial, multifactorial, primorial. David Harvey suggested the internal function `mpn_bdiv_dbm1', implementing division relevant to Toom multiplication. He also worked on fast assembly sequences, in particular on a fast AMD64 `mpn_mul_basecase'. He wrote the internal middle product functions `mpn_mulmid_basecase', `mpn_toom42_mulmid', `mpn_mulmid_n' and related helper routines. Martin Boij wrote `mpn_perfect_power_p'. Marc Glisse improved `gmpxx.h': use fewer temporaries (faster), specializations of `numeric_limits' and `common_type', C++11 features (move constructors, explicit bool conversion, UDL), make the conversion from `mpq_class' to `mpz_class' explicit, optimize operations where one argument is a small compile-time constant, replace some heap allocations by stack allocations. He also fixed the eofbit handling of C++ streams, and removed one division from `mpq/aors.c'. (This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell about the omission!) The development of floating point functions of GNU MP 2, were supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial System SOlving). The development of GMP 2, 3, and 4 was supported in part by the IDA Center for Computing Sciences. Thanks go to Hans Thorsen for donating an SGI system for the GMP test system environment.  File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top Appendix B References ********************* B.1 Books ========= * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity", Wiley, 1998. * Richard Crandall and Carl Pomerance, "Prime Numbers: A Computational Perspective", 2nd edition, Springer-Verlag, 2005. `http://www.math.dartmouth.edu/~carlp/' * Henri Cohen, "A Course in Computational Algebraic Number Theory", Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. `http://www.math.u-bordeaux.fr/~cohen/' * Donald E. Knuth, "The Art of Computer Programming", volume 2, "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998. `http://www-cs-faculty.stanford.edu/~knuth/taocp.html' * John D. Lipson, "Elements of Algebra and Algebraic Computing", The Benjamin Cummings Publishing Company Inc, 1981. * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, "Handbook of Applied Cryptography", `http://www.cacr.math.uwaterloo.ca/hac/' * Richard M. Stallman and the GCC Developer Community, "Using the GNU Compiler Collection", Free Software Foundation, 2008, available online `http://gcc.gnu.org/onlinedocs/', and in the GCC package `ftp://ftp.gnu.org/gnu/gcc/' B.2 Papers ========== * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP Square Root", Journal of Automated Reasoning, volume 29, 2002, pp. 225-252. Also available online as INRIA Research Report 4475, June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf' * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022' * Torbjörn Granlund and Peter L. Montgomery, "Division by Invariant Integers using Multiplication", in Proceedings of the SIGPLAN PLDI'94 Conference, June 1994. Also available `http://gmplib.org/~tege/divcnst-pldi94.pdf'. * Niels Möller and Torbjörn Granlund, "Improved division by invariant integers", IEEE Transactions on Computers, 11 June 2010. `http://gmplib.org/~tege/division-paper.pdf' * Torbjörn Granlund and Niels Möller, "Division of integers large and small", to appear. * Tudor Jebelean, "An algorithm for exact division", Journal of Symbolic Computation, volume 15, 1993, pp. 169-180. Research report version available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz' * Tudor Jebelean, "Exact Division with Karatsuba Complexity - Extended Abstract", RISC-Linz technical report 96-31, `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz' * Tudor Jebelean, "Practical Integer Division with Karatsuba Complexity", ISSAC 97, pp. 339-341. Technical report available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz' * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm", ISSAC 93, pp. 111-116. Technical report version available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz' * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD of Long Integers", Journal of Symbolic Computation, volume 19, 1995, pp. 145-157. Technical report version also available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz' * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer Division", Journal of Symbolic Computation, volume 21, 1996, pp. 441-455. Early technical report version also available `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz' * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator", ACM Transactions on Modelling and Computer Simulation, volume 8, January 1998, pp. 3-30. Available online `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz' (or .pdf) * R. Moenck and A. Borodin, "Fast Modular Transforms via Division", Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast Modular Transforms", Journal of Computer and System Sciences, volume 8, number 3, June 1974, pp. 366-386. * Niels Möller, "On Schönhage's algorithm and subquadratic integer GCD computation", in Mathematics of Computation, volume 77, January 2008, pp. 589-607. * Peter L. Montgomery, "Modular Multiplication Without Trial Division", in Mathematics of Computation, volume 44, number 170, April 1985. * Arnold Schönhage and Volker Strassen, "Schnelle Multiplikation grosser Zahlen", Computing 7, 1971, pp. 281-292. * Kenneth Weber, "The accelerated integer GCD algorithm", ACM Transactions on Mathematical Software, volume 21, number 1, March 1995, pp. 111-122. * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report 3805, November 1999, `http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf' * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root Implementations", `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz' * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11: IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. Reprinted as "More on Multiplying and Squaring Large Integers", IEEE Transactions on Computers, volume 43, number 8, August 1994, pp. 899-908.  File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top Appendix C GNU Free Documentation License ***************************************** Version 1.3, 3 November 2008 Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. `http://fsf.org/' Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. 1. APPLICABILITY AND DEFINITIONS This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you". You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. A "Modified Version" of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language. A "Secondary Section" is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) 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For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text. The "publisher" means any person or entity that distributes copies of the Document to the public. A section "Entitled XYZ" means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as "Acknowledgements", "Dedications", "Endorsements", or "History".) To "Preserve the Title" of such a section when you modify the Document means that it remains a section "Entitled XYZ" according to this definition. The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. 2. VERBATIM COPYING You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3. You may also lend copies, under the same conditions stated above, and you may publicly display copies. 3. 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If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages. If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public. It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document. 4. MODIFICATIONS You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version: A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission. B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement. C. State on the Title page the name of the publisher of the Modified Version, as the publisher. D. Preserve all the copyright notices of the Document. E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices. F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below. G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice. H. Include an unaltered copy of this License. I. Preserve the section Entitled "History", Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence. J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the "History" section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission. K. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein. L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles. M. Delete any section Entitled "Endorsements". Such a section may not be included in the Modified Version. N. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in title with any Invariant Section. O. Preserve any Warranty Disclaimers. If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles. You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties--for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one. The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version. 5. COMBINING DOCUMENTS You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements." 6. COLLECTIONS OF DOCUMENTS You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document. 7. AGGREGATION WITH INDEPENDENT WORKS A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate. 8. TRANSLATION Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title. 9. TERMINATION You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License. However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it. 10. FUTURE REVISIONS OF THIS LICENSE The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See `http://www.gnu.org/copyleft/'. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document. 11. RELICENSING "Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site. "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization. "Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document. An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008. The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing. ADDENDUM: How to use this License for your documents ==================================================== To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright (C) YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.  File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top Concept Index ************* [index] * Menu: * #include: Headers and Libraries. (line 6) * --build: Build Options. (line 52) * --disable-fft: Build Options. (line 314) * --disable-shared: Build Options. (line 45) * --disable-static: Build Options. (line 45) * --enable-alloca: Build Options. (line 275) * --enable-assert: Build Options. (line 320) * --enable-cxx: Build Options. (line 227) * --enable-fat: Build Options. (line 162) * --enable-profiling <1>: Build Options. (line 324) * --enable-profiling: Profiling. (line 6) * --exec-prefix: Build Options. (line 32) * --host: Build Options. (line 66) * --prefix: Build Options. (line 32) * -finstrument-functions: Profiling. (line 66) * 2exp functions: Efficiency. (line 43) * 68000: Notes for Particular Systems. (line 80) * 80x86: Notes for Particular Systems. (line 127) * ABI <1>: Build Options. (line 169) * ABI: ABI and ISA. (line 6) * About this manual: Introduction to GMP. (line 58) * AC_CHECK_LIB: Autoconf. (line 11) * AIX <1>: Notes for Particular Systems. (line 7) * AIX: ABI and ISA. (line 169) * Algorithms: Algorithms. (line 6) * alloca: Build Options. (line 275) * Allocation of memory: Custom Allocation. (line 6) * AMD64: ABI and ISA. (line 44) * Anonymous FTP of latest version: Introduction to GMP. (line 38) * Application Binary Interface: ABI and ISA. (line 6) * Arithmetic functions <1>: Rational Arithmetic. (line 6) * Arithmetic functions <2>: Float Arithmetic. (line 6) * Arithmetic functions: Integer Arithmetic. (line 6) * ARM: Notes for Particular Systems. (line 20) * Assembly cache handling: Assembly Cache Handling. (line 6) * Assembly carry propagation: Assembly Carry Propagation. (line 6) * Assembly code organisation: Assembly Code Organisation. (line 6) * Assembly coding: Assembly Coding. (line 6) * Assembly floating Point: Assembly Floating Point. (line 6) * Assembly loop unrolling: Assembly Loop Unrolling. (line 6) * Assembly SIMD: Assembly SIMD Instructions. (line 6) * Assembly software pipelining: Assembly Software Pipelining. (line 6) * Assembly writing guide: Assembly Writing Guide. (line 6) * Assertion checking <1>: Build Options. (line 320) * Assertion checking: Debugging. (line 79) * Assignment functions <1>: Assigning Integers. (line 6) * Assignment functions <2>: Initializing Rationals. (line 6) * Assignment functions <3>: Assigning Floats. (line 6) * Assignment functions <4>: Simultaneous Float Init & Assign. (line 6) * Assignment functions: Simultaneous Integer Init & Assign. (line 6) * Autoconf: Autoconf. (line 6) * Basics: GMP Basics. (line 6) * Binomial coefficient algorithm: Binomial Coefficients Algorithm. (line 6) * Binomial coefficient functions: Number Theoretic Functions. (line 124) * Binutils strip: Known Build Problems. (line 28) * Bit manipulation functions: Integer Logic and Bit Fiddling. (line 6) * Bit scanning functions: Integer Logic and Bit Fiddling. (line 38) * Bit shift left: Integer Arithmetic. (line 35) * Bit shift right: Integer Division. (line 53) * Bits per limb: Useful Macros and Constants. (line 7) * Bug reporting: Reporting Bugs. (line 6) * Build directory: Build Options. (line 19) * Build notes for binary packaging: Notes for Package Builds. (line 6) * Build notes for particular systems: Notes for Particular Systems. (line 6) * Build options: Build Options. (line 6) * Build problems known: Known Build Problems. (line 6) * Build system: Build Options. (line 52) * Building GMP: Installing GMP. (line 6) * Bus error: Debugging. (line 7) * C compiler: Build Options. (line 180) * C++ compiler: Build Options. (line 251) * C++ interface: C++ Class Interface. (line 6) * C++ interface internals: C++ Interface Internals. (line 6) * C++ istream input: C++ Formatted Input. (line 6) * C++ ostream output: C++ Formatted Output. (line 6) * C++ support: Build Options. (line 227) * CC: Build Options. (line 180) * CC_FOR_BUILD: Build Options. (line 214) * CFLAGS: Build Options. (line 180) * Checker: Debugging. (line 115) * checkergcc: Debugging. (line 122) * Code organisation: Assembly Code Organisation. (line 6) * Compaq C++: Notes for Particular Systems. (line 25) * Comparison functions <1>: Comparing Rationals. (line 6) * Comparison functions <2>: Float Comparison. (line 6) * Comparison functions: Integer Comparisons. (line 6) * Compatibility with older versions: Compatibility with older versions. (line 6) * Conditions for copying GNU MP: Copying. (line 6) * Configuring GMP: Installing GMP. (line 6) * Congruence algorithm: Exact Remainder. (line 30) * Congruence functions: Integer Division. (line 124) * Constants: Useful Macros and Constants. (line 6) * Contributors: Contributors. (line 6) * Conventions for parameters: Parameter Conventions. (line 6) * Conventions for variables: Variable Conventions. (line 6) * Conversion functions <1>: Converting Integers. (line 6) * Conversion functions <2>: Converting Floats. (line 6) * Conversion functions: Rational Conversions. (line 6) * Copying conditions: Copying. (line 6) * CPPFLAGS: Build Options. (line 206) * CPU types <1>: Introduction to GMP. (line 24) * CPU types: Build Options. (line 108) * Cross compiling: Build Options. (line 66) * Custom allocation: Custom Allocation. (line 6) * CXX: Build Options. (line 251) * CXXFLAGS: Build Options. (line 251) * Cygwin: Notes for Particular Systems. (line 43) * Darwin: Known Build Problems. (line 51) * Debugging: Debugging. (line 6) * Demonstration programs: Demonstration Programs. (line 6) * Digits in an integer: Miscellaneous Integer Functions. (line 23) * Divisibility algorithm: Exact Remainder. (line 30) * Divisibility functions: Integer Division. (line 124) * Divisibility testing: Efficiency. (line 91) * Division algorithms: Division Algorithms. (line 6) * Division functions <1>: Rational Arithmetic. (line 22) * Division functions <2>: Float Arithmetic. (line 33) * Division functions: Integer Division. (line 6) * DJGPP <1>: Notes for Particular Systems. (line 43) * DJGPP: Known Build Problems. (line 18) * DLLs: Notes for Particular Systems. (line 56) * DocBook: Build Options. (line 347) * Documentation formats: Build Options. (line 340) * Documentation license: GNU Free Documentation License. (line 6) * DVI: Build Options. (line 343) * Efficiency: Efficiency. (line 6) * Emacs: Emacs. (line 6) * Exact division functions: Integer Division. (line 102) * Exact remainder: Exact Remainder. (line 6) * Example programs: Demonstration Programs. (line 6) * Exec prefix: Build Options. (line 32) * Execution profiling <1>: Profiling. (line 6) * Execution profiling: Build Options. (line 324) * Exponentiation functions <1>: Float Arithmetic. (line 41) * Exponentiation functions: Integer Exponentiation. (line 6) * Export: Integer Import and Export. (line 45) * Expression parsing demo: Demonstration Programs. (line 18) * Extended GCD: Number Theoretic Functions. (line 49) * Factor removal functions: Number Theoretic Functions. (line 105) * Factorial algorithm: Factorial Algorithm. (line 6) * Factorial functions: Number Theoretic Functions. (line 113) * Factorization demo: Demonstration Programs. (line 25) * Fast Fourier Transform: FFT Multiplication. (line 6) * Fat binary: Build Options. (line 162) * FFT multiplication <1>: Build Options. (line 314) * FFT multiplication: FFT Multiplication. (line 6) * Fibonacci number algorithm: Fibonacci Numbers Algorithm. (line 6) * Fibonacci sequence functions: Number Theoretic Functions. (line 132) * Float arithmetic functions: Float Arithmetic. (line 6) * Float assignment functions <1>: Simultaneous Float Init & Assign. (line 6) * Float assignment functions: Assigning Floats. (line 6) * Float comparison functions: Float Comparison. (line 6) * Float conversion functions: Converting Floats. (line 6) * Float functions: Floating-point Functions. (line 6) * Float initialization functions <1>: Simultaneous Float Init & Assign. (line 6) * Float initialization functions: Initializing Floats. (line 6) * Float input and output functions: I/O of Floats. (line 6) * Float internals: Float Internals. (line 6) * Float miscellaneous functions: Miscellaneous Float Functions. (line 6) * Float random number functions: Miscellaneous Float Functions. (line 27) * Float rounding functions: Miscellaneous Float Functions. (line 9) * Float sign tests: Float Comparison. (line 33) * Floating point mode: Notes for Particular Systems. (line 34) * Floating-point functions: Floating-point Functions. (line 6) * Floating-point number: Nomenclature and Types. (line 21) * fnccheck: Profiling. (line 77) * Formatted input: Formatted Input. (line 6) * Formatted output: Formatted Output. (line 6) * Free Documentation License: GNU Free Documentation License. (line 6) * frexp <1>: Converting Integers. (line 42) * frexp: Converting Floats. (line 23) * FTP of latest version: Introduction to GMP. (line 38) * Function classes: Function Classes. (line 6) * FunctionCheck: Profiling. (line 77) * GCC Checker: Debugging. (line 115) * GCD algorithms: Greatest Common Divisor Algorithms. (line 6) * GCD extended: Number Theoretic Functions. (line 49) * GCD functions: Number Theoretic Functions. (line 32) * GDB: Debugging. (line 58) * Generic C: Build Options. (line 153) * GMP Perl module: Demonstration Programs. (line 35) * GMP version number: Useful Macros and Constants. (line 12) * gmp.h: Headers and Libraries. (line 6) * gmpxx.h: C++ Interface General. (line 8) * GNU Debugger: Debugging. (line 58) * GNU Free Documentation License: GNU Free Documentation License. (line 6) * GNU strip: Known Build Problems. (line 28) * gprof: Profiling. (line 41) * Greatest common divisor algorithms: Greatest Common Divisor Algorithms. (line 6) * Greatest common divisor functions: Number Theoretic Functions. (line 32) * Hardware floating point mode: Notes for Particular Systems. (line 34) * Headers: Headers and Libraries. (line 6) * Heap problems: Debugging. (line 24) * Home page: Introduction to GMP. (line 34) * Host system: Build Options. (line 66) * HP-UX: ABI and ISA. (line 107) * HPPA: ABI and ISA. (line 68) * I/O functions <1>: I/O of Floats. (line 6) * I/O functions <2>: I/O of Rationals. (line 6) * I/O functions: I/O of Integers. (line 6) * i386: Notes for Particular Systems. (line 127) * IA-64: ABI and ISA. (line 107) * Import: Integer Import and Export. (line 11) * In-place operations: Efficiency. (line 57) * Include files: Headers and Libraries. (line 6) * info-lookup-symbol: Emacs. (line 6) * Initialization functions <1>: Simultaneous Float Init & Assign. (line 6) * Initialization functions <2>: Random State Initialization. (line 6) * Initialization functions <3>: Initializing Floats. (line 6) * Initialization functions <4>: Simultaneous Integer Init & Assign. (line 6) * Initialization functions <5>: Initializing Rationals. (line 6) * Initialization functions: Initializing Integers. (line 6) * Initializing and clearing: Efficiency. (line 21) * Input functions <1>: I/O of Integers. (line 6) * Input functions <2>: I/O of Rationals. (line 6) * Input functions <3>: Formatted Input Functions. (line 6) * Input functions: I/O of Floats. (line 6) * Install prefix: Build Options. (line 32) * Installing GMP: Installing GMP. (line 6) * Instruction Set Architecture: ABI and ISA. (line 6) * instrument-functions: Profiling. (line 66) * Integer: Nomenclature and Types. (line 6) * Integer arithmetic functions: Integer Arithmetic. (line 6) * Integer assignment functions <1>: Assigning Integers. (line 6) * Integer assignment functions: Simultaneous Integer Init & Assign. (line 6) * Integer bit manipulation functions: Integer Logic and Bit Fiddling. (line 6) * Integer comparison functions: Integer Comparisons. (line 6) * Integer conversion functions: Converting Integers. (line 6) * Integer division functions: Integer Division. (line 6) * Integer exponentiation functions: Integer Exponentiation. (line 6) * Integer export: Integer Import and Export. (line 45) * Integer functions: Integer Functions. (line 6) * Integer import: Integer Import and Export. (line 11) * Integer initialization functions <1>: Simultaneous Integer Init & Assign. (line 6) * Integer initialization functions: Initializing Integers. (line 6) * Integer input and output functions: I/O of Integers. (line 6) * Integer internals: Integer Internals. (line 6) * Integer logical functions: Integer Logic and Bit Fiddling. (line 6) * Integer miscellaneous functions: Miscellaneous Integer Functions. (line 6) * Integer random number functions: Integer Random Numbers. (line 6) * Integer root functions: Integer Roots. (line 6) * Integer sign tests: Integer Comparisons. (line 28) * Integer special functions: Integer Special Functions. (line 6) * Interix: Notes for Particular Systems. (line 51) * Internals: Internals. (line 6) * Introduction: Introduction to GMP. (line 6) * Inverse modulo functions: Number Theoretic Functions. (line 74) * IRIX <1>: Known Build Problems. (line 38) * IRIX: ABI and ISA. (line 132) * ISA: ABI and ISA. (line 6) * istream input: C++ Formatted Input. (line 6) * Jacobi symbol algorithm: Jacobi Symbol. (line 6) * Jacobi symbol functions: Number Theoretic Functions. (line 81) * Karatsuba multiplication: Karatsuba Multiplication. (line 6) * Karatsuba square root algorithm: Square Root Algorithm. (line 6) * Kronecker symbol functions: Number Theoretic Functions. (line 93) * Language bindings: Language Bindings. (line 6) * Latest version of GMP: Introduction to GMP. (line 38) * LCM functions: Number Theoretic Functions. (line 69) * Least common multiple functions: Number Theoretic Functions. (line 69) * Legendre symbol functions: Number Theoretic Functions. (line 84) * libgmp: Headers and Libraries. (line 22) * libgmpxx: Headers and Libraries. (line 27) * Libraries: Headers and Libraries. (line 22) * Libtool: Headers and Libraries. (line 33) * Libtool versioning: Notes for Package Builds. (line 9) * License conditions: Copying. (line 6) * Limb: Nomenclature and Types. (line 31) * Limb size: Useful Macros and Constants. (line 7) * Linear congruential algorithm: Random Number Algorithms. (line 25) * Linear congruential random numbers: Random State Initialization. (line 32) * Linking: Headers and Libraries. (line 22) * Logical functions: Integer Logic and Bit Fiddling. (line 6) * Low-level functions: Low-level Functions. (line 6) * Lucas number algorithm: Lucas Numbers Algorithm. (line 6) * Lucas number functions: Number Theoretic Functions. (line 143) * MacOS X: Known Build Problems. (line 51) * Mailing lists: Introduction to GMP. (line 45) * Malloc debugger: Debugging. (line 30) * Malloc problems: Debugging. (line 24) * Memory allocation: Custom Allocation. (line 6) * Memory management: Memory Management. (line 6) * Mersenne twister algorithm: Random Number Algorithms. (line 17) * Mersenne twister random numbers: Random State Initialization. (line 13) * MINGW: Notes for Particular Systems. (line 43) * MIPS: ABI and ISA. (line 132) * Miscellaneous float functions: Miscellaneous Float Functions. (line 6) * Miscellaneous integer functions: Miscellaneous Integer Functions. (line 6) * MMX: Notes for Particular Systems. (line 133) * Modular inverse functions: Number Theoretic Functions. (line 74) * Most significant bit: Miscellaneous Integer Functions. (line 34) * MPN_PATH: Build Options. (line 328) * MS Windows: Notes for Particular Systems. (line 43) * MS-DOS: Notes for Particular Systems. (line 43) * Multi-threading: Reentrancy. (line 6) * Multiplication algorithms: Multiplication Algorithms. (line 6) * Nails: Low-level Functions. (line 485) * Native compilation: Build Options. (line 52) * NeXT: Known Build Problems. (line 57) * Next prime function: Number Theoretic Functions. (line 25) * Nomenclature: Nomenclature and Types. (line 6) * Non-Unix systems: Build Options. (line 11) * Nth root algorithm: Nth Root Algorithm. (line 6) * Number sequences: Efficiency. (line 147) * Number theoretic functions: Number Theoretic Functions. (line 6) * Numerator and denominator: Applying Integer Functions. (line 6) * obstack output: Formatted Output Functions. (line 81) * OpenBSD: Notes for Particular Systems. (line 86) * Optimizing performance: Performance optimization. (line 6) * ostream output: C++ Formatted Output. (line 6) * Other languages: Language Bindings. (line 6) * Output functions <1>: Formatted Output Functions. (line 6) * Output functions <2>: I/O of Rationals. (line 6) * Output functions <3>: I/O of Floats. (line 6) * Output functions: I/O of Integers. (line 6) * Packaged builds: Notes for Package Builds. (line 6) * Parameter conventions: Parameter Conventions. (line 6) * Parsing expressions demo: Demonstration Programs. (line 15) * Particular systems: Notes for Particular Systems. (line 6) * Past GMP versions: Compatibility with older versions. (line 6) * PDF: Build Options. (line 343) * Perfect power algorithm: Perfect Power Algorithm. (line 6) * Perfect power functions: Integer Roots. (line 27) * Perfect square algorithm: Perfect Square Algorithm. (line 6) * Perfect square functions: Integer Roots. (line 36) * perl: Demonstration Programs. (line 35) * Perl module: Demonstration Programs. (line 35) * Postscript: Build Options. (line 343) * Power/PowerPC <1>: Notes for Particular Systems. (line 92) * Power/PowerPC: Known Build Problems. (line 63) * Powering algorithms: Powering Algorithms. (line 6) * Powering functions <1>: Integer Exponentiation. (line 6) * Powering functions: Float Arithmetic. (line 41) * PowerPC: ABI and ISA. (line 167) * Precision of floats: Floating-point Functions. (line 6) * Precision of hardware floating point: Notes for Particular Systems. (line 34) * Prefix: Build Options. (line 32) * Prime testing algorithms: Prime Testing Algorithm. (line 6) * Prime testing functions: Number Theoretic Functions. (line 7) * Primorial functions: Number Theoretic Functions. (line 118) * printf formatted output: Formatted Output. (line 6) * Probable prime testing functions: Number Theoretic Functions. (line 7) * prof: Profiling. (line 24) * Profiling: Profiling. (line 6) * Radix conversion algorithms: Radix Conversion Algorithms. (line 6) * Random number algorithms: Random Number Algorithms. (line 6) * Random number functions <1>: Miscellaneous Float Functions. (line 27) * Random number functions <2>: Random Number Functions. (line 6) * Random number functions: Integer Random Numbers. (line 6) * Random number seeding: Random State Seeding. (line 6) * Random number state: Random State Initialization. (line 6) * Random state: Nomenclature and Types. (line 46) * Rational arithmetic: Efficiency. (line 113) * Rational arithmetic functions: Rational Arithmetic. (line 6) * Rational assignment functions: Initializing Rationals. (line 6) * Rational comparison functions: Comparing Rationals. (line 6) * Rational conversion functions: Rational Conversions. (line 6) * Rational initialization functions: Initializing Rationals. (line 6) * Rational input and output functions: I/O of Rationals. (line 6) * Rational internals: Rational Internals. (line 6) * Rational number: Nomenclature and Types. (line 16) * Rational number functions: Rational Number Functions. (line 6) * Rational numerator and denominator: Applying Integer Functions. (line 6) * Rational sign tests: Comparing Rationals. (line 27) * Raw output internals: Raw Output Internals. (line 6) * Reallocations: Efficiency. (line 30) * Reentrancy: Reentrancy. (line 6) * References: References. (line 6) * Remove factor functions: Number Theoretic Functions. (line 105) * Reporting bugs: Reporting Bugs. (line 6) * Root extraction algorithm: Nth Root Algorithm. (line 6) * Root extraction algorithms: Root Extraction Algorithms. (line 6) * Root extraction functions <1>: Integer Roots. (line 6) * Root extraction functions: Float Arithmetic. (line 37) * Root testing functions: Integer Roots. (line 36) * Rounding functions: Miscellaneous Float Functions. (line 9) * Sample programs: Demonstration Programs. (line 6) * Scan bit functions: Integer Logic and Bit Fiddling. (line 38) * scanf formatted input: Formatted Input. (line 6) * SCO: Known Build Problems. (line 38) * Seeding random numbers: Random State Seeding. (line 6) * Segmentation violation: Debugging. (line 7) * Sequent Symmetry: Known Build Problems. (line 68) * Services for Unix: Notes for Particular Systems. (line 51) * Shared library versioning: Notes for Package Builds. (line 9) * Sign tests <1>: Integer Comparisons. (line 28) * Sign tests <2>: Comparing Rationals. (line 27) * Sign tests: Float Comparison. (line 33) * Size in digits: Miscellaneous Integer Functions. (line 23) * Small operands: Efficiency. (line 7) * Solaris <1>: Known Build Problems. (line 72) * Solaris <2>: ABI and ISA. (line 199) * Solaris: Known Build Problems. (line 78) * Sparc: Notes for Particular Systems. (line 109) * Sparc V9: ABI and ISA. (line 199) * Special integer functions: Integer Special Functions. (line 6) * Square root algorithm: Square Root Algorithm. (line 6) * SSE2: Notes for Particular Systems. (line 133) * Stack backtrace: Debugging. (line 50) * Stack overflow <1>: Build Options. (line 275) * Stack overflow: Debugging. (line 7) * Static linking: Efficiency. (line 14) * stdarg.h: Headers and Libraries. (line 17) * stdio.h: Headers and Libraries. (line 11) * Stripped libraries: Known Build Problems. (line 28) * Sun: ABI and ISA. (line 199) * SunOS: Notes for Particular Systems. (line 121) * Systems: Notes for Particular Systems. (line 6) * Temporary memory: Build Options. (line 275) * Texinfo: Build Options. (line 340) * Text input/output: Efficiency. (line 153) * Thread safety: Reentrancy. (line 6) * Toom multiplication <1>: Other Multiplication. (line 6) * Toom multiplication <2>: Toom 3-Way Multiplication. (line 6) * Toom multiplication <3>: Higher degree Toom'n'half. (line 6) * Toom multiplication: Toom 4-Way Multiplication. (line 6) * Types: Nomenclature and Types. (line 6) * ui and si functions: Efficiency. (line 50) * Unbalanced multiplication: Unbalanced Multiplication. (line 6) * Upward compatibility: Compatibility with older versions. (line 6) * Useful macros and constants: Useful Macros and Constants. (line 6) * User-defined precision: Floating-point Functions. (line 6) * Valgrind: Debugging. (line 130) * Variable conventions: Variable Conventions. (line 6) * Version number: Useful Macros and Constants. (line 12) * Web page: Introduction to GMP. (line 34) * Windows: Notes for Particular Systems. (line 56) * x86: Notes for Particular Systems. (line 127) * x87: Notes for Particular Systems. (line 34) * XML: Build Options. (line 347)  File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top Function and Type Index *********************** [index] * Menu: * __GMP_CC: Useful Macros and Constants. (line 23) * __GMP_CFLAGS: Useful Macros and Constants. (line 24) * __GNU_MP_VERSION: Useful Macros and Constants. (line 10) * __GNU_MP_VERSION_MINOR: Useful Macros and Constants. (line 11) * __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants. (line 12) * _mpz_realloc: Integer Special Functions. (line 51) * abs <1>: C++ Interface Rationals. (line 49) * abs <2>: C++ Interface Floats. (line 83) * abs: C++ Interface Integers. (line 47) * ceil: C++ Interface Floats. (line 84) * cmp <1>: C++ Interface Rationals. (line 51) * cmp <2>: C++ Interface Integers. (line 49) * cmp <3>: C++ Interface Rationals. (line 50) * cmp: C++ Interface Floats. (line 86) * floor: C++ Interface Floats. (line 93) * gmp_asprintf: Formatted Output Functions. (line 65) * gmp_errno: Random State Initialization. (line 55) * GMP_ERROR_INVALID_ARGUMENT: Random State Initialization. (line 55) * GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization. (line 55) * gmp_fprintf: Formatted Output Functions. (line 29) * gmp_fscanf: Formatted Input Functions. (line 25) * GMP_LIMB_BITS: Low-level Functions. (line 515) * GMP_NAIL_BITS: Low-level Functions. (line 513) * GMP_NAIL_MASK: Low-level Functions. (line 523) * GMP_NUMB_BITS: Low-level Functions. (line 514) * GMP_NUMB_MASK: Low-level Functions. (line 524) * GMP_NUMB_MAX: Low-level Functions. (line 532) * gmp_obstack_printf: Formatted Output Functions. (line 79) * gmp_obstack_vprintf: Formatted Output Functions. (line 81) * gmp_printf: Formatted Output Functions. (line 24) * GMP_RAND_ALG_DEFAULT: Random State Initialization. (line 49) * GMP_RAND_ALG_LC: Random State Initialization. (line 49) * gmp_randclass: C++ Interface Random Numbers. (line 7) * gmp_randclass::get_f: C++ Interface Random Numbers. (line 46) * gmp_randclass::get_z_bits: C++ Interface Random Numbers. (line 38) * gmp_randclass::get_z_range: C++ Interface Random Numbers. (line 42) * gmp_randclass::gmp_randclass: C++ Interface Random Numbers. (line 13) * gmp_randclass::seed: C++ Interface Random Numbers. (line 33) * gmp_randclear: Random State Initialization. (line 62) * gmp_randinit: Random State Initialization. (line 47) * gmp_randinit_default: Random State Initialization. (line 7) * gmp_randinit_lc_2exp: Random State Initialization. (line 18) * gmp_randinit_lc_2exp_size: Random State Initialization. (line 32) * gmp_randinit_mt: Random State Initialization. (line 13) * gmp_randinit_set: Random State Initialization. (line 43) * gmp_randseed: Random State Seeding. (line 7) * gmp_randseed_ui: Random State Seeding. (line 9) * gmp_randstate_t: Nomenclature and Types. (line 46) * gmp_scanf: Formatted Input Functions. (line 21) * gmp_snprintf: Formatted Output Functions. (line 46) * gmp_sprintf: Formatted Output Functions. (line 34) * gmp_sscanf: Formatted Input Functions. (line 29) * gmp_urandomb_ui: Random State Miscellaneous. (line 8) * gmp_urandomm_ui: Random State Miscellaneous. (line 14) * gmp_vasprintf: Formatted Output Functions. (line 66) * gmp_version: Useful Macros and Constants. (line 18) * gmp_vfprintf: Formatted Output Functions. (line 30) * gmp_vfscanf: Formatted Input Functions. (line 26) * gmp_vprintf: Formatted Output Functions. (line 25) * gmp_vscanf: Formatted Input Functions. (line 22) * gmp_vsnprintf: Formatted Output Functions. (line 48) * gmp_vsprintf: Formatted Output Functions. (line 35) * gmp_vsscanf: Formatted Input Functions. (line 31) * hypot: C++ Interface Floats. (line 94) * mp_bitcnt_t: Nomenclature and Types. (line 42) * mp_bits_per_limb: Useful Macros and Constants. (line 7) * mp_exp_t: Nomenclature and Types. (line 27) * mp_get_memory_functions: Custom Allocation. (line 90) * mp_limb_t: Nomenclature and Types. (line 31) * mp_set_memory_functions: Custom Allocation. (line 18) * mp_size_t: Nomenclature and Types. (line 37) * mpf_abs: Float Arithmetic. (line 47) * mpf_add: Float Arithmetic. (line 7) * mpf_add_ui: Float Arithmetic. (line 9) * mpf_ceil: Miscellaneous Float Functions. (line 7) * mpf_class: C++ Interface General. (line 20) * mpf_class::fits_sint_p: C++ Interface Floats. (line 87) * mpf_class::fits_slong_p: C++ Interface Floats. (line 88) * mpf_class::fits_sshort_p: C++ Interface Floats. (line 89) * mpf_class::fits_uint_p: C++ Interface Floats. (line 90) * mpf_class::fits_ulong_p: C++ Interface Floats. (line 91) * mpf_class::fits_ushort_p: C++ Interface Floats. (line 92) * mpf_class::get_d: C++ Interface Floats. (line 95) * mpf_class::get_mpf_t: C++ Interface General. (line 66) * mpf_class::get_prec: C++ Interface Floats. (line 115) * mpf_class::get_si: C++ Interface Floats. (line 96) * mpf_class::get_str: C++ Interface Floats. (line 98) * mpf_class::get_ui: C++ Interface Floats. (line 99) * mpf_class::mpf_class: C++ Interface Floats. (line 12) * mpf_class::operator=: C++ Interface Floats. (line 60) * mpf_class::set_prec: C++ Interface Floats. (line 116) * mpf_class::set_prec_raw: C++ Interface Floats. (line 117) * mpf_class::set_str: C++ Interface Floats. (line 101) * mpf_class::swap: C++ Interface Floats. (line 104) * mpf_clear: Initializing Floats. (line 37) * mpf_clears: Initializing Floats. (line 41) * mpf_cmp: Float Comparison. (line 7) * mpf_cmp_d: Float Comparison. (line 8) * mpf_cmp_si: Float Comparison. (line 10) * mpf_cmp_ui: Float Comparison. (line 9) * mpf_div: Float Arithmetic. (line 29) * mpf_div_2exp: Float Arithmetic. (line 53) * mpf_div_ui: Float Arithmetic. (line 33) * mpf_eq: Float Comparison. (line 17) * mpf_fits_sint_p: Miscellaneous Float Functions. (line 20) * mpf_fits_slong_p: Miscellaneous Float Functions. (line 18) * mpf_fits_sshort_p: Miscellaneous Float Functions. (line 22) * mpf_fits_uint_p: Miscellaneous Float Functions. (line 19) * mpf_fits_ulong_p: Miscellaneous Float Functions. (line 17) * mpf_fits_ushort_p: Miscellaneous Float Functions. (line 21) * mpf_floor: Miscellaneous Float Functions. (line 8) * mpf_get_d: Converting Floats. (line 7) * mpf_get_d_2exp: Converting Floats. (line 16) * mpf_get_default_prec: Initializing Floats. (line 12) * mpf_get_prec: Initializing Floats. (line 62) * mpf_get_si: Converting Floats. (line 27) * mpf_get_str: Converting Floats. (line 37) * mpf_get_ui: Converting Floats. (line 28) * mpf_init: Initializing Floats. (line 19) * mpf_init2: Initializing Floats. (line 26) * mpf_init_set: Simultaneous Float Init & Assign. (line 16) * mpf_init_set_d: Simultaneous Float Init & Assign. (line 19) * mpf_init_set_si: Simultaneous Float Init & Assign. (line 18) * mpf_init_set_str: Simultaneous Float Init & Assign. (line 25) * mpf_init_set_ui: Simultaneous Float Init & Assign. (line 17) * mpf_inits: Initializing Floats. (line 31) * mpf_inp_str: I/O of Floats. (line 39) * mpf_integer_p: Miscellaneous Float Functions. (line 14) * mpf_mul: Float Arithmetic. (line 19) * mpf_mul_2exp: Float Arithmetic. (line 50) * mpf_mul_ui: Float Arithmetic. (line 21) * mpf_neg: Float Arithmetic. (line 44) * mpf_out_str: I/O of Floats. (line 19) * mpf_pow_ui: Float Arithmetic. (line 41) * mpf_random2: Miscellaneous Float Functions. (line 37) * mpf_reldiff: Float Comparison. (line 29) * mpf_set: Assigning Floats. (line 10) * mpf_set_d: Assigning Floats. (line 13) * mpf_set_default_prec: Initializing Floats. (line 7) * mpf_set_prec: Initializing Floats. (line 65) * mpf_set_prec_raw: Initializing Floats. (line 72) * mpf_set_q: Assigning Floats. (line 15) * mpf_set_si: Assigning Floats. (line 12) * mpf_set_str: Assigning Floats. (line 18) * mpf_set_ui: Assigning Floats. (line 11) * mpf_set_z: Assigning Floats. (line 14) * mpf_sgn: Float Comparison. (line 33) * mpf_sqrt: Float Arithmetic. (line 36) * mpf_sqrt_ui: Float Arithmetic. (line 37) * mpf_sub: Float Arithmetic. (line 12) * mpf_sub_ui: Float Arithmetic. (line 16) * mpf_swap: Assigning Floats. (line 52) * mpf_t: Nomenclature and Types. (line 21) * mpf_trunc: Miscellaneous Float Functions. (line 9) * mpf_ui_div: Float Arithmetic. (line 31) * mpf_ui_sub: Float Arithmetic. (line 14) * mpf_urandomb: Miscellaneous Float Functions. (line 27) * mpn_add: Low-level Functions. (line 69) * mpn_add_1: Low-level Functions. (line 64) * mpn_add_n: Low-level Functions. (line 54) * mpn_addmul_1: Low-level Functions. (line 148) * mpn_and_n: Low-level Functions. (line 427) * mpn_andn_n: Low-level Functions. (line 442) * mpn_cmp: Low-level Functions. (line 284) * mpn_com: Low-level Functions. (line 467) * mpn_copyd: Low-level Functions. (line 476) * mpn_copyi: Low-level Functions. (line 472) * mpn_divexact_by3: Low-level Functions. (line 229) * mpn_divexact_by3c: Low-level Functions. (line 231) * mpn_divmod: Low-level Functions. (line 224) * mpn_divmod_1: Low-level Functions. (line 208) * mpn_divrem: Low-level Functions. (line 182) * mpn_divrem_1: Low-level Functions. (line 206) * mpn_gcd: Low-level Functions. (line 289) * mpn_gcd_1: Low-level Functions. (line 299) * mpn_gcdext: Low-level Functions. (line 305) * mpn_get_str: Low-level Functions. (line 352) * mpn_hamdist: Low-level Functions. (line 416) * mpn_ior_n: Low-level Functions. (line 432) * mpn_iorn_n: Low-level Functions. (line 447) * mpn_lshift: Low-level Functions. (line 260) * mpn_mod_1: Low-level Functions. (line 255) * mpn_mul: Low-level Functions. (line 114) * mpn_mul_1: Low-level Functions. (line 133) * mpn_mul_n: Low-level Functions. (line 103) * mpn_nand_n: Low-level Functions. (line 452) * mpn_neg: Low-level Functions. (line 98) * mpn_nior_n: Low-level Functions. (line 457) * mpn_perfect_square_p: Low-level Functions. (line 422) * mpn_popcount: Low-level Functions. (line 412) * mpn_random: Low-level Functions. (line 401) * mpn_random2: Low-level Functions. (line 402) * mpn_rshift: Low-level Functions. (line 272) * mpn_scan0: Low-level Functions. (line 386) * mpn_scan1: Low-level Functions. (line 394) * mpn_set_str: Low-level Functions. (line 367) * mpn_sqr: Low-level Functions. (line 125) * mpn_sqrtrem: Low-level Functions. (line 334) * mpn_sub: Low-level Functions. (line 90) * mpn_sub_1: Low-level Functions. (line 85) * mpn_sub_n: Low-level Functions. (line 76) * mpn_submul_1: Low-level Functions. (line 159) * mpn_tdiv_qr: Low-level Functions. (line 171) * mpn_xnor_n: Low-level Functions. (line 462) * mpn_xor_n: Low-level Functions. (line 437) * mpn_zero: Low-level Functions. (line 479) * mpq_abs: Rational Arithmetic. (line 31) * mpq_add: Rational Arithmetic. (line 7) * mpq_canonicalize: Rational Number Functions. (line 22) * mpq_class: C++ Interface General. (line 19) * mpq_class::canonicalize: C++ Interface Rationals. (line 43) * mpq_class::get_d: C++ Interface Rationals. (line 52) * mpq_class::get_den: C++ Interface Rationals. (line 66) * mpq_class::get_den_mpz_t: C++ Interface Rationals. (line 76) * mpq_class::get_mpq_t: C++ Interface General. (line 65) * mpq_class::get_num: C++ Interface Rationals. (line 65) * mpq_class::get_num_mpz_t: C++ Interface Rationals. (line 75) * mpq_class::get_str: C++ Interface Rationals. (line 53) * mpq_class::mpq_class: C++ Interface Rationals. (line 23) * mpq_class::set_str: C++ Interface Rationals. (line 54) * mpq_class::swap: C++ Interface Rationals. (line 57) * mpq_clear: Initializing Rationals. (line 16) * mpq_clears: Initializing Rationals. (line 20) * mpq_cmp: Comparing Rationals. (line 7) * mpq_cmp_si: Comparing Rationals. (line 17) * mpq_cmp_ui: Comparing Rationals. (line 15) * mpq_denref: Applying Integer Functions. (line 18) * mpq_div: Rational Arithmetic. (line 22) * mpq_div_2exp: Rational Arithmetic. (line 25) * mpq_equal: Comparing Rationals. (line 33) * mpq_get_d: Rational Conversions. (line 7) * mpq_get_den: Applying Integer Functions. (line 24) * mpq_get_num: Applying Integer Functions. (line 23) * mpq_get_str: Rational Conversions. (line 22) * mpq_init: Initializing Rationals. (line 7) * mpq_inits: Initializing Rationals. (line 12) * mpq_inp_str: I/O of Rationals. (line 26) * mpq_inv: Rational Arithmetic. (line 34) * mpq_mul: Rational Arithmetic. (line 15) * mpq_mul_2exp: Rational Arithmetic. (line 18) * mpq_neg: Rational Arithmetic. (line 28) * mpq_numref: Applying Integer Functions. (line 17) * mpq_out_str: I/O of Rationals. (line 18) * mpq_set: Initializing Rationals. (line 24) * mpq_set_d: Rational Conversions. (line 17) * mpq_set_den: Applying Integer Functions. (line 26) * mpq_set_f: Rational Conversions. (line 18) * mpq_set_num: Applying Integer Functions. (line 25) * mpq_set_si: Initializing Rationals. (line 31) * mpq_set_str: Initializing Rationals. (line 36) * mpq_set_ui: Initializing Rationals. (line 29) * mpq_set_z: Initializing Rationals. (line 25) * mpq_sgn: Comparing Rationals. (line 27) * mpq_sub: Rational Arithmetic. (line 11) * mpq_swap: Initializing Rationals. (line 56) * mpq_t: Nomenclature and Types. (line 16) * mpz_2fac_ui: Number Theoretic Functions. (line 111) * mpz_abs: Integer Arithmetic. (line 42) * mpz_add: Integer Arithmetic. (line 7) * mpz_add_ui: Integer Arithmetic. (line 9) * mpz_addmul: Integer Arithmetic. (line 25) * mpz_addmul_ui: Integer Arithmetic. (line 27) * mpz_and: Integer Logic and Bit Fiddling. (line 11) * mpz_array_init: Integer Special Functions. (line 11) * mpz_bin_ui: Number Theoretic Functions. (line 122) * mpz_bin_uiui: Number Theoretic Functions. (line 124) * mpz_cdiv_q: Integer Division. (line 13) * mpz_cdiv_q_2exp: Integer Division. (line 24) * mpz_cdiv_q_ui: Integer Division. (line 17) * mpz_cdiv_qr: Integer Division. (line 15) * mpz_cdiv_qr_ui: Integer Division. (line 21) * mpz_cdiv_r: Integer Division. (line 14) * mpz_cdiv_r_2exp: Integer Division. (line 25) * mpz_cdiv_r_ui: Integer Division. (line 19) * mpz_cdiv_ui: Integer Division. (line 23) * mpz_class: C++ Interface General. (line 18) * mpz_class::fits_sint_p: C++ Interface Integers. (line 50) * mpz_class::fits_slong_p: C++ Interface Integers. (line 51) * mpz_class::fits_sshort_p: C++ Interface Integers. (line 52) * mpz_class::fits_uint_p: C++ Interface Integers. (line 53) * mpz_class::fits_ulong_p: C++ Interface Integers. (line 54) * mpz_class::fits_ushort_p: C++ Interface Integers. (line 55) * mpz_class::get_d: C++ Interface Integers. (line 56) * mpz_class::get_mpz_t: C++ Interface General. (line 64) * mpz_class::get_si: C++ Interface Integers. (line 57) * mpz_class::get_str: C++ Interface Integers. (line 58) * mpz_class::get_ui: C++ Interface Integers. (line 59) * mpz_class::mpz_class: C++ Interface Integers. (line 7) * mpz_class::set_str: C++ Interface Integers. (line 60) * mpz_class::swap: C++ Interface Integers. (line 64) * mpz_clear: Initializing Integers. (line 49) * mpz_clears: Initializing Integers. (line 53) * mpz_clrbit: Integer Logic and Bit Fiddling. (line 54) * mpz_cmp: Integer Comparisons. (line 7) * mpz_cmp_d: Integer Comparisons. (line 8) * mpz_cmp_si: Integer Comparisons. (line 9) * mpz_cmp_ui: Integer Comparisons. (line 10) * mpz_cmpabs: Integer Comparisons. (line 18) * mpz_cmpabs_d: Integer Comparisons. (line 19) * mpz_cmpabs_ui: Integer Comparisons. (line 20) * mpz_com: Integer Logic and Bit Fiddling. (line 20) * mpz_combit: Integer Logic and Bit Fiddling. (line 57) * mpz_congruent_2exp_p: Integer Division. (line 124) * mpz_congruent_p: Integer Division. (line 121) * mpz_congruent_ui_p: Integer Division. (line 123) * mpz_divexact: Integer Division. (line 101) * mpz_divexact_ui: Integer Division. (line 102) * mpz_divisible_2exp_p: Integer Division. (line 112) * mpz_divisible_p: Integer Division. (line 110) * mpz_divisible_ui_p: Integer Division. (line 111) * mpz_even_p: Miscellaneous Integer Functions. (line 18) * mpz_export: Integer Import and Export. (line 45) * mpz_fac_ui: Number Theoretic Functions. (line 110) * mpz_fdiv_q: Integer Division. (line 27) * mpz_fdiv_q_2exp: Integer Division. (line 38) * mpz_fdiv_q_ui: Integer Division. (line 31) * mpz_fdiv_qr: Integer Division. (line 29) * mpz_fdiv_qr_ui: Integer Division. (line 35) * mpz_fdiv_r: Integer Division. (line 28) * mpz_fdiv_r_2exp: Integer Division. (line 39) * mpz_fdiv_r_ui: Integer Division. (line 33) * mpz_fdiv_ui: Integer Division. (line 37) * mpz_fib2_ui: Number Theoretic Functions. (line 132) * mpz_fib_ui: Number Theoretic Functions. (line 130) * mpz_fits_sint_p: Miscellaneous Integer Functions. (line 10) * mpz_fits_slong_p: Miscellaneous Integer Functions. (line 8) * mpz_fits_sshort_p: Miscellaneous Integer Functions. (line 12) * mpz_fits_uint_p: Miscellaneous Integer Functions. (line 9) * mpz_fits_ulong_p: Miscellaneous Integer Functions. (line 7) * mpz_fits_ushort_p: Miscellaneous Integer Functions. (line 11) * mpz_gcd: Number Theoretic Functions. (line 32) * mpz_gcd_ui: Number Theoretic Functions. (line 39) * mpz_gcdext: Number Theoretic Functions. (line 49) * mpz_get_d: Converting Integers. (line 27) * mpz_get_d_2exp: Converting Integers. (line 35) * mpz_get_si: Converting Integers. (line 18) * mpz_get_str: Converting Integers. (line 46) * mpz_get_ui: Converting Integers. (line 11) * mpz_getlimbn: Integer Special Functions. (line 60) * mpz_hamdist: Integer Logic and Bit Fiddling. (line 29) * mpz_import: Integer Import and Export. (line 11) * mpz_init: Initializing Integers. (line 26) * mpz_init2: Initializing Integers. (line 33) * mpz_init_set: Simultaneous Integer Init & Assign. (line 27) * mpz_init_set_d: Simultaneous Integer Init & Assign. (line 30) * mpz_init_set_si: Simultaneous Integer Init & Assign. (line 29) * mpz_init_set_str: Simultaneous Integer Init & Assign. (line 34) * mpz_init_set_ui: Simultaneous Integer Init & Assign. (line 28) * mpz_inits: Initializing Integers. (line 29) * mpz_inp_raw: I/O of Integers. (line 61) * mpz_inp_str: I/O of Integers. (line 30) * mpz_invert: Number Theoretic Functions. (line 74) * mpz_ior: Integer Logic and Bit Fiddling. (line 14) * mpz_jacobi: Number Theoretic Functions. (line 81) * mpz_kronecker: Number Theoretic Functions. (line 89) * mpz_kronecker_si: Number Theoretic Functions. (line 90) * mpz_kronecker_ui: Number Theoretic Functions. (line 91) * mpz_lcm: Number Theoretic Functions. (line 68) * mpz_lcm_ui: Number Theoretic Functions. (line 69) * mpz_legendre: Number Theoretic Functions. (line 84) * mpz_lucnum2_ui: Number Theoretic Functions. (line 143) * mpz_lucnum_ui: Number Theoretic Functions. (line 141) * mpz_mfac_uiui: Number Theoretic Functions. (line 113) * mpz_mod: Integer Division. (line 91) * mpz_mod_ui: Integer Division. (line 93) * mpz_mul: Integer Arithmetic. (line 19) * mpz_mul_2exp: Integer Arithmetic. (line 35) * mpz_mul_si: Integer Arithmetic. (line 20) * mpz_mul_ui: Integer Arithmetic. (line 22) * mpz_neg: Integer Arithmetic. (line 39) * mpz_nextprime: Number Theoretic Functions. (line 25) * mpz_odd_p: Miscellaneous Integer Functions. (line 17) * mpz_out_raw: I/O of Integers. (line 45) * mpz_out_str: I/O of Integers. (line 18) * mpz_perfect_power_p: Integer Roots. (line 27) * mpz_perfect_square_p: Integer Roots. (line 36) * mpz_popcount: Integer Logic and Bit Fiddling. (line 23) * mpz_pow_ui: Integer Exponentiation. (line 31) * mpz_powm: Integer Exponentiation. (line 8) * mpz_powm_sec: Integer Exponentiation. (line 18) * mpz_powm_ui: Integer Exponentiation. (line 10) * mpz_primorial_ui: Number Theoretic Functions. (line 118) * mpz_probab_prime_p: Number Theoretic Functions. (line 7) * mpz_random: Integer Random Numbers. (line 42) * mpz_random2: Integer Random Numbers. (line 51) * mpz_realloc2: Initializing Integers. (line 57) * mpz_remove: Number Theoretic Functions. (line 105) * mpz_root: Integer Roots. (line 7) * mpz_rootrem: Integer Roots. (line 13) * mpz_rrandomb: Integer Random Numbers. (line 31) * mpz_scan0: Integer Logic and Bit Fiddling. (line 37) * mpz_scan1: Integer Logic and Bit Fiddling. (line 38) * mpz_set: Assigning Integers. (line 10) * mpz_set_d: Assigning Integers. (line 13) * mpz_set_f: Assigning Integers. (line 15) * mpz_set_q: Assigning Integers. (line 14) * mpz_set_si: Assigning Integers. (line 12) * mpz_set_str: Assigning Integers. (line 21) * mpz_set_ui: Assigning Integers. (line 11) * mpz_setbit: Integer Logic and Bit Fiddling. (line 51) * mpz_sgn: Integer Comparisons. (line 28) * mpz_si_kronecker: Number Theoretic Functions. (line 92) * mpz_size: Integer Special Functions. (line 68) * mpz_sizeinbase: Miscellaneous Integer Functions. (line 23) * mpz_sqrt: Integer Roots. (line 17) * mpz_sqrtrem: Integer Roots. (line 20) * mpz_sub: Integer Arithmetic. (line 12) * mpz_sub_ui: Integer Arithmetic. (line 14) * mpz_submul: Integer Arithmetic. (line 30) * mpz_submul_ui: Integer Arithmetic. (line 32) * mpz_swap: Assigning Integers. (line 37) * mpz_t: Nomenclature and Types. (line 6) * mpz_tdiv_q: Integer Division. (line 41) * mpz_tdiv_q_2exp: Integer Division. (line 52) * mpz_tdiv_q_ui: Integer Division. (line 45) * mpz_tdiv_qr: Integer Division. (line 43) * mpz_tdiv_qr_ui: Integer Division. (line 49) * mpz_tdiv_r: Integer Division. (line 42) * mpz_tdiv_r_2exp: Integer Division. (line 53) * mpz_tdiv_r_ui: Integer Division. (line 47) * mpz_tdiv_ui: Integer Division. (line 51) * mpz_tstbit: Integer Logic and Bit Fiddling. (line 60) * mpz_ui_kronecker: Number Theoretic Functions. (line 93) * mpz_ui_pow_ui: Integer Exponentiation. (line 33) * mpz_ui_sub: Integer Arithmetic. (line 16) * mpz_urandomb: Integer Random Numbers. (line 14) * mpz_urandomm: Integer Random Numbers. (line 23) * mpz_xor: Integer Logic and Bit Fiddling. (line 17) * operator"" <1>: C++ Interface Integers. (line 30) * operator"" <2>: C++ Interface Floats. (line 56) * operator"": C++ Interface Rationals. (line 38) * operator%: C++ Interface Integers. (line 35) * operator/: C++ Interface Integers. (line 34) * operator<<: C++ Formatted Output. (line 20) * operator>> <1>: C++ Interface Rationals. (line 85) * operator>>: C++ Formatted Input. (line 25) * sgn <1>: C++ Interface Rationals. (line 56) * sgn <2>: C++ Interface Integers. (line 62) * sgn: C++ Interface Floats. (line 102) * sqrt <1>: C++ Interface Integers. (line 63) * sqrt: C++ Interface Floats. (line 103) * swap <1>: C++ Interface Floats. (line 105) * swap <2>: C++ Interface Integers. (line 65) * swap: C++ Interface Rationals. (line 58) * trunc: C++ Interface Floats. (line 106)  Local Variables: coding: iso-8859-1 End: gmp-doc-5.1.2/doc/Makefile.in0000644000175000000620000005717712146435163014713 0ustar stevestaff# Makefile.in generated by automake 1.11.6 from Makefile.am. # @configure_input@ # Copyright (C) 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, # 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software # Foundation, Inc. # This Makefile.in is free software; the Free Software Foundation # gives unlimited permission to copy and/or distribute it, # with or without modifications, as long as this notice is preserved. # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY, to the extent permitted by law; without # even the implied warranty of MERCHANTABILITY or FITNESS FOR A # PARTICULAR PURPOSE. @SET_MAKE@ # Copyright 2003 Free Software Foundation, Inc. # # This file is part of the GNU MP Library. # # The GNU MP Library is free software; you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation; either version 3 of the License, or (at your # option) any later version. # # The GNU MP Library is distributed in the hope that it will be useful, but # WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY # or FITNESS FOR A PARTICULAR PURPOSE. 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GMP Itemized Development Tasks

Copyright 2000, 2001, 2002, 2003, 2004, 2006, 2008, 2009 Free Software
Foundation, Inc.

This file is part of the GNU MP Library.

The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published
by the Free Software Foundation; either version 3 of the License, or (at
your option) any later version.

The GNU MP Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library.  If not, see http://www.gnu.org/licenses/.

This file current as of 5 Dec 2011. An up-to-date version is available at http://gmplib.org/tasks.html. Please send comments about this page to gmp-devel@gmplib.org.

These are itemized GMP development tasks. Not all the tasks listed here are suitable for volunteers, but many of them are. Please see the projects file for more sizeable projects.

CAUTION: This file needs updating. Many of the tasks here have either already been taken care of, or have become irrelevant.

Correctness and Completeness

  • _LONG_LONG_LIMB in gmp.h is not namespace clean. Reported by Patrick Pelissier.
    We sort of mentioned _LONG_LONG_LIMB in past releases, so need to be careful about changing it. It used to be a define applications had to set for long long limb systems, but that in particular is no longer relevant now that it's established automatically.
  • The various reuse.c tests need to force reallocation by calling _mpz_realloc with a small (1 limb) size.
  • One reuse case is missing from mpX/tests/reuse.c: mpz_XXX(a,a,a).
  • Make the string reading functions allow the `0x' prefix when the base is explicitly 16. They currently only allow that prefix when the base is unspecified (zero).
  • mpf_eq is not always correct, when one operand is 1000000000... and the other operand is 0111111111..., i.e., extremely close. There is a special case in mpf_sub for this situation; put similar code in mpf_eq. [In progress.]
  • mpf_eq doesn't implement what gmp.texi specifies. It should not use just whole limbs, but partial limbs. [In progress.]
  • mpf_set_str doesn't validate it's exponent, for instance garbage 123.456eX789X is accepted (and an exponent 0 used), and overflow of a long is not detected.
  • mpf_add doesn't check for a carry from truncated portions of the inputs, and in that respect doesn't implement the "infinite precision followed by truncate" specified in the manual.
  • Windows DLLs: tests/mpz/reuse.c and tests/mpf/reuse.c initialize global variables with pointers to mpz_add etc, which doesn't work when those routines are coming from a DLL (because they're effectively function pointer global variables themselves). Need to rearrange perhaps to a set of calls to a test function rather than iterating over an array.
  • mpz_pow_ui: Detect when the result would be more memory than a size_t can represent and raise some suitable exception, probably an alloc call asking for SIZE_T_MAX, and if that somehow succeeds then an abort. Various size overflows of this kind are not handled gracefully, probably resulting in segvs.
    In mpz_n_pow_ui, detect when the count of low zero bits exceeds an unsigned long. There's a (small) chance of this happening but still having enough memory to represent the value. Reported by Winfried Dreckmann in for instance mpz_ui_pow_ui (x, 4UL, 1431655766UL).
  • mpf: Detect exponent overflow and raise some exception. It'd be nice to allow the full mp_exp_t range since that's how it's been in the past, but maybe dropping one bit would make it easier to test if e1+e2 goes out of bounds.

Machine Independent Optimization

  • mpf_cmp: For better cache locality, don't test for low zero limbs until the high limbs fail to give an ordering. Reduce code size by turning the three mpn_cmp's into a single loop stopping when the end of one operand is reached (and then looking for a non-zero in the rest of the other).
  • mpf_mul_2exp, mpf_div_2exp: The use of mpn_lshift for any size<=prec means repeated mul_2exp and div_2exp calls accumulate low zero limbs until size==prec+1 is reached. Those zeros will slow down subsequent operations, especially if the value is otherwise only small. If low bits of the low limb are zero, use mpn_rshift so as to not increase the size.
  • mpn_dc_sqrtrem, mpn_sqrtrem2: Don't use mpn_add_1 and mpn_sub_1 for 1 limb operations, instead ADDC_LIMB and SUBC_LIMB.
  • mpn_sqrtrem2: Use plain variables for sp[0] and rp[0] calculations, so the compiler needn't worry about aliasing between sp and rp.
  • mpn_sqrtrem: Some work can be saved in the last step when the remainder is not required, as noted in Paul's paper.
  • mpq_add, mpq_sub: The gcd fits a single limb with high probability and in this case binvert_limb could be used to calculate the inverse just once for the two exact divisions "op1.den / gcd" and "op2.den / gcd", rather than letting mpn_bdiv_q_1 do it each time. This would require calling mpn_pi1_bdiv_q_1.
  • mpn_gcdext: Don't test count_leading_zeros for zero, instead check the high bit of the operand and avoid invoking count_leading_zeros. This is an optimization on all machines, and significant on machines with slow count_leading_zeros, though it's possible an already normalized operand might not be encountered very often.
  • Rewrite umul_ppmm to use floating-point for generating the most significant limb (if GMP_LIMB_BITS <= 52 bits). (Peter Montgomery has some ideas on this subject.)
  • Improve the default umul_ppmm code in longlong.h: Add partial products with fewer operations.
  • Consider inlining mpz_set_ui. This would be both small and fast, especially for compile-time constants, but would make application binaries depend on having 1 limb allocated to an mpz_t, preventing the "lazy" allocation scheme below.
  • Consider inlining mpz_[cft]div_ui and maybe mpz_[cft]div_r_ui. A __gmp_divide_by_zero would be needed for the divide by zero test, unless that could be left to mpn_mod_1 (not sure currently whether all the risc chips provoke the right exception there if using mul-by-inverse).
  • Consider inlining: mpz_fits_s*_p. The setups for LONG_MAX etc would need to go into gmp.h, and on Cray it might, unfortunately, be necessary to forcibly include <limits.h> since there's no apparent way to get SHRT_MAX with an expression (since short and unsigned short can be different sizes).
  • mpz_powm and mpz_powm_ui aren't very fast on one or two limb moduli, due to a lot of function call overheads. These could perhaps be handled as special cases.
  • Make sure mpz_powm_ui is never slower than the corresponding computation using mpz_powm.
  • mpz_powm REDC should do multiplications by g[] using the division method when they're small, since the REDC form of a small multiplier is normally a full size product. Probably would need a new tuned parameter to say what size multiplier is "small", as a function of the size of the modulus.
  • mpn_gcd might be able to be sped up on small to moderate sizes by improving find_a, possibly just by providing an alternate implementation for CPUs with slowish count_leading_zeros.
  • mpf_set_str produces low zero limbs when a string has a fraction but is exactly representable, eg. 0.5 in decimal. These could be stripped to save work in later operations.
  • mpz_and, mpz_ior and mpz_xor should use mpn_and_n etc for the benefit of the small number of targets with native versions of those routines. Need to be careful not to pass size==0. Is some code sharing possible between the mpz routines?
  • mpf_add: Don't do a copy to avoid overlapping operands unless it's really necessary (currently only sizes are tested, not whether r really is u or v).
  • mpf_add: Under the check for v having no effect on the result, perhaps test for r==u and do nothing in that case, rather than currently it looks like an MPN_COPY_INCR will be done to reduce prec+1 limbs to prec.
  • mpf_div_ui: Instead of padding with low zeros, call mpn_divrem_1 asking for fractional quotient limbs.
  • mpf_div_ui: Eliminate TMP_ALLOC. When r!=u there's no overlap and the division can be called on those operands. When r==u and is prec+1 limbs, then it's an in-place division. If r==u and not prec+1 limbs, then move the available limbs up to prec+1 and do an in-place there.
  • mpf_div_ui: Whether the high quotient limb is zero can be determined by testing the dividend for high<divisor. When non-zero, the division can be done on prec dividend limbs instead of prec+1. The result size is also known before the division, so that can be a tail call (once the TMP_ALLOC is eliminated).
  • mpn_divrem_2 could usefully accept unnormalized divisors and shift the dividend on-the-fly, since this should cost nothing on superscalar processors and avoid the need for temporary copying in mpn_tdiv_qr.
  • mpf_sqrt: If r!=u, and if u doesn't need to be padded with zeros, then there's no need for the tp temporary.
  • mpq_cmp_ui could form the num1*den2 and num2*den1 products limb-by-limb from high to low and look at each step for values differing by more than the possible carry bit from the uncalculated portion.
  • mpq_cmp could do the same high-to-low progressive multiply and compare. The benefits of karatsuba and higher multiplication algorithms are lost, but if it's assumed only a few high limbs will be needed to determine an order then that's fine.
  • mpn_add_1, mpn_sub_1, mpn_add, mpn_sub: Internally use __GMPN_ADD_1 etc instead of the functions, so they get inlined on all compilers, not just gcc and others with inline recognised in gmp.h. __GMPN_ADD_1 etc are meant mostly to support application inline mpn_add_1 etc and if they don't come out good for internal uses then special forms can be introduced, for instance many internal uses are in-place. Sometimes a block of code is executed based on the carry-out, rather than using it arithmetically, and those places might want to do their own loops entirely.
  • __gmp_extract_double on 64-bit systems could use just one bitfield for the mantissa extraction, not two, when endianness permits. Might depend on the compiler allowing long long bit fields when that's the only actual 64-bit type.
  • tal-notreent.c could keep a block of memory permanently allocated. Currently the last nested TMP_FREE releases all memory, so there's an allocate and free every time a top-level function using TMP is called. Would need mp_set_memory_functions to tell tal-notreent.c to release any cached memory when changing allocation functions though.
  • __gmp_tmp_alloc from tal-notreent.c could be partially inlined. If the current chunk has enough room then a couple of pointers can be updated. Only if more space is required then a call to some sort of __gmp_tmp_increase would be needed. The requirement that TMP_ALLOC is an expression might make the implementation a bit ugly and/or a bit sub-optimal.
    #define TMP_ALLOC(n)
      ((ROUND_UP(n) > current->end - current->point ?
         __gmp_tmp_increase (ROUND_UP (n)) : 0),
         current->point += ROUND_UP (n),
         current->point - ROUND_UP (n))
    
  • __mp_bases has a lot of data for bases which are pretty much never used. Perhaps the table should just go up to base 16, and have code to generate data above that, if and when required. Naturally this assumes the code would be smaller than the data saved.
  • __mp_bases field big_base_inverted is only used if USE_PREINV_DIVREM_1 is true, and could be omitted otherwise, to save space.
  • mpz_get_str, mtox: For power-of-2 bases, which are of course fast, it seems a little silly to make a second pass over the mpn_get_str output to convert to ASCII. Perhaps combine that with the bit extractions.
  • mpz_gcdext: If the caller requests only the S cofactor (of A), and A<B, then the code ends up generating the cofactor T (of B) and deriving S from that. Perhaps it'd be possible to arrange to get S in the first place by calling mpn_gcdext with A+B,B. This might only be an advantage if A and B are about the same size.
  • mpz_n_pow_ui does a good job with small bases and stripping powers of 2, but it's perhaps a bit too complicated for what it gains. The simpler mpn_pow_1 is a little faster on small exponents. (Note some of the ugliness in mpz_n_pow_ui is due to supporting mpn_mul_2.)
    Perhaps the stripping of 2s in mpz_n_pow_ui should be confined to single limb operands for simplicity and since that's where the greatest gain would be.
    Ideally mpn_pow_1 and mpz_n_pow_ui would be merged. The reason mpz_n_pow_ui writes to an mpz_t is that its callers leave it to make a good estimate of the result size. Callers of mpn_pow_1 already know the size by separate means (mp_bases).
  • mpz_invert should call mpn_gcdext directly.

Machine Dependent Optimization

  • invert_limb on various processors might benefit from the little Newton iteration done for alpha and ia64.
  • Alpha 21264: mpn_addlsh1_n could be implemented with mpn_addmul_1, since that code at 3.5 is a touch faster than a separate lshift and add_n at 1.75+2.125=3.875. Or very likely some specific addlsh1_n code could beat both.
  • Alpha 21264: Improve feed-in code for mpn_mul_1, mpn_addmul_1, and mpn_submul_1.
  • Alpha 21164: Rewrite mpn_mul_1, mpn_addmul_1, and mpn_submul_1 for the 21164. This should use both integer multiplies and floating-point multiplies. For the floating-point operations, the single-limb multiplier should be split into three 21-bit chunks, or perhaps even better in four 16-bit chunks. Probably possible to reach 9 cycles/limb.
  • Alpha: GCC 3.4 will introduce __builtin_ctzl, __builtin_clzl and __builtin_popcountl using the corresponding CIX ct instructions, and __builtin_alpha_cmpbge. These should give GCC more information about scheduling etc than the asm blocks currently used in longlong.h and gmp-impl.h.
  • Alpha Unicos: Apparently there's no alloca on this system, making configure choose the slower malloc-reentrant allocation method. Is there a better way? Maybe variable-length arrays per notes below.
  • Alpha Unicos 21164, 21264: .align is not used since it pads with garbage. Does the code get the intended slotting required for the claimed speeds? .align at the start of a function would presumably be safe no matter how it pads.
  • ARM V5: count_leading_zeros can use the clz instruction. For GCC 3.4 and up, do this via __builtin_clzl since then gcc knows it's "predicable".
  • Itanium: GCC 3.4 introduces __builtin_popcount which can be used instead of an asm block. The builtin should give gcc more opportunities for scheduling, bundling and predication. __builtin_ctz similarly (it just uses popcount as per current longlong.h).
  • UltraSPARC/64: Optimize mpn_mul_1, mpn_addmul_1, for s2 < 2^32 (or perhaps for any zero 16-bit s2 chunk). Not sure how much this can improve the speed, though, since the symmetry that we rely on is lost. Perhaps we can just gain cycles when s2 < 2^16, or more accurately, when two 16-bit s2 chunks which are 16 bits apart are zero.
  • UltraSPARC/64: Write native mpn_submul_1, analogous to mpn_addmul_1.
  • UltraSPARC/64: Write umul_ppmm. Using four "mulx"s either with an asm block or via the generic C code is about 90 cycles. Try using fp operations, and also try using karatsuba for just three "mulx"s.
  • UltraSPARC/32: Rewrite mpn_lshift, mpn_rshift. Will give 2 cycles/limb. Trivial modifications of mpn/sparc64 should do.
  • UltraSPARC/32: Write special mpn_Xmul_1 loops for s2 < 2^16.
  • UltraSPARC/32: Use mulx for umul_ppmm if possible (see commented out code in longlong.h). This is unlikely to save more than a couple of cycles, so perhaps isn't worth bothering with.
  • UltraSPARC/32: On Solaris gcc doesn't give us __sparc_v9__ or anything to indicate V9 support when -mcpu=v9 is selected. See gcc/config/sol2-sld-64.h. Will need to pass something through from ./configure to select the right code in longlong.h. (Currently nothing is lost because mulx for multiplying is commented out.)
  • UltraSPARC/32: mpn_divexact_1 and mpn_modexact_1c_odd can use a 64-bit inverse and take 64-bits at a time from the dividend, as per the 32-bit divisor case in mpn/sparc64/mode1o.c. This must be done in assembler, since the full 64-bit registers (%gN) are not available from C.
  • UltraSPARC/32: mpn_divexact_by3c can work 64-bits at a time using mulx, in assembler. This would be the same as for sparc64.
  • UltraSPARC: binvert_limb might save a few cycles from masking down to just the useful bits at each point in the calculation, since mulx speed depends on the highest bit set. Either explicit masks or small types like short and int ought to work.
  • Sparc64 HAL R1 popc: This chip reputedly implements popc properly (see gcc sparc.md). Would need to recognise it as sparchalr1 or something in configure / config.sub / config.guess. popc_limb in gmp-impl.h could use this (per commented out code). count_trailing_zeros could use it too.
  • PA64: Improve mpn_addmul_1, mpn_submul_1, and mpn_mul_1. The current code runs at 11 cycles/limb. It should be possible to saturate the cache, which will happen at 8 cycles/limb (7.5 for mpn_mul_1). Write special loops for s2 < 2^32; it should be possible to make them run at about 5 cycles/limb.
  • PPC601: See which of the power or powerpc32 code runs better. Currently the powerpc32 is used, but only because it's the default for powerpc*.
  • PPC630: Rewrite mpn_addmul_1, mpn_submul_1, and mpn_mul_1. Use both integer and floating-point operations, possibly two floating-point and one integer limb per loop. Split operands into four 16-bit chunks for fast fp operations. Should easily reach 9 cycles/limb (using one int + one fp), but perhaps even 7 cycles/limb (using one int + two fp).
  • PPC630: mpn_rshift could do the same sort of unrolled loop as mpn_lshift. Some judicious use of m4 might let the two share source code, or with a register to control the loop direction perhaps even share object code.
  • Implement mpn_mul_basecase and mpn_sqr_basecase for important machines. Helping the generic sqr_basecase.c with an mpn_sqr_diagonal might be enough for some of the RISCs.
  • POWER2/POWER2SC: Schedule mpn_lshift/mpn_rshift. Will bring time from 1.75 to 1.25 cycles/limb.
  • X86: Optimize non-MMX mpn_lshift for shifts by 1. (See Pentium code.)
  • X86: Good authority has it that in the past an inline rep movs would upset GCC register allocation for the whole function. Is this still true in GCC 3? It uses rep movs itself for __builtin_memcpy. Examine the code for some simple and complex functions to find out. Inlining rep movs would be desirable, it'd be both smaller and faster.
  • Pentium P54: mpn_lshift and mpn_rshift can come down from 6.0 c/l to 5.5 or 5.375 by paying attention to pairing after shrdl and shldl, see mpn/x86/pentium/README.
  • Pentium P55 MMX: mpn_lshift and mpn_rshift might benefit from some destination prefetching.
  • PentiumPro: mpn_divrem_1 might be able to use a mul-by-inverse, hoping for maybe 30 c/l.
  • K7: mpn_lshift and mpn_rshift might be able to do something branch-free for unaligned startups, and shaving one insn from the loop with alternative indexing might save a cycle.
  • PPC32: Try using fewer registers in the current mpn_lshift. The pipeline is now extremely deep, perhaps unnecessarily deep.
  • Fujitsu VPP: Vectorize main functions, perhaps in assembly language.
  • Fujitsu VPP: Write mpn_mul_basecase and mpn_sqr_basecase. This should use a "vertical multiplication method", to avoid carry propagation. splitting one of the operands in 11-bit chunks.
  • Pentium: mpn_lshift by 31 should use the special rshift by 1 code, and vice versa mpn_rshift by 31 should use the special lshift by 1. This would be best as a jump across to the other routine, could let both live in lshift.asm and omit rshift.asm on finding mpn_rshift already provided.
  • Cray T3E: Experiment with optimization options. In particular, -hpipeline3 seems promising. We should at least up -O to -O2 or -O3.
  • Cray: mpn_com and mpn_and_n etc very probably wants a pragma like MPN_COPY_INCR.
  • Cray vector systems: mpn_lshift, mpn_rshift, mpn_popcount and mpn_hamdist are nice and small and could be inlined to avoid function calls.
  • Cray: Variable length arrays seem to be faster than the tal-notreent.c scheme. Not sure why, maybe they merely give the compiler more information about aliasing (or the lack thereof). Would like to modify TMP_ALLOC to use them, or introduce a new scheme. Memory blocks wanted unconditionally are easy enough, those wanted only sometimes are a problem. Perhaps a special size calculation to ask for a dummy length 1 when unwanted, or perhaps an inlined subroutine duplicating code under each conditional. Don't really want to turn everything into a dog's dinner just because Cray don't offer an alloca.
  • Cray: mpn_get_str on power-of-2 bases ought to vectorize. Does it? bits_per_digit and the inner loop over bits in a limb might prevent it. Perhaps special cases for binary, octal and hex would be worthwhile (very possibly for all processors too).
  • S390: BSWAP_LIMB_FETCH looks like it could be done with lrvg, as per glibc sysdeps/s390/s390-64/bits/byteswap.h. This is only for 64-bit mode or something is it, since 32-bit mode has other code? Also, is it worth using for BSWAP_LIMB too, or would that mean a store and re-fetch? Presumably that's what comes out in glibc.
  • Improve count_leading_zeros for 64-bit machines:
    	   if ((x >> 32) == 0) { x <<= 32; cnt += 32; }
    	   if ((x >> 48) == 0) { x <<= 16; cnt += 16; }
    	   ... 
  • IRIX 6 MIPSpro compiler has an __inline which could perhaps be used in __GMP_EXTERN_INLINE. What would be the right way to identify suitable versions of that compiler?
  • IRIX cc is rumoured to have an _int_mult_upper (in <intrinsics.h> like Cray), but it didn't seem to exist on some IRIX 6.5 systems tried. If it does actually exist somewhere it would very likely be an improvement over a function call to umul.asm.
  • mpn_get_str final divisions by the base with udiv_qrnd_unnorm could use some sort of multiply-by-inverse on suitable machines. This ends up happening for decimal by presenting the compiler with a run-time constant, but the same for other bases would be good. Perhaps use could be made of the fact base<256.
  • mpn_umul_ppmm, mpn_udiv_qrnnd: Return a structure like div_t to avoid going through memory, in particular helping RISCs that don't do store-to-load forwarding. Clearly this is only possible if the ABI returns a structure of two mp_limb_ts in registers.
    On PowerPC, structures are returned in memory on AIX and Darwin. In SVR4 they're returned in registers, except that draft SVR4 had said memory, so it'd be prudent to check which is done. We can jam the compiler into the right mode if we know how, since all this is purely internal to libgmp. (gcc has an option, though of course gcc doesn't matter since we use inline asm there.)

New Functionality

  • Maybe add mpz_crr (Chinese Remainder Reconstruction).
  • Let `0b' and `0B' mean binary input everywhere.
  • mpz_init and mpq_init could do lazy allocation. Set ALLOC(var) to 0 to indicate nothing allocated, and let _mpz_realloc do the initial alloc. Set z->_mp_d to a dummy that mpz_get_ui and similar can unconditionally fetch from. Niels Möller has had a go at this.
    The advantages of the lazy scheme would be:
    • Initial allocate would be the size required for the first value stored, rather than getting 1 limb in mpz_init and then more or less immediately reallocating.
    • mpz_init would only store magic values in the mpz_t fields, and could be inlined.
    • A fixed initializer could even be used by applications, like mpz_t z = MPZ_INITIALIZER;, which might be convenient for globals.
    The advantages of the current scheme are:
    • mpz_set_ui and other similar routines needn't check the size allocated and can just store unconditionally.
    • mpz_set_ui and perhaps others like mpz_tdiv_r_ui and a prospective mpz_set_ull could be inlined.
  • Add mpf_out_raw and mpf_inp_raw. Make sure format is portable between 32-bit and 64-bit machines, and between little-endian and big-endian machines. A format which MPFR can use too would be good.
  • mpn_and_n ... mpn_copyd: Perhaps make the mpn logops and copys available in gmp.h, either as library functions or inlines, with the availability of library functions instantiated in the generated gmp.h at build time.
  • mpz_set_str etc variants taking string lengths rather than null-terminators.
  • mpz_andn, mpz_iorn, mpz_nand, mpz_nior, mpz_xnor might be useful additions, if they could share code with the current such functions (which should be possible).
  • mpz_and_ui etc might be of use sometimes. Suggested by Niels Möller.
  • mpf_set_str and mpf_inp_str could usefully accept 0x, 0b etc when base==0. Perhaps the exponent could default to decimal in this case, with a further 0x, 0b etc allowed there. Eg. 0xFFAA@0x5A. A leading "0" for octal would match the integers, but probably something like "0.123" ought not mean octal.
  • GMP_LONG_LONG_LIMB or some such could become a documented feature of gmp.h, so applications could know whether to printf a limb using %lu or %Lu.
  • GMP_PRIdMP_LIMB and similar defines following C99 <inttypes.h> might be of use to applications printing limbs. But if GMP_LONG_LONG_LIMB or whatever is added then perhaps this can easily enough be left to applications.
  • gmp_printf could accept %b for binary output. It'd be nice if it worked for plain int etc too, not just mpz_t etc.
  • gmp_printf in fact could usefully accept an arbitrary base, for both integer and float conversions. A base either in the format string or as a parameter with * should be allowed. Maybe &13b (b for base) or something like that.
  • gmp_printf could perhaps accept mpq_t for float conversions, eg. "%.4Qf". This would be merely for convenience, but still might be useful. Rounding would be the same as for an mpf_t (ie. currently round-to-nearest, but not actually documented). Alternately, perhaps a separate mpq_get_str_point or some such might be more use. Suggested by Pedro Gimeno.
  • mpz_rscan0 or mpz_revscan0 or some such searching towards the low end of an integer might match mpz_scan0 nicely. Likewise for scan1. Suggested by Roberto Bagnara.
  • mpz_bit_subset or some such to test whether one integer is a bitwise subset of another might be of use. Some sort of return value indicating whether it's a proper or non-proper subset would be good and wouldn't cost anything in the implementation. Suggested by Roberto Bagnara.
  • mpf_get_ld, mpf_set_ld: Conversions between mpf_t and long double, suggested by Dan Christensen. Other long double routines might be desirable too, but mpf would be a start.
    long double is an ANSI-ism, so everything involving it would need to be suppressed on a K&R compiler.
    There'd be some work to be done by configure to recognise the format in use, MPFR has a start on this. Often long double is the same as double, which is easy but pretty pointless. A single float format detector macro could look at double then long double
    Sometimes there's a compiler option for the size of a long double, eg. xlc on AIX can use either 64-bit or 128-bit. It's probably simplest to regard this as a compiler compatibility issue, and leave it to users or sysadmins to ensure application and library code is built the same.
  • mpz_sqrt_if_perfect_square: When mpz_perfect_square_p does its tests it calculates a square root and then discards it. For some applications it might be useful to return that root. Suggested by Jason Moxham.
  • mpz_get_ull, mpz_set_ull, mpz_get_sll, mpz_get_sll: Conversions for long long. These would aid interoperability, though a mixture of GMP and long long would probably not be too common. Since long long is not always available (it's in C99 and GCC though), disadvantages of using long long in libgmp.a would be
    • Library contents vary according to the build compiler.
    • gmp.h would need an ugly #ifdef block to decide if the application compiler could take the long long prototypes.
    • Some sort of LIBGMP_HAS_LONGLONG might be wanted to indicate whether the functions are available. (Applications using autoconf could probe the library too.)
    It'd be possible to defer the need for long long to application compile time, by having something like mpz_set_2ui called with two halves of a long long. Disadvantages of this would be,
    • Bigger code in the application, though perhaps not if a long long is normally passed as two halves anyway.
    • mpz_get_ull would be a rather big inline, or would have to be two function calls.
    • mpz_get_sll would be a worse inline, and would put the treatment of -0x10..00 into applications (see mpz_get_si correctness above).
    • Although having libgmp.a independent of the build compiler is nice, it sort of sacrifices the capabilities of a good compiler to uniformity with inferior ones.
    Plain use of long long is probably the lesser evil, if only because it makes best use of gcc. In fact perhaps it would suffice to guarantee long long conversions only when using GCC for both application and library. That would cover free software, and we can worry about selected vendor compilers later.
    In C++ the situation is probably clearer, we demand fairly recent C++ so long long should be available always. We'd probably prefer to have the C and C++ the same in respect of long long support, but it would be possible to have it unconditionally in gmpxx.h, by some means or another.
  • mpz_strtoz parsing the same as strtol. Suggested by Alexander Kruppa.

Configuration

  • Alpha ev7, ev79: Add code to config.guess to detect these. Believe ev7 will be "3-1307" in the current switch, but need to verify that. (On OSF, current configfsf.guess identifies ev7 using psrinfo, we need to do it ourselves for other systems.)
  • Alpha OSF: Libtool (version 1.5) doesn't seem to recognise this system is "pic always" and ends up running gcc twice with the same options. This is wasteful, but harmless. Perhaps a newer libtool will be better.
  • ARM: umul_ppmm in longlong.h always uses umull, but is that available only for M series chips or some such? Perhaps it should be configured in some way.
  • HPPA: config.guess should recognize 7000, 7100, 7200, and 8x00.
  • HPPA: gcc 3.2 introduces a -mschedule=7200 etc parameter, which could be driven by an exact hppa cpu type.
  • Mips: config.guess should say mipsr3000, mipsr4000, mipsr10000, etc. "hinv -c processor" gives lots of information on Irix. Standard config.guess appends "el" to indicate endianness, but AC_C_BIGENDIAN seems the best way to handle that for GMP.
  • PowerPC: The function descriptor nonsense for AIX is currently driven by *-*-aix*. It might be more reliable to do some sort of feature test, examining the compiler output perhaps. It might also be nice to merge the aix.m4 files into powerpc-defs.m4.
  • config.m4 is generated only by the configure script, it won't be regenerated by config.status. Creating it as an AC_OUTPUT would work, but it might upset "make" to have things like L$ get into the Makefiles through AC_SUBST. AC_CONFIG_COMMANDS would be the alternative. With some careful m4 quoting the changequote calls might not be needed, which might free up the order in which things had to be output.
  • Automake: Latest automake has a CCAS, CCASFLAGS scheme. Though we probably wouldn't be using its assembler support we could try to use those variables in compatible ways.
  • GMP_LDFLAGS could probably be done with plain LDFLAGS already used by automake for all linking. But with a bit of luck the next libtool will pass pretty much all CFLAGS through to the compiler when linking, making GMP_LDFLAGS unnecessary.
  • mpn/Makeasm.am uses -c and -o together in the .S and .asm rules, but apparently that isn't completely portable (there's an autoconf AC_PROG_CC_C_O test for it). So far we've not had problems, but perhaps the rules could be rewritten to use "foo.s" as the temporary, or to do a suitable "mv" of the result. The only danger from using foo.s would be if a compile failed and the temporary foo.s then looked like the primary source. Hopefully if the SUFFIXES are ordered to have .S and .asm ahead of .s that wouldn't happen. Might need to check.

Random Numbers

  • _gmp_rand is not particularly fast on the linear congruential algorithm and could stand various improvements.
    • Make a second seed area within gmp_randstate_t (or _mp_algdata rather) to save some copying.
    • Make a special case for a single limb 2exp modulus, to avoid mpn_mul calls. Perhaps the same for two limbs.
    • Inline the lc code, to avoid a function call and TMP_ALLOC for every chunk.
    • Perhaps the 2exp and general LC cases should be split, for clarity (if the general case is retained).
  • gmp_randstate_t used for parameters perhaps should become gmp_randstate_ptr the same as other types.
  • Some of the empirical randomness tests could be included in a "make check". They ought to work everywhere, for a given seed at least.

C++

  • mpz_class(string), etc: Use the C++ global locale to identify whitespace.
    mpf_class(string): Use the C++ global locale decimal point, rather than the C one.
    Consider making these variant mpz_set_str etc forms available for mpz_t too, not just mpz_class etc.
  • mpq_class operator+=: Don't emit an unnecssary mpq_set(q,q) before mpz_addmul etc.
  • Put various bits of gmpxx.h into libgmpxx, to avoid excessive inlining. Candidates for this would be,
    • mpz_class(const char *), etc: since they're normally not fast anyway, and we can hide the exception throw.
    • mpz_class(string), etc: to hide the cstr needed to get to the C conversion function.
    • mpz_class string, char* etc constructors: likewise to hide the throws and conversions.
    • mpz_class::get_str, etc: to hide the char* to string conversion and free. Perhaps mpz_get_str can write directly into a string, to avoid copying.
      Consider making such string returning variants available for use with plain mpz_t etc too.

Miscellaneous

  • mpz_gcdext and mpn_gcdext ought to document what range of values the generated cofactors can take, and preferably ensure the definition uniquely specifies the cofactors for given inputs. A basic extended Euclidean algorithm or multi-step variant leads to |x|<|b| and |y|<|a| or something like that, but there's probably two solutions under just those restrictions.
  • demos/factorize.c: use mpz_divisible_ui_p rather than mpz_tdiv_qr_ui. (Of course dividing multiple primes at a time would be better still.)
  • The various test programs use quite a bit of the main libgmp. This establishes good cross-checks, but it might be better to use simple reference routines where possible. Where it's not possible some attention could be paid to the order of the tests, so a libgmp routine is only used for tests once it seems to be good.
  • MUL_FFT_THRESHOLD etc: the FFT thresholds should allow a return to a previous k at certain sizes. This arises basically due to the step effect caused by size multiples effectively used for each k. Looking at a graph makes it fairly clear.
  • __gmp_doprnt_mpf does a rather unattractive round-to-nearest on the string returned by mpf_get_str. Perhaps some variant of mpf_get_str could be made which would better suit.

Aids to Development

  • Add ASSERTs at the start of each user-visible mpz/mpq/mpf function to check the validity of each mp?_t parameter, in particular to check they've been mp?_inited. This might catch elementary mistakes in user programs. Care would need to be taken over MPZ_TMP_INITed variables used internally. If nothing else then consistency checks like size<=alloc, ptr not NULL and ptr+size not wrapping around the address space, would be possible. A more sophisticated scheme could track _mp_d pointers and ensure only a valid one is used. Such a scheme probably wouldn't be reentrant, not without some help from the system.
  • tune/time.c could try to determine at runtime whether getrusage and gettimeofday are reliable. Currently we pretend in configure that the dodgy m68k netbsd 1.4.1 getrusage doesn't exist. If a test might take a long time to run then perhaps cache the result in a file somewhere.
  • tune/time.c could choose the default precision based on the speed_unittime determined, independent of the method in use.
  • Cray vector systems: CPU frequency could be determined from sysconf(_SC_CLK_TCK), since it seems to be clock cycle based. Is this true for all Cray systems? Would like some documentation or something to confirm.

Documentation

  • mpz_inp_str (etc) doesn't say when it stops reading digits.
  • mpn_get_str isn't terribly clear about how many digits it produces. It'd probably be possible to say at most one leading zero, which is what both it and mpz_get_str currently do. But want to be careful not to bind ourselves to something that might not suit another implementation.
  • va_arg doesn't do the right thing with mpz_t etc directly, but instead needs a pointer type like MP_INT*. It'd be good to show how to do this, but we'd either need to document mpz_ptr and friends, or perhaps fallback on something slightly nasty with void*.

Bright Ideas

The following may or may not be feasible, and aren't likely to get done in the near future, but are at least worth thinking about.

  • Reorganize longlong.h so that we can inline the operations even for the system compiler. When there is no such compiler feature, make calls to stub functions. Write such stub functions for as many machines as possible.
  • longlong.h could declare when it's using, or would like to use, mpn_umul_ppmm, and the corresponding umul.asm file could be included in libgmp only in that case, the same as is effectively done for __clz_tab. Likewise udiv.asm and perhaps cntlz.asm. This would only be a very small space saving, so perhaps not worth the complexity.
  • longlong.h could be built at configure time by concatenating or #including fragments from each directory in the mpn path. This would select CPU specific macros the same way as CPU specific assembler code. Code used would no longer depend on cpp predefines, and the current nested conditionals could be flattened out.
  • mpz_get_si returns 0x80000000 for -0x100000000, whereas it's sort of supposed to return the low 31 (or 63) bits. But this is undocumented, and perhaps not too important.
  • mpz_init_set* and mpz_realloc could allocate say an extra 16 limbs over what's needed, so as to reduce the chance of having to do a reallocate if the mpz_t grows a bit more. This could only be an option, since it'd badly bloat memory usage in applications using many small values.
  • mpq functions could perhaps check for numerator or denominator equal to 1, on the assumption that integers or denominator-only values might be expected to occur reasonably often.
  • count_trailing_zeros is used on more or less uniformly distributed numbers in a couple of places. For some CPUs count_trailing_zeros is slow and it's probably worth handling the frequently occurring 0 to 2 trailing zeros cases specially.
  • mpf_t might like to let the exponent be undefined when size==0, instead of requiring it 0 as now. It should be possible to do size==0 tests before paying attention to the exponent. The advantage is not needing to set exp in the various places a zero result can arise, which avoids some tedium but is otherwise perhaps not too important. Currently mpz_set_f and mpf_cmp_ui depend on exp==0, maybe elsewhere too.
  • __gmp_allocate_func: Could use GCC __attribute__ ((malloc)) on this, though don't know if it'd do much. GCC 3.0 allows that attribute on functions, but not function pointers (see info node "Attribute Syntax"), so would need a new autoconf test. This can wait until there's a GCC that supports it.
  • mpz_add_ui contains two __GMPN_COPYs, one from mpn_add_1 and one from mpn_sub_1. If those two routines were opened up a bit maybe that code could be shared. When a copy needs to be done there's no carry to append for the add, and if the copy is non-empty no high zero for the sub.

Old and Obsolete Stuff

The following tasks apply to chips or systems that are old and/or obsolete. It's unlikely anything will be done about them unless anyone is actively using them.

  • Sparc32: The integer based udiv_nfp.asm used to be selected by configure --nfp but that option is gone now that autoconf is used. The file could go somewhere suitable in the mpn search if any chips might benefit from it, though it's possible we don't currently differentiate enough exact cpu types to do this properly.
  • VAX D and G format double floats are straightforward and could perhaps be handled directly in __gmp_extract_double and maybe in mpn_get_d, rather than falling back on the generic code. (Both formats are detected by configure.)

gmp-doc-5.1.2/doc/projects.html0000644000175000000620000005574412146435154015363 0ustar stevestaff GMP Development Projects

GMP Development Projects

Copyright 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2008, 2009, 2010, 2011
Free Software Foundation, Inc.

This file is part of the GNU MP Library.

The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published
by the Free Software Foundation; either version 3 of the License, or (at
your option) any later version.

The GNU MP Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library.  If not, see http://www.gnu.org/licenses/.

This file current as of 5 Dec 2011. An up-to-date version is available at http://gmplib.org/projects.html. Please send comments about this page to gmp-devel@gmplib.org.

This file lists projects suitable for volunteers. Please see the tasks file for smaller tasks.

If you want to work on any of the projects below, please let gmp-devel@gmplib.org know. If you want to help with a project that already somebody else is working on, you will get in touch through gmp-devel@gmplib.org. (There are no email addresses of volunteers below, due to spamming problems.)

  • Faster multiplication
    1. Work on the algorithm selection code for unbalanced multiplication.
    2. Implement an FFT variant computing the coefficients mod m different limb size primes of the form l*2^k+1. i.e., compute m separate FFTs. The wanted coefficients will at the end be found by lifting with CRT (Chinese Remainder Theorem). If we let m = 3, i.e., use 3 primes, we can split the operands into coefficients at limb boundaries, and if our machine uses b-bit limbs, we can multiply numbers with close to 2^b limbs without coefficient overflow. For smaller multiplication, we might perhaps let m = 1, and instead of splitting our operands at limb boundaries, split them in much smaller pieces. We might also use 4 or more primes, and split operands into bigger than b-bit chunks. By using more primes, the gain in shorter transform length, but lose in having to do more FFTs, but that is a slight total save. We then lose in more expensive CRT.

      [We now have two implementations of this algorithm, one by Tommy Färnqvist and one by Niels Möller.]

    3. Work on short products. Our mullo and mulmid are probably K, but we lack mulhi.

    Another possibility would be an optimized cube. In the basecase that should definitely be able to save cross-products in a similar fashion to squaring, but some investigation might be needed for how best to adapt the higher-order algorithms. Not sure whether cubing or further small powers have any particularly important uses though.

  • Assembly routines

    Write new and improve existing assembly routines. The tests/devel programs and the tune/speed.c and tune/many.pl programs are useful for testing and timing the routines you write. See the README files in those directories for more information.

    Please make sure your new routines are fast for these three situations:

    1. Small operands of less than, say, 10 limbs.
    2. Medium size operands, that fit into the cache.
    3. Huge operands that does not fit into the cache.

    The most important routines are mpn_addmul_1, mpn_mul_basecase and mpn_sqr_basecase. The latter two don't exist for all machines, while mpn_addmul_1 exists for almost all machines.

    Standard techniques for these routines are unrolling, software pipelining, and specialization for common operand values. For machines with poor integer multiplication, it is sometimes possible to remedy the situation using floating-point operations or SIMD operations such as MMX (x86) (x86), SSE (x86), VMX (PowerPC), VIS (Sparc).

    Using floating-point operations is interesting but somewhat tricky. Since IEEE double has 53 bit of mantissa, one has to split the operands in small pieces, so that no intermediates are greater than 2^53. For 32-bit computers, splitting one operand into 16-bit pieces works. For 64-bit machines, one operand can be split into 21-bit pieces and the other into 32-bit pieces. (A 64-bit operand can be split into just three 21-bit pieces if one allows the split operands to be negative!)

  • Faster sqrt

    The current code uses divisions, which are reasonably fast, but it'd be possible to use only multiplications by computing 1/sqrt(A) using this iteration:

    				    2
    		   x   = x  (3 − A x )/2
    		    i+1	  i	    i  
    The square root can then be computed like this:
    		     sqrt(A) = A x
    				  n  

    That final multiply might be the full size of the input (though it might only need the high half of that), so there may or may not be any speedup overall.

    We should probably allow a special exponent-like parameter, to speed computations of a precise square root of a small number in mpf and mpfr.

  • Nth root

    Improve mpn_rootrem. The current code is not too bad, but its time complexity is a function of the input, while it is possible to make the average complexity a function of the output.

  • Fat binaries

    Add more functions to the set of fat functions.

    The speed of multipliciaton is today highly dependent on combination functions like addlsh1_n. A fat binary will never use any such functions, since they are classified as optional. Ideally, we should use them, but making the current compile-time selections of optional functions become run-time selections for fat binaries.

    If we make fat binaries work really well, we should move away frm tehe current configure scheme (at least by default) and instead include all code always.

  • Exceptions

    Some sort of scheme for exceptions handling would be desirable. Presently the only thing documented is that divide by zero in GMP functions provokes a deliberate machine divide by zero (on those systems where such a thing exists at least). The global gmp_errno is not actually documented, except for the old gmp_randinit function. Being currently just a plain global means it's not thread-safe.

    The basic choices for exceptions are returning an error code or having a handler function to be called. The disadvantage of error returns is they have to be checked, leading to tedious and rarely executed code, and strictly speaking such a scheme wouldn't be source or binary compatible. The disadvantage of a handler function is that a longjmp or similar recovery from it may be difficult. A combination would be possible, for instance by allowing the handler to return an error code.

    Divide-by-zero, sqrt-of-negative, and similar operand range errors can normally be detected at the start of functions, so exception handling would have a clean state. What's worth considering though is that the GMP function detecting the exception may have been called via some third party library or self contained application module, and hence have various bits of state to be cleaned up above it. It'd be highly desirable for an exceptions scheme to allow for such cleanups.

    The C++ destructor mechanism could help with cleanups both internally and externally, but being a plain C library we don't want to depend on that.

    A C++ throw might be a good optional extra exceptions mechanism, perhaps under a build option. For GCC -fexceptions will add the necessary frame information to plain C code, or GMP could be compiled as C++.

    Out-of-memory exceptions are expected to be handled by the mp_set_memory_functions routines, rather than being a prospective part of divide-by-zero etc. Some similar considerations apply but what differs is that out-of-memory can arise deep within GMP internals. Even fundamental routines like mpn_add_n and mpn_addmul_1 can use temporary memory (for instance on Cray vector systems). Allowing for an error code return would require an awful lot of checking internally. Perhaps it'd still be worthwhile, but it'd be a lot of changes and the extra code would probably be rather rarely executed in normal usages.

    A longjmp recovery for out-of-memory will currently, in general, lead to memory leaks and may leave GMP variables operated on in inconsistent states. Maybe it'd be possible to record recovery information for use by the relevant allocate or reallocate function, but that too would be a lot of changes.

    One scheme for out-of-memory would be to note that all GMP allocations go through the mp_set_memory_functions routines. So if the application has an intended setjmp recovery point it can record memory activity by GMP and abandon space allocated and variables initialized after that point. This might be as simple as directing the allocation functions to a separate pool, but in general would have the disadvantage of needing application-level bookkeeping on top of the normal system malloc. An advantage however is that it needs nothing from GMP itself and on that basis doesn't burden applications not needing recovery. Note that there's probably some details to be worked out here about reallocs of existing variables, and perhaps about copying or swapping between "permanent" and "temporary" variables.

    Applications desiring a fine-grained error control, for instance a language interpreter, would very possibly not be well served by a scheme requiring longjmp. Wrapping every GMP function call with a setjmp would be very inconvenient.

    Another option would be to let mpz_t etc hold a sort of NaN, a special value indicating an out-of-memory or other failure. This would be similar to NaNs in mpfr. Unfortunately such a scheme could only be used by programs prepared to handle such special values, since for instance a program waiting for some condition to be satisfied could become an infinite loop if it wasn't also watching for NaNs. The work to implement this would be significant too, lots of checking of inputs and intermediate results. And if mpn routines were to participate in this (which they would have to internally) a lot of new return values would need to be added, since of course there's no mpz_t etc structure for them to indicate failure in.

    Stack overflow is another possible exception, but perhaps not one that can be easily detected in general. On i386 GNU/Linux for instance GCC normally doesn't generate stack probes for an alloca, but merely adjusts %esp. A big enough alloca can miss the stack redzone and hit arbitrary data. GMP stack usage is normally a function of operand size, which might be enough for some applications to know they'll be safe. Otherwise a fixed maximum usage can probably be obtained by building with --enable-alloca=malloc-reentrant (or notreentrant). Arranging the default to be alloca only on blocks up to a certain size and malloc thereafter might be a better approach and would have the advantage of not having calculations limited by available stack.

    Actually recovering from stack overflow is of course another problem. It might be possible to catch a SIGSEGV in the stack redzone and do something in a sigaltstack, on systems which have that, but recovery might otherwise not be possible. This is worth bearing in mind because there's no point worrying about tight and careful out-of-memory recovery if an out-of-stack is fatal.

    Operand overflow is another exception to be addressed. It's easy for instance to ask mpz_pow_ui for a result bigger than an mpz_t can possibly represent. Currently overflows in limb or byte count calculations will go undetected. Often they'll still end up asking the memory functions for blocks bigger than available memory, but that's by no means certain and results are unpredictable in general. It'd be desirable to tighten up such size calculations. Probably only selected routines would need checks, if it's assumed say that no input will be more than half of all memory and hence size additions like say mpz_mul won't overflow.

  • Performance Tool

    It'd be nice to have some sort of tool for getting an overview of performance. Clearly a great many things could be done, but some primary uses would be,

    1. Checking speed variations between compilers.
    2. Checking relative performance between systems or CPUs.

    A combination of measuring some fundamental routines and some representative application routines might satisfy these.

    The tune/time.c routines would be the easiest way to get good accurate measurements on lots of different systems. The high level speed_measure may or may not suit, but the basic speed_starttime and speed_endtime would cover lots of portability and accuracy questions.

  • Using restrict

    There might be some value in judicious use of C99 style restrict on various pointers, but this would need some careful thought about what it implies for the various operand overlaps permitted in GMP.

    Rumour has it some pre-C99 compilers had restrict, but expressing tighter (or perhaps looser) requirements. Might be worth investigating that before using restrict unconditionally.

    Loops are presumably where the greatest benefit would be had, by allowing the compiler to advance reads ahead of writes, perhaps as part of loop unrolling. However critical loops are generally coded in assembler, so there might not be very much to gain. And on Cray systems the explicit use of _Pragma gives an equivalent effect.

    One thing to note is that Microsoft C headers (on ia64 at least) contain __declspec(restrict), so a #define of restrict should be avoided. It might be wisest to setup a gmp_restrict.

  • Factorial

    Rewrite for simplicty and speed. Work is in progress.

  • Binomial Coefficients

    Rewrite for simplicty and speed. Work is in progress.

  • Prime Testing

    GMP is not really a number theory library and probably shouldn't have large amounts of code dedicated to sophisticated prime testing algorithms, but basic things well-implemented would suit. Tests offering certainty are probably all too big or too slow (or both!) to justify inclusion in the main library. Demo programs showing some possibilities would be good though.

    The present "repetitions" argument to mpz_probab_prime_p is rather specific to the Miller-Rabin tests of the current implementation. Better would be some sort of parameter asking perhaps for a maximum chance 1/2^x of a probable prime in fact being composite. If applications follow the advice that the present reps gives 1/4^reps chance then perhaps such a change is unnecessary, but an explicitly described 1/2^x would allow for changes in the implementation or even for new proofs about the theory.

    mpz_probab_prime_p always initializes a new gmp_randstate_t for randomized tests, which unfortunately means it's not really very random and in particular always runs the same tests for a given input. Perhaps a new interface could accept an rstate to use, so successive tests could increase confidence in the result.

    mpn_mod_34lsub1 is an obvious and easy improvement to the trial divisions. And since the various prime factors are constants, the remainder can be tested with something like

    #define MP_LIMB_DIVISIBLE_7_P(n) \
      ((n) * MODLIMB_INVERSE_7 <= MP_LIMB_T_MAX/7)
    
    Which would help compilers that don't know how to optimize divisions by constants, and is even an improvement on current gcc 3.2 code. This technique works for any modulus, see Granlund and Montgomery "Division by Invariant Integers" section 9.

    The trial divisions are done with primes generated and grouped at runtime. This could instead be a table of data, with pre-calculated inverses too. Storing deltas, ie. amounts to add, rather than actual primes would save space. udiv_qrnnd_preinv style inverses can be made to exist by adding dummy factors of 2 if necessary. Some thought needs to be given as to how big such a table should be, based on how much dividing would be profitable for what sort of size inputs. The data could be shared by the perfect power testing.

    Jason Moxham points out that if a sqrt(-1) mod N exists then any factor of N must be == 1 mod 4, saving half the work in trial dividing. (If x^2==-1 mod N then for a prime factor p we have x^2==-1 mod p and so the jacobi symbol (-1/p)=1. But also (-1/p)=(-1)^((p-1)/2), hence must have p==1 mod 4.) But knowing whether sqrt(-1) mod N exists is not too easy. A strong pseudoprime test can reveal one, so perhaps such a test could be inserted part way though the dividing.

    Jon Grantham "Frobenius Pseudoprimes" (www.pseudoprime.com) describes a quadratic pseudoprime test taking about 3x longer than a plain test, but with only a 1/7710 chance of error (whereas 3 plain Miller-Rabin tests would offer only (1/4)^3 == 1/64). Such a test needs completely random parameters to satisfy the theory, though single-limb values would run faster. It's probably best to do at least one plain Miller-Rabin before any quadratic tests, since that can identify composites in less total time.

    Some thought needs to be given to the structure of which tests (trial division, Miller-Rabin, quadratic) and how many are done, based on what sort of inputs we expect, with a view to minimizing average time.

    It might be a good idea to break out subroutines for the various tests, so that an application can combine them in ways it prefers, if sensible defaults in mpz_probab_prime_p don't suit. In particular this would let applications skip tests it knew would be unprofitable, like trial dividing when an input is already known to have no small factors.

    For small inputs, combinations of theory and explicit search make it relatively easy to offer certainty. For instance numbers up to 2^32 could be handled with a strong pseudoprime test and table lookup. But it's rather doubtful whether a smallnum prime test belongs in a bignum library. Perhaps if it had other internal uses.

    An mpz_nthprime might be cute, but is almost certainly impractical for anything but small n.

  • Intra-Library Calls

    On various systems, calls within libgmp still go through the PLT, TOC or other mechanism, which makes the code bigger and slower than it needs to be.

    The theory would be to have all GMP intra-library calls resolved directly to the routines in the library. An application wouldn't be able to replace a routine, the way it can normally, but there seems no good reason to do that, in normal circumstances.

    The visibility attribute in recent gcc is good for this, because it lets gcc omit unnecessary GOT pointer setups or whatever if it finds all calls are local and there's no global data references. Documented entrypoints would be protected, and purely internal things not wanted by test programs or anything can be internal.

    Unfortunately, on i386 it seems protected ends up causing text segment relocations within libgmp.so, meaning the library code can't be shared between processes, defeating the purpose of a shared library. Perhaps this is just a gremlin in binutils (debian packaged 2.13.90.0.16-1).

    The linker can be told directly (with a link script, or options) to do the same sort of thing. This doesn't change the code emitted by gcc of course, but it does mean calls are resolved directly to their targets, avoiding a PLT entry.

    Keeping symbols private to libgmp.so is probably a good thing in general too, to stop anyone even attempting to access them. But some undocumented things will need or want to be kept visible, for use by mpfr, or the test and tune programs. Libtool has a standard option for selecting public symbols (used now for libmp).

  • Math functions for the mpf layer

    Implement the functions of math.h for the GMP mpf layer! Check the book "Pi and the AGM" by Borwein and Borwein for ideas how to do this. These functions are desirable: acos, acosh, asin, asinh, atan, atanh, atan2, cos, cosh, exp, log, log10, pow, sin, sinh, tan, tanh.

    Note that the mpfr functions already provide these functions, and that we usually recommend new programs to use mpfr instead of mpf.


gmp-doc-5.1.2/doc/gmp.info0000644000175000000620000001366012146435202014265 0ustar stevestaffThis is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from ../../gmp/doc/gmp.texi. This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover Texts being "You have freedom to copy and modify this GNU Manual, like GNU software". A copy of the license is included in *note GNU Free Documentation License::. INFO-DIR-SECTION GNU libraries START-INFO-DIR-ENTRY * gmp: (gmp). GNU Multiple Precision Arithmetic Library. END-INFO-DIR-ENTRY  Indirect: gmp.info-1: 997 gmp.info-2: 299989  Tag Table: (Indirect) Node: Top997 Node: Copying3183 Node: Introduction to GMP5034 Node: Installing GMP7745 Node: Build Options8477 Node: ABI and ISA24261 Node: Notes for Package Builds33832 Node: Notes for Particular Systems36919 Node: Known Build Problems43516 Node: Performance optimization47050 Node: GMP Basics48179 Node: Headers and Libraries48827 Node: Nomenclature and Types50251 Node: Function Classes52247 Node: Variable Conventions53781 Node: Parameter Conventions55390 Node: Memory Management57446 Node: Reentrancy58574 Node: Useful Macros and Constants60447 Node: Compatibility with older versions61438 Node: Demonstration Programs62349 Node: Efficiency64214 Node: Debugging71838 Node: Profiling78754 Node: Autoconf82745 Node: Emacs84524 Node: Reporting Bugs85130 Node: Integer Functions87673 Node: Initializing Integers88449 Node: Assigning Integers90825 Node: Simultaneous Integer Init & Assign92412 Node: Converting Integers94037 Node: Integer Arithmetic96961 Node: Integer Division98547 Node: Integer Exponentiation104857 Node: Integer Roots106297 Node: Number Theoretic Functions107971 Node: Integer Comparisons115256 Node: Integer Logic and Bit Fiddling116634 Node: I/O of Integers119181 Node: Integer Random Numbers122150 Node: Integer Import and Export124761 Node: Miscellaneous Integer Functions128771 Node: Integer Special Functions130631 Node: Rational Number Functions133718 Node: Initializing Rationals134911 Node: Rational Conversions137372 Node: Rational Arithmetic139104 Node: Comparing Rationals140408 Node: Applying Integer Functions141774 Node: I/O of Rationals143257 Node: Floating-point Functions145299 Node: Initializing Floats148184 Node: Assigning Floats152271 Node: Simultaneous Float Init & Assign154838 Node: Converting Floats156366 Node: Float Arithmetic159616 Node: Float Comparison161629 Node: I/O of Floats163209 Node: Miscellaneous Float Functions165892 Node: Low-level Functions167834 Node: Random Number Functions192351 Node: Random State Initialization193419 Node: Random State Seeding196278 Node: Random State Miscellaneous197667 Node: Formatted Output198309 Node: Formatted Output Strings198554 Node: Formatted Output Functions203933 Node: C++ Formatted Output208008 Node: Formatted Input210690 Node: Formatted Input Strings210926 Node: Formatted Input Functions215578 Node: C++ Formatted Input218547 Node: C++ Class Interface220450 Node: C++ Interface General221444 Node: C++ Interface Integers224514 Node: C++ Interface Rationals228229 Node: C++ Interface Floats232246 Node: C++ Interface Random Numbers238250 Node: C++ Interface Limitations240652 Node: Custom Allocation243472 Node: Language Bindings247691 Node: Algorithms251646 Node: Multiplication Algorithms252346 Node: Basecase Multiplication253435 Node: Karatsuba Multiplication255343 Node: Toom 3-Way Multiplication258969 Node: Toom 4-Way Multiplication265388 Node: Higher degree Toom'n'half266767 Node: FFT Multiplication268052 Node: Other Multiplication273387 Node: Unbalanced Multiplication275861 Node: Division Algorithms276649 Node: Single Limb Division277028 Node: Basecase Division279918 Node: Divide and Conquer Division281121 Node: Block-Wise Barrett Division283190 Node: Exact Division283842 Node: Exact Remainder287007 Node: Small Quotient Division289257 Node: Greatest Common Divisor Algorithms290855 Node: Binary GCD291152 Node: Lehmer's Algorithm294001 Node: Subquadratic GCD296220 Node: Extended GCD298677 Node: Jacobi Symbol299989 Node: Powering Algorithms301004 Node: Normal Powering Algorithm301267 Node: Modular Powering Algorithm301795 Node: Root Extraction Algorithms302577 Node: Square Root Algorithm302892 Node: Nth Root Algorithm305033 Node: Perfect Square Algorithm305818 Node: Perfect Power Algorithm307905 Node: Radix Conversion Algorithms308526 Node: Binary to Radix308902 Node: Radix to Binary312832 Node: Other Algorithms314920 Node: Prime Testing Algorithm315272 Node: Factorial Algorithm316456 Node: Binomial Coefficients Algorithm318846 Node: Fibonacci Numbers Algorithm319740 Node: Lucas Numbers Algorithm322214 Node: Random Number Algorithms322935 Node: Assembly Coding325057 Node: Assembly Code Organisation326017 Node: Assembly Basics326984 Node: Assembly Carry Propagation328134 Node: Assembly Cache Handling329965 Node: Assembly Functional Units332126 Node: Assembly Floating Point333739 Node: Assembly SIMD Instructions337517 Node: Assembly Software Pipelining338499 Node: Assembly Loop Unrolling339561 Node: Assembly Writing Guide341776 Node: Internals344541 Node: Integer Internals345053 Node: Rational Internals347309 Node: Float Internals348547 Node: Raw Output Internals355961 Node: C++ Interface Internals357155 Node: Contributors360441 Node: References366191 Node: GNU Free Documentation License371946 Node: Concept Index397115 Node: Function Index443004  End Tag Table  Local Variables: coding: iso-8859-1 End: gmp-doc-5.1.2/doc/texinfo.tex0000644000175000000620000105530712146435154015036 0ustar stevestaff% texinfo.tex -- TeX macros to handle Texinfo files. % % Load plain if necessary, i.e., if running under initex. \expandafter\ifx\csname fmtname\endcsname\relax\input plain\fi % \def\texinfoversion{2009-11-15.11} % % Copyright (C) 1985, 1986, 1988, 1990, 1991, 1992, 1993, 1994, 1995, % 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, % 2007, 2008 Free Software Foundation, Inc. % % This texinfo.tex file is free software: you can redistribute it and/or % modify it under the terms of the GNU General Public License as % published by the Free Software Foundation, either version 3 of the % License, or (at your option) any later version. % % This texinfo.tex file is distributed in the hope that it will be % useful, but WITHOUT ANY WARRANTY; without even the implied warranty % of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU % General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % % As a special exception, when this file is read by TeX when processing % a Texinfo source document, you may use the result without % restriction. (This has been our intent since Texinfo was invented.) % % Please try the latest version of texinfo.tex before submitting bug % reports; you can get the latest version from: % http://www.gnu.org/software/texinfo/ (the Texinfo home page), or % ftp://tug.org/tex/texinfo.tex % (and all CTAN mirrors, see http://www.ctan.org). % The texinfo.tex in any given distribution could well be out % of date, so if that's what you're using, please check. % % Send bug reports to bug-texinfo@gnu.org. Please include including a % complete document in each bug report with which we can reproduce the % problem. Patches are, of course, greatly appreciated. % % To process a Texinfo manual with TeX, it's most reliable to use the % texi2dvi shell script that comes with the distribution. For a simple % manual foo.texi, however, you can get away with this: % tex foo.texi % texindex foo.?? % tex foo.texi % tex foo.texi % dvips foo.dvi -o # or whatever; this makes foo.ps. % The extra TeX runs get the cross-reference information correct. % Sometimes one run after texindex suffices, and sometimes you need more % than two; texi2dvi does it as many times as necessary. % % It is possible to adapt texinfo.tex for other languages, to some % extent. You can get the existing language-specific files from the % full Texinfo distribution. % % The GNU Texinfo home page is http://www.gnu.org/software/texinfo. \message{Loading texinfo [version \texinfoversion]:} % If in a .fmt file, print the version number % and turn on active characters that we couldn't do earlier because % they might have appeared in the input file name. \everyjob{\message{[Texinfo version \texinfoversion]}% \catcode`+=\active \catcode`\_=\active} \chardef\other=12 % We never want plain's \outer definition of \+ in Texinfo. % For @tex, we can use \tabalign. \let\+ = \relax % Save some plain tex macros whose names we will redefine. \let\ptexb=\b \let\ptexbullet=\bullet \let\ptexc=\c \let\ptexcomma=\, \let\ptexdot=\. \let\ptexdots=\dots \let\ptexend=\end \let\ptexequiv=\equiv \let\ptexexclam=\! \let\ptexfootnote=\footnote \let\ptexgtr=> \let\ptexhat=^ \let\ptexi=\i \let\ptexindent=\indent \let\ptexinsert=\insert \let\ptexlbrace=\{ \let\ptexless=< \let\ptexnewwrite\newwrite \let\ptexnoindent=\noindent \let\ptexplus=+ \let\ptexrbrace=\} \let\ptexslash=\/ \let\ptexstar=\* \let\ptext=\t \let\ptextop=\top % If this character appears in an error message or help string, it % starts a new line in the output. \newlinechar = `^^J % Use TeX 3.0's \inputlineno to get the line number, for better error % messages, but if we're using an old version of TeX, don't do anything. % \ifx\inputlineno\thisisundefined \let\linenumber = \empty % Pre-3.0. \else \def\linenumber{l.\the\inputlineno:\space} \fi % Set up fixed words for English if not already set. \ifx\putwordAppendix\undefined \gdef\putwordAppendix{Appendix}\fi \ifx\putwordChapter\undefined \gdef\putwordChapter{Chapter}\fi \ifx\putwordfile\undefined \gdef\putwordfile{file}\fi \ifx\putwordin\undefined \gdef\putwordin{in}\fi \ifx\putwordIndexIsEmpty\undefined \gdef\putwordIndexIsEmpty{(Index is empty)}\fi \ifx\putwordIndexNonexistent\undefined \gdef\putwordIndexNonexistent{(Index is nonexistent)}\fi \ifx\putwordInfo\undefined \gdef\putwordInfo{Info}\fi \ifx\putwordInstanceVariableof\undefined \gdef\putwordInstanceVariableof{Instance Variable of}\fi \ifx\putwordMethodon\undefined \gdef\putwordMethodon{Method on}\fi \ifx\putwordNoTitle\undefined \gdef\putwordNoTitle{No Title}\fi \ifx\putwordof\undefined \gdef\putwordof{of}\fi \ifx\putwordon\undefined \gdef\putwordon{on}\fi \ifx\putwordpage\undefined \gdef\putwordpage{page}\fi \ifx\putwordsection\undefined \gdef\putwordsection{section}\fi \ifx\putwordSection\undefined \gdef\putwordSection{Section}\fi \ifx\putwordsee\undefined \gdef\putwordsee{see}\fi \ifx\putwordSee\undefined \gdef\putwordSee{See}\fi \ifx\putwordShortTOC\undefined \gdef\putwordShortTOC{Short Contents}\fi \ifx\putwordTOC\undefined \gdef\putwordTOC{Table of Contents}\fi % \ifx\putwordMJan\undefined \gdef\putwordMJan{January}\fi \ifx\putwordMFeb\undefined \gdef\putwordMFeb{February}\fi \ifx\putwordMMar\undefined \gdef\putwordMMar{March}\fi \ifx\putwordMApr\undefined \gdef\putwordMApr{April}\fi \ifx\putwordMMay\undefined \gdef\putwordMMay{May}\fi \ifx\putwordMJun\undefined \gdef\putwordMJun{June}\fi \ifx\putwordMJul\undefined \gdef\putwordMJul{July}\fi \ifx\putwordMAug\undefined \gdef\putwordMAug{August}\fi \ifx\putwordMSep\undefined \gdef\putwordMSep{September}\fi \ifx\putwordMOct\undefined \gdef\putwordMOct{October}\fi \ifx\putwordMNov\undefined \gdef\putwordMNov{November}\fi \ifx\putwordMDec\undefined \gdef\putwordMDec{December}\fi % \ifx\putwordDefmac\undefined \gdef\putwordDefmac{Macro}\fi \ifx\putwordDefspec\undefined \gdef\putwordDefspec{Special Form}\fi \ifx\putwordDefvar\undefined \gdef\putwordDefvar{Variable}\fi \ifx\putwordDefopt\undefined \gdef\putwordDefopt{User Option}\fi \ifx\putwordDeffunc\undefined \gdef\putwordDeffunc{Function}\fi % Since the category of space is not known, we have to be careful. \chardef\spacecat = 10 \def\spaceisspace{\catcode`\ =\spacecat} % sometimes characters are active, so we need control sequences. \chardef\colonChar = `\: \chardef\commaChar = `\, \chardef\dashChar = `\- \chardef\dotChar = `\. \chardef\exclamChar= `\! \chardef\lquoteChar= `\` \chardef\questChar = `\? \chardef\rquoteChar= `\' \chardef\semiChar = `\; \chardef\underChar = `\_ % Ignore a token. % \def\gobble#1{} % The following is used inside several \edef's. \def\makecsname#1{\expandafter\noexpand\csname#1\endcsname} % Hyphenation fixes. \hyphenation{ Flor-i-da Ghost-script Ghost-view Mac-OS Post-Script ap-pen-dix bit-map bit-maps data-base data-bases eshell fall-ing half-way long-est man-u-script man-u-scripts mini-buf-fer mini-buf-fers over-view par-a-digm par-a-digms rath-er rec-tan-gu-lar ro-bot-ics se-vere-ly set-up spa-ces spell-ing spell-ings stand-alone strong-est time-stamp time-stamps which-ever white-space wide-spread wrap-around } % Margin to add to right of even pages, to left of odd pages. \newdimen\bindingoffset \newdimen\normaloffset \newdimen\pagewidth \newdimen\pageheight % For a final copy, take out the rectangles % that mark overfull boxes (in case you have decided % that the text looks ok even though it passes the margin). % \def\finalout{\overfullrule=0pt} % @| inserts a changebar to the left of the current line. It should % surround any changed text. This approach does *not* work if the % change spans more than two lines of output. To handle that, we would % have adopt a much more difficult approach (putting marks into the main % vertical list for the beginning and end of each change). % \def\|{% % \vadjust can only be used in horizontal mode. \leavevmode % % Append this vertical mode material after the current line in the output. \vadjust{% % We want to insert a rule with the height and depth of the current % leading; that is exactly what \strutbox is supposed to record. \vskip-\baselineskip % % \vadjust-items are inserted at the left edge of the type. So % the \llap here moves out into the left-hand margin. \llap{% % % For a thicker or thinner bar, change the `1pt'. \vrule height\baselineskip width1pt % % This is the space between the bar and the text. \hskip 12pt }% }% } % Sometimes it is convenient to have everything in the transcript file % and nothing on the terminal. We don't just call \tracingall here, % since that produces some useless output on the terminal. We also make % some effort to order the tracing commands to reduce output in the log % file; cf. trace.sty in LaTeX. % \def\gloggingall{\begingroup \globaldefs = 1 \loggingall \endgroup}% \def\loggingall{% \tracingstats2 \tracingpages1 \tracinglostchars2 % 2 gives us more in etex \tracingparagraphs1 \tracingoutput1 \tracingmacros2 \tracingrestores1 \showboxbreadth\maxdimen \showboxdepth\maxdimen \ifx\eTeXversion\undefined\else % etex gives us more logging \tracingscantokens1 \tracingifs1 \tracinggroups1 \tracingnesting2 \tracingassigns1 \fi \tracingcommands3 % 3 gives us more in etex \errorcontextlines16 }% % add check for \lastpenalty to plain's definitions. If the last thing % we did was a \nobreak, we don't want to insert more space. % \def\smallbreak{\ifnum\lastpenalty<10000\par\ifdim\lastskip<\smallskipamount \removelastskip\penalty-50\smallskip\fi\fi} \def\medbreak{\ifnum\lastpenalty<10000\par\ifdim\lastskip<\medskipamount \removelastskip\penalty-100\medskip\fi\fi} \def\bigbreak{\ifnum\lastpenalty<10000\par\ifdim\lastskip<\bigskipamount \removelastskip\penalty-200\bigskip\fi\fi} % For @cropmarks command. % Do @cropmarks to get crop marks. % \newif\ifcropmarks \let\cropmarks = \cropmarkstrue % % Dimensions to add cropmarks at corners. % Added by P. A. MacKay, 12 Nov. 1986 % \newdimen\outerhsize \newdimen\outervsize % set by the paper size routines \newdimen\cornerlong \cornerlong=1pc \newdimen\cornerthick \cornerthick=.3pt \newdimen\topandbottommargin \topandbottommargin=.75in % Output a mark which sets \thischapter, \thissection and \thiscolor. % We dump everything together because we only have one kind of mark. % This works because we only use \botmark / \topmark, not \firstmark. % % A mark contains a subexpression of the \ifcase ... \fi construct. % \get*marks macros below extract the needed part using \ifcase. % % Another complication is to let the user choose whether \thischapter % (\thissection) refers to the chapter (section) in effect at the top % of a page, or that at the bottom of a page. The solution is % described on page 260 of The TeXbook. It involves outputting two % marks for the sectioning macros, one before the section break, and % one after. I won't pretend I can describe this better than DEK... \def\domark{% \toks0=\expandafter{\lastchapterdefs}% \toks2=\expandafter{\lastsectiondefs}% \toks4=\expandafter{\prevchapterdefs}% \toks6=\expandafter{\prevsectiondefs}% \toks8=\expandafter{\lastcolordefs}% \mark{% \the\toks0 \the\toks2 \noexpand\or \the\toks4 \the\toks6 \noexpand\else \the\toks8 }% } % \topmark doesn't work for the very first chapter (after the title % page or the contents), so we use \firstmark there -- this gets us % the mark with the chapter defs, unless the user sneaks in, e.g., % @setcolor (or @url, or @link, etc.) between @contents and the very % first @chapter. \def\gettopheadingmarks{% \ifcase0\topmark\fi \ifx\thischapter\empty \ifcase0\firstmark\fi \fi } \def\getbottomheadingmarks{\ifcase1\botmark\fi} \def\getcolormarks{\ifcase2\topmark\fi} % Avoid "undefined control sequence" errors. \def\lastchapterdefs{} \def\lastsectiondefs{} \def\prevchapterdefs{} \def\prevsectiondefs{} \def\lastcolordefs{} % Main output routine. \chardef\PAGE = 255 \output = {\onepageout{\pagecontents\PAGE}} \newbox\headlinebox \newbox\footlinebox % \onepageout takes a vbox as an argument. Note that \pagecontents % does insertions, but you have to call it yourself. \def\onepageout#1{% \ifcropmarks \hoffset=0pt \else \hoffset=\normaloffset \fi % \ifodd\pageno \advance\hoffset by \bindingoffset \else \advance\hoffset by -\bindingoffset\fi % % Do this outside of the \shipout so @code etc. will be expanded in % the headline as they should be, not taken literally (outputting ''code). \ifodd\pageno \getoddheadingmarks \else \getevenheadingmarks \fi \setbox\headlinebox = \vbox{\let\hsize=\pagewidth \makeheadline}% \ifodd\pageno \getoddfootingmarks \else \getevenfootingmarks \fi \setbox\footlinebox = \vbox{\let\hsize=\pagewidth \makefootline}% % {% % Have to do this stuff outside the \shipout because we want it to % take effect in \write's, yet the group defined by the \vbox ends % before the \shipout runs. % \indexdummies % don't expand commands in the output. \normalturnoffactive % \ in index entries must not stay \, e.g., if % the page break happens to be in the middle of an example. % We don't want .vr (or whatever) entries like this: % \entry{{\tt \indexbackslash }acronym}{32}{\code {\acronym}} % "\acronym" won't work when it's read back in; % it needs to be % {\code {{\tt \backslashcurfont }acronym} \shipout\vbox{% % Do this early so pdf references go to the beginning of the page. \ifpdfmakepagedest \pdfdest name{\the\pageno} xyz\fi % \ifcropmarks \vbox to \outervsize\bgroup \hsize = \outerhsize \vskip-\topandbottommargin \vtop to0pt{% \line{\ewtop\hfil\ewtop}% \nointerlineskip \line{% \vbox{\moveleft\cornerthick\nstop}% \hfill \vbox{\moveright\cornerthick\nstop}% }% \vss}% \vskip\topandbottommargin \line\bgroup \hfil % center the page within the outer (page) hsize. \ifodd\pageno\hskip\bindingoffset\fi \vbox\bgroup \fi % \unvbox\headlinebox \pagebody{#1}% \ifdim\ht\footlinebox > 0pt % Only leave this space if the footline is nonempty. % (We lessened \vsize for it in \oddfootingyyy.) % The \baselineskip=24pt in plain's \makefootline has no effect. \vskip 24pt \unvbox\footlinebox \fi % \ifcropmarks \egroup % end of \vbox\bgroup \hfil\egroup % end of (centering) \line\bgroup \vskip\topandbottommargin plus1fill minus1fill \boxmaxdepth = \cornerthick \vbox to0pt{\vss \line{% \vbox{\moveleft\cornerthick\nsbot}% \hfill \vbox{\moveright\cornerthick\nsbot}% }% \nointerlineskip \line{\ewbot\hfil\ewbot}% }% \egroup % \vbox from first cropmarks clause \fi }% end of \shipout\vbox }% end of group with \indexdummies \advancepageno \ifnum\outputpenalty>-20000 \else\dosupereject\fi } \newinsert\margin \dimen\margin=\maxdimen \def\pagebody#1{\vbox to\pageheight{\boxmaxdepth=\maxdepth #1}} {\catcode`\@ =11 \gdef\pagecontents#1{\ifvoid\topins\else\unvbox\topins\fi % marginal hacks, juha@viisa.uucp (Juha Takala) \ifvoid\margin\else % marginal info is present \rlap{\kern\hsize\vbox to\z@{\kern1pt\box\margin \vss}}\fi \dimen@=\dp#1\relax \unvbox#1\relax \ifvoid\footins\else\vskip\skip\footins\footnoterule \unvbox\footins\fi \ifr@ggedbottom \kern-\dimen@ \vfil \fi} } % Here are the rules for the cropmarks. Note that they are % offset so that the space between them is truly \outerhsize or \outervsize % (P. A. MacKay, 12 November, 1986) % \def\ewtop{\vrule height\cornerthick depth0pt width\cornerlong} \def\nstop{\vbox {\hrule height\cornerthick depth\cornerlong width\cornerthick}} \def\ewbot{\vrule height0pt depth\cornerthick width\cornerlong} \def\nsbot{\vbox {\hrule height\cornerlong depth\cornerthick width\cornerthick}} % Parse an argument, then pass it to #1. The argument is the rest of % the input line (except we remove a trailing comment). #1 should be a % macro which expects an ordinary undelimited TeX argument. % \def\parsearg{\parseargusing{}} \def\parseargusing#1#2{% \def\argtorun{#2}% \begingroup \obeylines \spaceisspace #1% \parseargline\empty% Insert the \empty token, see \finishparsearg below. } {\obeylines % \gdef\parseargline#1^^M{% \endgroup % End of the group started in \parsearg. \argremovecomment #1\comment\ArgTerm% }% } % First remove any @comment, then any @c comment. \def\argremovecomment#1\comment#2\ArgTerm{\argremovec #1\c\ArgTerm} \def\argremovec#1\c#2\ArgTerm{\argcheckspaces#1\^^M\ArgTerm} % Each occurrence of `\^^M' or `\^^M' is replaced by a single space. % % \argremovec might leave us with trailing space, e.g., % @end itemize @c foo % This space token undergoes the same procedure and is eventually removed % by \finishparsearg. % \def\argcheckspaces#1\^^M{\argcheckspacesX#1\^^M \^^M} \def\argcheckspacesX#1 \^^M{\argcheckspacesY#1\^^M} \def\argcheckspacesY#1\^^M#2\^^M#3\ArgTerm{% \def\temp{#3}% \ifx\temp\empty % Do not use \next, perhaps the caller of \parsearg uses it; reuse \temp: \let\temp\finishparsearg \else \let\temp\argcheckspaces \fi % Put the space token in: \temp#1 #3\ArgTerm } % If a _delimited_ argument is enclosed in braces, they get stripped; so % to get _exactly_ the rest of the line, we had to prevent such situation. % We prepended an \empty token at the very beginning and we expand it now, % just before passing the control to \argtorun. % (Similarly, we have to think about #3 of \argcheckspacesY above: it is % either the null string, or it ends with \^^M---thus there is no danger % that a pair of braces would be stripped. % % But first, we have to remove the trailing space token. % \def\finishparsearg#1 \ArgTerm{\expandafter\argtorun\expandafter{#1}} % \parseargdef\foo{...} % is roughly equivalent to % \def\foo{\parsearg\Xfoo} % \def\Xfoo#1{...} % % Actually, I use \csname\string\foo\endcsname, ie. \\foo, as it is my % favourite TeX trick. --kasal, 16nov03 \def\parseargdef#1{% \expandafter \doparseargdef \csname\string#1\endcsname #1% } \def\doparseargdef#1#2{% \def#2{\parsearg#1}% \def#1##1% } % Several utility definitions with active space: { \obeyspaces \gdef\obeyedspace{ } % Make each space character in the input produce a normal interword % space in the output. Don't allow a line break at this space, as this % is used only in environments like @example, where each line of input % should produce a line of output anyway. % \gdef\sepspaces{\obeyspaces\let =\tie} % If an index command is used in an @example environment, any spaces % therein should become regular spaces in the raw index file, not the % expansion of \tie (\leavevmode \penalty \@M \ ). \gdef\unsepspaces{\let =\space} } \def\flushcr{\ifx\par\lisppar \def\next##1{}\else \let\next=\relax \fi \next} % Define the framework for environments in texinfo.tex. It's used like this: % % \envdef\foo{...} % \def\Efoo{...} % % It's the responsibility of \envdef to insert \begingroup before the % actual body; @end closes the group after calling \Efoo. \envdef also % defines \thisenv, so the current environment is known; @end checks % whether the environment name matches. The \checkenv macro can also be % used to check whether the current environment is the one expected. % % Non-false conditionals (@iftex, @ifset) don't fit into this, so they % are not treated as environments; they don't open a group. (The % implementation of @end takes care not to call \endgroup in this % special case.) % At run-time, environments start with this: \def\startenvironment#1{\begingroup\def\thisenv{#1}} % initialize \let\thisenv\empty % ... but they get defined via ``\envdef\foo{...}'': \long\def\envdef#1#2{\def#1{\startenvironment#1#2}} \def\envparseargdef#1#2{\parseargdef#1{\startenvironment#1#2}} % Check whether we're in the right environment: \def\checkenv#1{% \def\temp{#1}% \ifx\thisenv\temp \else \badenverr \fi } % Environment mismatch, #1 expected: \def\badenverr{% \errhelp = \EMsimple \errmessage{This command can appear only \inenvironment\temp, not \inenvironment\thisenv}% } \def\inenvironment#1{% \ifx#1\empty out of any environment% \else in environment \expandafter\string#1% \fi } % @end foo executes the definition of \Efoo. % But first, it executes a specialized version of \checkenv % \parseargdef\end{% \if 1\csname iscond.#1\endcsname \else % The general wording of \badenverr may not be ideal, but... --kasal, 06nov03 \expandafter\checkenv\csname#1\endcsname \csname E#1\endcsname \endgroup \fi } \newhelp\EMsimple{Press RETURN to continue.} %% Simple single-character @ commands % @@ prints an @ % Kludge this until the fonts are right (grr). \def\@{{\tt\char64}} % This is turned off because it was never documented % and you can use @w{...} around a quote to suppress ligatures. %% Define @` and @' to be the same as ` and ' %% but suppressing ligatures. %\def\`{{`}} %\def\'{{'}} % Used to generate quoted braces. \def\mylbrace {{\tt\char123}} \def\myrbrace {{\tt\char125}} \let\{=\mylbrace \let\}=\myrbrace \begingroup % Definitions to produce \{ and \} commands for indices, % and @{ and @} for the aux/toc files. \catcode`\{ = \other \catcode`\} = \other \catcode`\[ = 1 \catcode`\] = 2 \catcode`\! = 0 \catcode`\\ = \other !gdef!lbracecmd[\{]% !gdef!rbracecmd[\}]% !gdef!lbraceatcmd[@{]% !gdef!rbraceatcmd[@}]% !endgroup % @comma{} to avoid , parsing problems. \let\comma = , % Accents: @, @dotaccent @ringaccent @ubaraccent @udotaccent % Others are defined by plain TeX: @` @' @" @^ @~ @= @u @v @H. \let\, = \c \let\dotaccent = \. \def\ringaccent#1{{\accent23 #1}} \let\tieaccent = \t \let\ubaraccent = \b \let\udotaccent = \d % Other special characters: @questiondown @exclamdown @ordf @ordm % Plain TeX defines: @AA @AE @O @OE @L (plus lowercase versions) @ss. \def\questiondown{?`} \def\exclamdown{!`} \def\ordf{\leavevmode\raise1ex\hbox{\selectfonts\lllsize \underbar{a}}} \def\ordm{\leavevmode\raise1ex\hbox{\selectfonts\lllsize \underbar{o}}} % Dotless i and dotless j, used for accents. \def\imacro{i} \def\jmacro{j} \def\dotless#1{% \def\temp{#1}% \ifx\temp\imacro \ifmmode\imath \else\ptexi \fi \else\ifx\temp\jmacro \ifmmode\jmath \else\j \fi \else \errmessage{@dotless can be used only with i or j}% \fi\fi } % The \TeX{} logo, as in plain, but resetting the spacing so that a % period following counts as ending a sentence. (Idea found in latex.) % \edef\TeX{\TeX \spacefactor=1000 } % @LaTeX{} logo. Not quite the same results as the definition in % latex.ltx, since we use a different font for the raised A; it's most % convenient for us to use an explicitly smaller font, rather than using % the \scriptstyle font (since we don't reset \scriptstyle and % \scriptscriptstyle). % \def\LaTeX{% L\kern-.36em {\setbox0=\hbox{T}% \vbox to \ht0{\hbox{\selectfonts\lllsize A}\vss}}% \kern-.15em \TeX } % Be sure we're in horizontal mode when doing a tie, since we make space % equivalent to this in @example-like environments. Otherwise, a space % at the beginning of a line will start with \penalty -- and % since \penalty is valid in vertical mode, we'd end up putting the % penalty on the vertical list instead of in the new paragraph. {\catcode`@ = 11 % Avoid using \@M directly, because that causes trouble % if the definition is written into an index file. \global\let\tiepenalty = \@M \gdef\tie{\leavevmode\penalty\tiepenalty\ } } % @: forces normal size whitespace following. \def\:{\spacefactor=1000 } % @* forces a line break. \def\*{\hfil\break\hbox{}\ignorespaces} % @/ allows a line break. \let\/=\allowbreak % @. is an end-of-sentence period. \def\.{.\spacefactor=\endofsentencespacefactor\space} % @! is an end-of-sentence bang. \def\!{!\spacefactor=\endofsentencespacefactor\space} % @? is an end-of-sentence query. \def\?{?\spacefactor=\endofsentencespacefactor\space} % @frenchspacing on|off says whether to put extra space after punctuation. % \def\onword{on} \def\offword{off} % \parseargdef\frenchspacing{% \def\temp{#1}% \ifx\temp\onword \plainfrenchspacing \else\ifx\temp\offword \plainnonfrenchspacing \else \errhelp = \EMsimple \errmessage{Unknown @frenchspacing option `\temp', must be on/off}% \fi\fi } % @w prevents a word break. Without the \leavevmode, @w at the % beginning of a paragraph, when TeX is still in vertical mode, would % produce a whole line of output instead of starting the paragraph. \def\w#1{\leavevmode\hbox{#1}} % @group ... @end group forces ... to be all on one page, by enclosing % it in a TeX vbox. We use \vtop instead of \vbox to construct the box % to keep its height that of a normal line. According to the rules for % \topskip (p.114 of the TeXbook), the glue inserted is % max (\topskip - \ht (first item), 0). If that height is large, % therefore, no glue is inserted, and the space between the headline and % the text is small, which looks bad. % % Another complication is that the group might be very large. This can % cause the glue on the previous page to be unduly stretched, because it % does not have much material. In this case, it's better to add an % explicit \vfill so that the extra space is at the bottom. The % threshold for doing this is if the group is more than \vfilllimit % percent of a page (\vfilllimit can be changed inside of @tex). % \newbox\groupbox \def\vfilllimit{0.7} % \envdef\group{% \ifnum\catcode`\^^M=\active \else \errhelp = \groupinvalidhelp \errmessage{@group invalid in context where filling is enabled}% \fi \startsavinginserts % \setbox\groupbox = \vtop\bgroup % Do @comment since we are called inside an environment such as % @example, where each end-of-line in the input causes an % end-of-line in the output. We don't want the end-of-line after % the `@group' to put extra space in the output. Since @group % should appear on a line by itself (according to the Texinfo % manual), we don't worry about eating any user text. \comment } % % The \vtop produces a box with normal height and large depth; thus, TeX puts % \baselineskip glue before it, and (when the next line of text is done) % \lineskip glue after it. Thus, space below is not quite equal to space % above. But it's pretty close. \def\Egroup{% % To get correct interline space between the last line of the group % and the first line afterwards, we have to propagate \prevdepth. \endgraf % Not \par, as it may have been set to \lisppar. \global\dimen1 = \prevdepth \egroup % End the \vtop. % \dimen0 is the vertical size of the group's box. \dimen0 = \ht\groupbox \advance\dimen0 by \dp\groupbox % \dimen2 is how much space is left on the page (more or less). \dimen2 = \pageheight \advance\dimen2 by -\pagetotal % if the group doesn't fit on the current page, and it's a big big % group, force a page break. \ifdim \dimen0 > \dimen2 \ifdim \pagetotal < \vfilllimit\pageheight \page \fi \fi \box\groupbox \prevdepth = \dimen1 \checkinserts } % % TeX puts in an \escapechar (i.e., `@') at the beginning of the help % message, so this ends up printing `@group can only ...'. % \newhelp\groupinvalidhelp{% group can only be used in environments such as @example,^^J% where each line of input produces a line of output.} % @need space-in-mils % forces a page break if there is not space-in-mils remaining. \newdimen\mil \mil=0.001in % Old definition--didn't work. %\parseargdef\need{\par % %% This method tries to make TeX break the page naturally %% if the depth of the box does not fit. %{\baselineskip=0pt% %\vtop to #1\mil{\vfil}\kern -#1\mil\nobreak %\prevdepth=-1000pt %}} \parseargdef\need{% % Ensure vertical mode, so we don't make a big box in the middle of a % paragraph. \par % % If the @need value is less than one line space, it's useless. \dimen0 = #1\mil \dimen2 = \ht\strutbox \advance\dimen2 by \dp\strutbox \ifdim\dimen0 > \dimen2 % % Do a \strut just to make the height of this box be normal, so the % normal leading is inserted relative to the preceding line. % And a page break here is fine. \vtop to #1\mil{\strut\vfil}% % % TeX does not even consider page breaks if a penalty added to the % main vertical list is 10000 or more. But in order to see if the % empty box we just added fits on the page, we must make it consider % page breaks. On the other hand, we don't want to actually break the % page after the empty box. So we use a penalty of 9999. % % There is an extremely small chance that TeX will actually break the % page at this \penalty, if there are no other feasible breakpoints in % sight. (If the user is using lots of big @group commands, which % almost-but-not-quite fill up a page, TeX will have a hard time doing % good page breaking, for example.) However, I could not construct an % example where a page broke at this \penalty; if it happens in a real % document, then we can reconsider our strategy. \penalty9999 % % Back up by the size of the box, whether we did a page break or not. \kern -#1\mil % % Do not allow a page break right after this kern. \nobreak \fi } % @br forces paragraph break (and is undocumented). \let\br = \par % @page forces the start of a new page. % \def\page{\par\vfill\supereject} % @exdent text.... % outputs text on separate line in roman font, starting at standard page margin % This records the amount of indent in the innermost environment. % That's how much \exdent should take out. \newskip\exdentamount % This defn is used inside fill environments such as @defun. \parseargdef\exdent{\hfil\break\hbox{\kern -\exdentamount{\rm#1}}\hfil\break} % This defn is used inside nofill environments such as @example. \parseargdef\nofillexdent{{\advance \leftskip by -\exdentamount \leftline{\hskip\leftskip{\rm#1}}}} % @inmargin{WHICH}{TEXT} puts TEXT in the WHICH margin next to the current % paragraph. For more general purposes, use the \margin insertion % class. WHICH is `l' or `r'. % \newskip\inmarginspacing \inmarginspacing=1cm \def\strutdepth{\dp\strutbox} % \def\doinmargin#1#2{\strut\vadjust{% \nobreak \kern-\strutdepth \vtop to \strutdepth{% \baselineskip=\strutdepth \vss % if you have multiple lines of stuff to put here, you'll need to % make the vbox yourself of the appropriate size. \ifx#1l% \llap{\ignorespaces #2\hskip\inmarginspacing}% \else \rlap{\hskip\hsize \hskip\inmarginspacing \ignorespaces #2}% \fi \null }% }} \def\inleftmargin{\doinmargin l} \def\inrightmargin{\doinmargin r} % % @inmargin{TEXT [, RIGHT-TEXT]} % (if RIGHT-TEXT is given, use TEXT for left page, RIGHT-TEXT for right; % else use TEXT for both). % \def\inmargin#1{\parseinmargin #1,,\finish} \def\parseinmargin#1,#2,#3\finish{% not perfect, but better than nothing. \setbox0 = \hbox{\ignorespaces #2}% \ifdim\wd0 > 0pt \def\lefttext{#1}% have both texts \def\righttext{#2}% \else \def\lefttext{#1}% have only one text \def\righttext{#1}% \fi % \ifodd\pageno \def\temp{\inrightmargin\righttext}% odd page -> outside is right margin \else \def\temp{\inleftmargin\lefttext}% \fi \temp } % @include FILE -- \input text of FILE. % \def\include{\parseargusing\filenamecatcodes\includezzz} \def\includezzz#1{% \pushthisfilestack \def\thisfile{#1}% {% \makevalueexpandable % we want to expand any @value in FILE. \turnoffactive % and allow special characters in the expansion \edef\temp{\noexpand\input #1 }% % % This trickery is to read FILE outside of a group, in case it makes % definitions, etc. \expandafter }\temp \popthisfilestack } \def\filenamecatcodes{% \catcode`\\=\other \catcode`~=\other \catcode`^=\other \catcode`_=\other \catcode`|=\other \catcode`<=\other \catcode`>=\other \catcode`+=\other \catcode`-=\other } \def\pushthisfilestack{% \expandafter\pushthisfilestackX\popthisfilestack\StackTerm } \def\pushthisfilestackX{% \expandafter\pushthisfilestackY\thisfile\StackTerm } \def\pushthisfilestackY #1\StackTerm #2\StackTerm {% \gdef\popthisfilestack{\gdef\thisfile{#1}\gdef\popthisfilestack{#2}}% } \def\popthisfilestack{\errthisfilestackempty} \def\errthisfilestackempty{\errmessage{Internal error: the stack of filenames is empty.}} \def\thisfile{} % @center line % outputs that line, centered. % \parseargdef\center{% \ifhmode \let\next\centerH \else \let\next\centerV \fi \next{\hfil \ignorespaces#1\unskip \hfil}% } \def\centerH#1{% {% \hfil\break \advance\hsize by -\leftskip \advance\hsize by -\rightskip \line{#1}% \break }% } \def\centerV#1{\line{\kern\leftskip #1\kern\rightskip}} % @sp n outputs n lines of vertical space \parseargdef\sp{\vskip #1\baselineskip} % @comment ...line which is ignored... % @c is the same as @comment % @ignore ... @end ignore is another way to write a comment \def\comment{\begingroup \catcode`\^^M=\other% \catcode`\@=\other \catcode`\{=\other \catcode`\}=\other% \commentxxx} {\catcode`\^^M=\other \gdef\commentxxx#1^^M{\endgroup}} \let\c=\comment % @paragraphindent NCHARS % We'll use ems for NCHARS, close enough. % NCHARS can also be the word `asis' or `none'. % We cannot feasibly implement @paragraphindent asis, though. % \def\asisword{asis} % no translation, these are keywords \def\noneword{none} % \parseargdef\paragraphindent{% \def\temp{#1}% \ifx\temp\asisword \else \ifx\temp\noneword \defaultparindent = 0pt \else \defaultparindent = #1em \fi \fi \parindent = \defaultparindent } % @exampleindent NCHARS % We'll use ems for NCHARS like @paragraphindent. % It seems @exampleindent asis isn't necessary, but % I preserve it to make it similar to @paragraphindent. \parseargdef\exampleindent{% \def\temp{#1}% \ifx\temp\asisword \else \ifx\temp\noneword \lispnarrowing = 0pt \else \lispnarrowing = #1em \fi \fi } % @firstparagraphindent WORD % If WORD is `none', then suppress indentation of the first paragraph % after a section heading. If WORD is `insert', then do indent at such % paragraphs. % % The paragraph indentation is suppressed or not by calling % \suppressfirstparagraphindent, which the sectioning commands do. % We switch the definition of this back and forth according to WORD. % By default, we suppress indentation. % \def\suppressfirstparagraphindent{\dosuppressfirstparagraphindent} \def\insertword{insert} % \parseargdef\firstparagraphindent{% \def\temp{#1}% \ifx\temp\noneword \let\suppressfirstparagraphindent = \dosuppressfirstparagraphindent \else\ifx\temp\insertword \let\suppressfirstparagraphindent = \relax \else \errhelp = \EMsimple \errmessage{Unknown @firstparagraphindent option `\temp'}% \fi\fi } % Here is how we actually suppress indentation. Redefine \everypar to % \kern backwards by \parindent, and then reset itself to empty. % % We also make \indent itself not actually do anything until the next % paragraph. % \gdef\dosuppressfirstparagraphindent{% \gdef\indent{% \restorefirstparagraphindent \indent }% \gdef\noindent{% \restorefirstparagraphindent \noindent }% \global\everypar = {% \kern -\parindent \restorefirstparagraphindent }% } \gdef\restorefirstparagraphindent{% \global \let \indent = \ptexindent \global \let \noindent = \ptexnoindent \global \everypar = {}% } % @asis just yields its argument. Used with @table, for example. % \def\asis#1{#1} % @math outputs its argument in math mode. % % One complication: _ usually means subscripts, but it could also mean % an actual _ character, as in @math{@var{some_variable} + 1}. So make % _ active, and distinguish by seeing if the current family is \slfam, % which is what @var uses. { \catcode`\_ = \active \gdef\mathunderscore{% \catcode`\_=\active \def_{\ifnum\fam=\slfam \_\else\sb\fi}% } } % Another complication: we want \\ (and @\) to output a \ character. % FYI, plain.tex uses \\ as a temporary control sequence (why?), but % this is not advertised and we don't care. Texinfo does not % otherwise define @\. % % The \mathchar is class=0=ordinary, family=7=ttfam, position=5C=\. \def\mathbackslash{\ifnum\fam=\ttfam \mathchar"075C \else\backslash \fi} % \def\math{% \tex \mathunderscore \let\\ = \mathbackslash \mathactive % make the texinfo accent commands work in math mode \let\"=\ddot \let\'=\acute \let\==\bar \let\^=\hat \let\`=\grave \let\u=\breve \let\v=\check \let\~=\tilde \let\dotaccent=\dot $\finishmath } \def\finishmath#1{#1$\endgroup} % Close the group opened by \tex. % Some active characters (such as <) are spaced differently in math. % We have to reset their definitions in case the @math was an argument % to a command which sets the catcodes (such as @item or @section). % { \catcode`^ = \active \catcode`< = \active \catcode`> = \active \catcode`+ = \active \gdef\mathactive{% \let^ = \ptexhat \let< = \ptexless \let> = \ptexgtr \let+ = \ptexplus } } % Some math mode symbols. \def\bullet{$\ptexbullet$} \def\geq{\ifmmode \ge\else $\ge$\fi} \def\leq{\ifmmode \le\else $\le$\fi} \def\minus{\ifmmode -\else $-$\fi} % @dots{} outputs an ellipsis using the current font. % We do .5em per period so that it has the same spacing in the cm % typewriter fonts as three actual period characters; on the other hand, % in other typewriter fonts three periods are wider than 1.5em. So do % whichever is larger. % \def\dots{% \leavevmode \setbox0=\hbox{...}% get width of three periods \ifdim\wd0 > 1.5em \dimen0 = \wd0 \else \dimen0 = 1.5em \fi \hbox to \dimen0{% \hskip 0pt plus.25fil .\hskip 0pt plus1fil .\hskip 0pt plus1fil .\hskip 0pt plus.5fil }% } % @enddots{} is an end-of-sentence ellipsis. % \def\enddots{% \dots \spacefactor=\endofsentencespacefactor } % @comma{} is so commas can be inserted into text without messing up % Texinfo's parsing. % \let\comma = , % @refill is a no-op. \let\refill=\relax % If working on a large document in chapters, it is convenient to % be able to disable indexing, cross-referencing, and contents, for test runs. % This is done with @novalidate (before @setfilename). % \newif\iflinks \linkstrue % by default we want the aux files. \let\novalidate = \linksfalse % @setfilename is done at the beginning of every texinfo file. % So open here the files we need to have open while reading the input. % This makes it possible to make a .fmt file for texinfo. \def\setfilename{% \fixbackslash % Turn off hack to swallow `\input texinfo'. \iflinks \tryauxfile % Open the new aux file. TeX will close it automatically at exit. \immediate\openout\auxfile=\jobname.aux \fi % \openindices needs to do some work in any case. \openindices \let\setfilename=\comment % Ignore extra @setfilename cmds. % % If texinfo.cnf is present on the system, read it. % Useful for site-wide @afourpaper, etc. \openin 1 texinfo.cnf \ifeof 1 \else \input texinfo.cnf \fi \closein 1 % \comment % Ignore the actual filename. } % Called from \setfilename. % \def\openindices{% \newindex{cp}% \newcodeindex{fn}% \newcodeindex{vr}% \newcodeindex{tp}% \newcodeindex{ky}% \newcodeindex{pg}% } % @bye. \outer\def\bye{\pagealignmacro\tracingstats=1\ptexend} \message{pdf,} % adobe `portable' document format \newcount\tempnum \newcount\lnkcount \newtoks\filename \newcount\filenamelength \newcount\pgn \newtoks\toksA \newtoks\toksB \newtoks\toksC \newtoks\toksD \newbox\boxA \newcount\countA \newif\ifpdf \newif\ifpdfmakepagedest % when pdftex is run in dvi mode, \pdfoutput is defined (so \pdfoutput=1 % can be set). So we test for \relax and 0 as well as \undefined, % borrowed from ifpdf.sty. \ifx\pdfoutput\undefined \else \ifx\pdfoutput\relax \else \ifcase\pdfoutput \else \pdftrue \fi \fi \fi % PDF uses PostScript string constants for the names of xref targets, % for display in the outlines, and in other places. Thus, we have to % double any backslashes. Otherwise, a name like "\node" will be % interpreted as a newline (\n), followed by o, d, e. Not good. % http://www.ntg.nl/pipermail/ntg-pdftex/2004-July/000654.html % (and related messages, the final outcome is that it is up to the TeX % user to double the backslashes and otherwise make the string valid, so % that's what we do). % double active backslashes. % {\catcode`\@=0 \catcode`\\=\active @gdef@activebackslashdouble{% @catcode`@\=@active @let\=@doublebackslash} } % To handle parens, we must adopt a different approach, since parens are % not active characters. hyperref.dtx (which has the same problem as % us) handles it with this amazing macro to replace tokens, with minor % changes for Texinfo. It is included here under the GPL by permission % from the author, Heiko Oberdiek. % % #1 is the tokens to replace. % #2 is the replacement. % #3 is the control sequence with the string. % \def\HyPsdSubst#1#2#3{% \def\HyPsdReplace##1#1##2\END{% ##1% \ifx\\##2\\% \else #2% \HyReturnAfterFi{% \HyPsdReplace##2\END }% \fi }% \xdef#3{\expandafter\HyPsdReplace#3#1\END}% } \long\def\HyReturnAfterFi#1\fi{\fi#1} % #1 is a control sequence in which to do the replacements. \def\backslashparens#1{% \xdef#1{#1}% redefine it as its expansion; the definition is simply % \lastnode when called from \setref -> \pdfmkdest. \HyPsdSubst{(}{\realbackslash(}{#1}% \HyPsdSubst{)}{\realbackslash)}{#1}% } \newhelp\nopdfimagehelp{Texinfo supports .png, .jpg, .jpeg, and .pdf images with PDF output, and none of those formats could be found. (.eps cannot be supported due to the design of the PDF format; use regular TeX (DVI output) for that.)} \ifpdf % % Color manipulation macros based on pdfcolor.tex. \def\cmykDarkRed{0.28 1 1 0.35} \def\cmykBlack{0 0 0 1} % \def\pdfsetcolor#1{\pdfliteral{#1 k}} % Set color, and create a mark which defines \thiscolor accordingly, % so that \makeheadline knows which color to restore. \def\setcolor#1{% \xdef\lastcolordefs{\gdef\noexpand\thiscolor{#1}}% \domark \pdfsetcolor{#1}% } % \def\maincolor{\cmykBlack} \pdfsetcolor{\maincolor} \edef\thiscolor{\maincolor} \def\lastcolordefs{} % \def\makefootline{% \baselineskip24pt \line{\pdfsetcolor{\maincolor}\the\footline}% } % \def\makeheadline{% \vbox to 0pt{% \vskip-22.5pt \line{% \vbox to8.5pt{}% % Extract \thiscolor definition from the marks. \getcolormarks % Typeset the headline with \maincolor, then restore the color. \pdfsetcolor{\maincolor}\the\headline\pdfsetcolor{\thiscolor}% }% \vss }% \nointerlineskip } % % \pdfcatalog{/PageMode /UseOutlines} % % #1 is image name, #2 width (might be empty/whitespace), #3 height (ditto). \def\dopdfimage#1#2#3{% \def\imagewidth{#2}\setbox0 = \hbox{\ignorespaces #2}% \def\imageheight{#3}\setbox2 = \hbox{\ignorespaces #3}% % % pdftex (and the PDF format) support .png, .jpg, .pdf (among % others). Let's try in that order. \let\pdfimgext=\empty \begingroup \openin 1 #1.png \ifeof 1 \openin 1 #1.jpg \ifeof 1 \openin 1 #1.jpeg \ifeof 1 \openin 1 #1.JPG \ifeof 1 \openin 1 #1.pdf \ifeof 1 \openin 1 #1.PDF \ifeof 1 \errhelp = \nopdfimagehelp \errmessage{Could not find image file #1 for pdf}% \else \gdef\pdfimgext{PDF}% \fi \else \gdef\pdfimgext{pdf}% \fi \else \gdef\pdfimgext{JPG}% \fi \else \gdef\pdfimgext{jpeg}% \fi \else \gdef\pdfimgext{jpg}% \fi \else \gdef\pdfimgext{png}% \fi \closein 1 \endgroup % % without \immediate, ancient pdftex seg faults when the same image is % included twice. (Version 3.14159-pre-1.0-unofficial-20010704.) \ifnum\pdftexversion < 14 \immediate\pdfimage \else \immediate\pdfximage \fi \ifdim \wd0 >0pt width \imagewidth \fi \ifdim \wd2 >0pt height \imageheight \fi \ifnum\pdftexversion<13 #1.\pdfimgext \else {#1.\pdfimgext}% \fi \ifnum\pdftexversion < 14 \else \pdfrefximage \pdflastximage \fi} % \def\pdfmkdest#1{{% % We have to set dummies so commands such as @code, and characters % such as \, aren't expanded when present in a section title. \indexnofonts \turnoffactive \activebackslashdouble \makevalueexpandable \def\pdfdestname{#1}% \backslashparens\pdfdestname \safewhatsit{\pdfdest name{\pdfdestname} xyz}% }} % % used to mark target names; must be expandable. \def\pdfmkpgn#1{#1} % % by default, use a color that is dark enough to print on paper as % nearly black, but still distinguishable for online viewing. \def\urlcolor{\cmykDarkRed} \def\linkcolor{\cmykDarkRed} \def\endlink{\setcolor{\maincolor}\pdfendlink} % % Adding outlines to PDF; macros for calculating structure of outlines % come from Petr Olsak \def\expnumber#1{\expandafter\ifx\csname#1\endcsname\relax 0% \else \csname#1\endcsname \fi} \def\advancenumber#1{\tempnum=\expnumber{#1}\relax \advance\tempnum by 1 \expandafter\xdef\csname#1\endcsname{\the\tempnum}} % % #1 is the section text, which is what will be displayed in the % outline by the pdf viewer. #2 is the pdf expression for the number % of subentries (or empty, for subsubsections). #3 is the node text, % which might be empty if this toc entry had no corresponding node. % #4 is the page number % \def\dopdfoutline#1#2#3#4{% % Generate a link to the node text if that exists; else, use the % page number. We could generate a destination for the section % text in the case where a section has no node, but it doesn't % seem worth the trouble, since most documents are normally structured. \def\pdfoutlinedest{#3}% \ifx\pdfoutlinedest\empty \def\pdfoutlinedest{#4}% \else % Doubled backslashes in the name. {\activebackslashdouble \xdef\pdfoutlinedest{#3}% \backslashparens\pdfoutlinedest}% \fi % % Also double the backslashes in the display string. {\activebackslashdouble \xdef\pdfoutlinetext{#1}% \backslashparens\pdfoutlinetext}% % \pdfoutline goto name{\pdfmkpgn{\pdfoutlinedest}}#2{\pdfoutlinetext}% } % \def\pdfmakeoutlines{% \begingroup % Thanh's hack / proper braces in bookmarks \edef\mylbrace{\iftrue \string{\else}\fi}\let\{=\mylbrace \edef\myrbrace{\iffalse{\else\string}\fi}\let\}=\myrbrace % % Read toc silently, to get counts of subentries for \pdfoutline. \def\numchapentry##1##2##3##4{% \def\thischapnum{##2}% \def\thissecnum{0}% \def\thissubsecnum{0}% }% \def\numsecentry##1##2##3##4{% \advancenumber{chap\thischapnum}% \def\thissecnum{##2}% \def\thissubsecnum{0}% }% \def\numsubsecentry##1##2##3##4{% \advancenumber{sec\thissecnum}% \def\thissubsecnum{##2}% }% \def\numsubsubsecentry##1##2##3##4{% \advancenumber{subsec\thissubsecnum}% }% \def\thischapnum{0}% \def\thissecnum{0}% \def\thissubsecnum{0}% % % use \def rather than \let here because we redefine \chapentry et % al. a second time, below. \def\appentry{\numchapentry}% \def\appsecentry{\numsecentry}% \def\appsubsecentry{\numsubsecentry}% \def\appsubsubsecentry{\numsubsubsecentry}% \def\unnchapentry{\numchapentry}% \def\unnsecentry{\numsecentry}% \def\unnsubsecentry{\numsubsecentry}% \def\unnsubsubsecentry{\numsubsubsecentry}% \readdatafile{toc}% % % Read toc second time, this time actually producing the outlines. % The `-' means take the \expnumber as the absolute number of % subentries, which we calculated on our first read of the .toc above. % % We use the node names as the destinations. \def\numchapentry##1##2##3##4{% \dopdfoutline{##1}{count-\expnumber{chap##2}}{##3}{##4}}% \def\numsecentry##1##2##3##4{% \dopdfoutline{##1}{count-\expnumber{sec##2}}{##3}{##4}}% \def\numsubsecentry##1##2##3##4{% \dopdfoutline{##1}{count-\expnumber{subsec##2}}{##3}{##4}}% \def\numsubsubsecentry##1##2##3##4{% count is always zero \dopdfoutline{##1}{}{##3}{##4}}% % % PDF outlines are displayed using system fonts, instead of % document fonts. Therefore we cannot use special characters, % since the encoding is unknown. For example, the eogonek from % Latin 2 (0xea) gets translated to a | character. Info from % Staszek Wawrykiewicz, 19 Jan 2004 04:09:24 +0100. % % xx to do this right, we have to translate 8-bit characters to % their "best" equivalent, based on the @documentencoding. Right % now, I guess we'll just let the pdf reader have its way. \indexnofonts \setupdatafile \catcode`\\=\active \otherbackslash \input \tocreadfilename \endgroup } % \def\skipspaces#1{\def\PP{#1}\def\D{|}% \ifx\PP\D\let\nextsp\relax \else\let\nextsp\skipspaces \ifx\p\space\else\addtokens{\filename}{\PP}% \advance\filenamelength by 1 \fi \fi \nextsp} \def\getfilename#1{\filenamelength=0\expandafter\skipspaces#1|\relax} \ifnum\pdftexversion < 14 \let \startlink \pdfannotlink \else \let \startlink \pdfstartlink \fi % make a live url in pdf output. \def\pdfurl#1{% \begingroup % it seems we really need yet another set of dummies; have not % tried to figure out what each command should do in the context % of @url. for now, just make @/ a no-op, that's the only one % people have actually reported a problem with. % \normalturnoffactive \def\@{@}% \let\/=\empty \makevalueexpandable \leavevmode\setcolor{\urlcolor}% \startlink attr{/Border [0 0 0]}% user{/Subtype /Link /A << /S /URI /URI (#1) >>}% \endgroup} \def\pdfgettoks#1.{\setbox\boxA=\hbox{\toksA={#1.}\toksB={}\maketoks}} \def\addtokens#1#2{\edef\addtoks{\noexpand#1={\the#1#2}}\addtoks} \def\adn#1{\addtokens{\toksC}{#1}\global\countA=1\let\next=\maketoks} \def\poptoks#1#2|ENDTOKS|{\let\first=#1\toksD={#1}\toksA={#2}} \def\maketoks{% \expandafter\poptoks\the\toksA|ENDTOKS|\relax \ifx\first0\adn0 \else\ifx\first1\adn1 \else\ifx\first2\adn2 \else\ifx\first3\adn3 \else\ifx\first4\adn4 \else\ifx\first5\adn5 \else\ifx\first6\adn6 \else\ifx\first7\adn7 \else\ifx\first8\adn8 \else\ifx\first9\adn9 \else \ifnum0=\countA\else\makelink\fi \ifx\first.\let\next=\done\else \let\next=\maketoks \addtokens{\toksB}{\the\toksD} \ifx\first,\addtokens{\toksB}{\space}\fi \fi \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi \next} \def\makelink{\addtokens{\toksB}% {\noexpand\pdflink{\the\toksC}}\toksC={}\global\countA=0} \def\pdflink#1{% \startlink attr{/Border [0 0 0]} goto name{\pdfmkpgn{#1}} \setcolor{\linkcolor}#1\endlink} \def\done{\edef\st{\global\noexpand\toksA={\the\toksB}}\st} \else \let\pdfmkdest = \gobble \let\pdfurl = \gobble \let\endlink = \relax \let\setcolor = \gobble \let\pdfsetcolor = \gobble \let\pdfmakeoutlines = \relax \fi % \ifx\pdfoutput \message{fonts,} % Change the current font style to #1, remembering it in \curfontstyle. % For now, we do not accumulate font styles: @b{@i{foo}} prints foo in % italics, not bold italics. % \def\setfontstyle#1{% \def\curfontstyle{#1}% not as a control sequence, because we are \edef'd. \csname ten#1\endcsname % change the current font } % Select #1 fonts with the current style. % \def\selectfonts#1{\csname #1fonts\endcsname \csname\curfontstyle\endcsname} \def\rm{\fam=0 \setfontstyle{rm}} \def\it{\fam=\itfam \setfontstyle{it}} \def\sl{\fam=\slfam \setfontstyle{sl}} \def\bf{\fam=\bffam \setfontstyle{bf}}\def\bfstylename{bf} \def\tt{\fam=\ttfam \setfontstyle{tt}} % Texinfo sort of supports the sans serif font style, which plain TeX does not. % So we set up a \sf. \newfam\sffam \def\sf{\fam=\sffam \setfontstyle{sf}} \let\li = \sf % Sometimes we call it \li, not \sf. % We don't need math for this font style. \def\ttsl{\setfontstyle{ttsl}} % Default leading. \newdimen\textleading \textleading = 13.2pt % Set the baselineskip to #1, and the lineskip and strut size % correspondingly. There is no deep meaning behind these magic numbers % used as factors; they just match (closely enough) what Knuth defined. % \def\lineskipfactor{.08333} \def\strutheightpercent{.70833} \def\strutdepthpercent {.29167} % % can get a sort of poor man's double spacing by redefining this. \def\baselinefactor{1} % \def\setleading#1{% \dimen0 = #1\relax \normalbaselineskip = \baselinefactor\dimen0 \normallineskip = \lineskipfactor\normalbaselineskip \normalbaselines \setbox\strutbox =\hbox{% \vrule width0pt height\strutheightpercent\baselineskip depth \strutdepthpercent \baselineskip }% } % PDF CMaps. See also LaTeX's t1.cmap. % % do nothing with this by default. \expandafter\let\csname cmapOT1\endcsname\gobble \expandafter\let\csname cmapOT1IT\endcsname\gobble \expandafter\let\csname cmapOT1TT\endcsname\gobble % if we are producing pdf, and we have \pdffontattr, then define cmaps. % (\pdffontattr was introduced many years ago, but people still run % older pdftex's; it's easy to conditionalize, so we do.) \ifpdf \ifx\pdffontattr\undefined \else \begingroup \catcode`\^^M=\active \def^^M{^^J}% Output line endings as the ^^J char. \catcode`\%=12 \immediate\pdfobj stream {%!PS-Adobe-3.0 Resource-CMap %%DocumentNeededResources: ProcSet (CIDInit) %%IncludeResource: ProcSet (CIDInit) %%BeginResource: CMap (TeX-OT1-0) %%Title: (TeX-OT1-0 TeX OT1 0) %%Version: 1.000 %%EndComments /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (TeX) /Ordering (OT1) /Supplement 0 >> def /CMapName /TeX-OT1-0 def /CMapType 2 def 1 begincodespacerange <00> <7F> endcodespacerange 8 beginbfrange <00> <01> <0393> <09> <0A> <03A8> <23> <26> <0023> <28> <3B> <0028> <3F> <5B> <003F> <5D> <5E> <005D> <61> <7A> <0061> <7B> <7C> <2013> endbfrange 40 beginbfchar <02> <0398> <03> <039B> <04> <039E> <05> <03A0> <06> <03A3> <07> <03D2> <08> <03A6> <0B> <00660066> <0C> <00660069> <0D> <0066006C> <0E> <006600660069> <0F> <00660066006C> <10> <0131> <11> <0237> <12> <0060> <13> <00B4> <14> <02C7> <15> <02D8> <16> <00AF> <17> <02DA> <18> <00B8> <19> <00DF> <1A> <00E6> <1B> <0153> <1C> <00F8> <1D> <00C6> <1E> <0152> <1F> <00D8> <21> <0021> <22> <201D> <27> <2019> <3C> <00A1> <3D> <003D> <3E> <00BF> <5C> <201C> <5F> <02D9> <60> <2018> <7D> <02DD> <7E> <007E> <7F> <00A8> endbfchar endcmap CMapName currentdict /CMap defineresource pop end end %%EndResource %%EOF }\endgroup \expandafter\edef\csname cmapOT1\endcsname#1{% \pdffontattr#1{/ToUnicode \the\pdflastobj\space 0 R}% }% % % \cmapOT1IT \begingroup \catcode`\^^M=\active \def^^M{^^J}% Output line endings as the ^^J char. \catcode`\%=12 \immediate\pdfobj stream {%!PS-Adobe-3.0 Resource-CMap %%DocumentNeededResources: ProcSet (CIDInit) %%IncludeResource: ProcSet (CIDInit) %%BeginResource: CMap (TeX-OT1IT-0) %%Title: (TeX-OT1IT-0 TeX OT1IT 0) %%Version: 1.000 %%EndComments /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (TeX) /Ordering (OT1IT) /Supplement 0 >> def /CMapName /TeX-OT1IT-0 def /CMapType 2 def 1 begincodespacerange <00> <7F> endcodespacerange 8 beginbfrange <00> <01> <0393> <09> <0A> <03A8> <25> <26> <0025> <28> <3B> <0028> <3F> <5B> <003F> <5D> <5E> <005D> <61> <7A> <0061> <7B> <7C> <2013> endbfrange 42 beginbfchar <02> <0398> <03> <039B> <04> <039E> <05> <03A0> <06> <03A3> <07> <03D2> <08> <03A6> <0B> <00660066> <0C> <00660069> <0D> <0066006C> <0E> <006600660069> <0F> <00660066006C> <10> <0131> <11> <0237> <12> <0060> <13> <00B4> <14> <02C7> <15> <02D8> <16> <00AF> <17> <02DA> <18> <00B8> <19> <00DF> <1A> <00E6> <1B> <0153> <1C> <00F8> <1D> <00C6> <1E> <0152> <1F> <00D8> <21> <0021> <22> <201D> <23> <0023> <24> <00A3> <27> <2019> <3C> <00A1> <3D> <003D> <3E> <00BF> <5C> <201C> <5F> <02D9> <60> <2018> <7D> <02DD> <7E> <007E> <7F> <00A8> endbfchar endcmap CMapName currentdict /CMap defineresource pop end end %%EndResource %%EOF }\endgroup \expandafter\edef\csname cmapOT1IT\endcsname#1{% \pdffontattr#1{/ToUnicode \the\pdflastobj\space 0 R}% }% % % \cmapOT1TT \begingroup \catcode`\^^M=\active \def^^M{^^J}% Output line endings as the ^^J char. \catcode`\%=12 \immediate\pdfobj stream {%!PS-Adobe-3.0 Resource-CMap %%DocumentNeededResources: ProcSet (CIDInit) %%IncludeResource: ProcSet (CIDInit) %%BeginResource: CMap (TeX-OT1TT-0) %%Title: (TeX-OT1TT-0 TeX OT1TT 0) %%Version: 1.000 %%EndComments /CIDInit /ProcSet findresource begin 12 dict begin begincmap /CIDSystemInfo << /Registry (TeX) /Ordering (OT1TT) /Supplement 0 >> def /CMapName /TeX-OT1TT-0 def /CMapType 2 def 1 begincodespacerange <00> <7F> endcodespacerange 5 beginbfrange <00> <01> <0393> <09> <0A> <03A8> <21> <26> <0021> <28> <5F> <0028> <61> <7E> <0061> endbfrange 32 beginbfchar <02> <0398> <03> <039B> <04> <039E> <05> <03A0> <06> <03A3> <07> <03D2> <08> <03A6> <0B> <2191> <0C> <2193> <0D> <0027> <0E> <00A1> <0F> <00BF> <10> <0131> <11> <0237> <12> <0060> <13> <00B4> <14> <02C7> <15> <02D8> <16> <00AF> <17> <02DA> <18> <00B8> <19> <00DF> <1A> <00E6> <1B> <0153> <1C> <00F8> <1D> <00C6> <1E> <0152> <1F> <00D8> <20> <2423> <27> <2019> <60> <2018> <7F> <00A8> endbfchar endcmap CMapName currentdict /CMap defineresource pop end end %%EndResource %%EOF }\endgroup \expandafter\edef\csname cmapOT1TT\endcsname#1{% \pdffontattr#1{/ToUnicode \the\pdflastobj\space 0 R}% }% \fi\fi % Set the font macro #1 to the font named #2, adding on the % specified font prefix (normally `cm'). % #3 is the font's design size, #4 is a scale factor, #5 is the CMap % encoding (currently only OT1, OT1IT and OT1TT are allowed, pass % empty to omit). \def\setfont#1#2#3#4#5{% \font#1=\fontprefix#2#3 scaled #4 \csname cmap#5\endcsname#1% } % This is what gets called when #5 of \setfont is empty. \let\cmap\gobble % emacs-page end of cmaps % Use cm as the default font prefix. % To specify the font prefix, you must define \fontprefix % before you read in texinfo.tex. \ifx\fontprefix\undefined \def\fontprefix{cm} \fi % Support font families that don't use the same naming scheme as CM. \def\rmshape{r} \def\rmbshape{bx} %where the normal face is bold \def\bfshape{b} \def\bxshape{bx} \def\ttshape{tt} \def\ttbshape{tt} \def\ttslshape{sltt} \def\itshape{ti} \def\itbshape{bxti} \def\slshape{sl} \def\slbshape{bxsl} \def\sfshape{ss} \def\sfbshape{ss} \def\scshape{csc} \def\scbshape{csc} % Definitions for a main text size of 11pt. This is the default in % Texinfo. % \def\definetextfontsizexi{% % Text fonts (11.2pt, magstep1). \def\textnominalsize{11pt} \edef\mainmagstep{\magstephalf} \setfont\textrm\rmshape{10}{\mainmagstep}{OT1} \setfont\texttt\ttshape{10}{\mainmagstep}{OT1TT} \setfont\textbf\bfshape{10}{\mainmagstep}{OT1} \setfont\textit\itshape{10}{\mainmagstep}{OT1IT} \setfont\textsl\slshape{10}{\mainmagstep}{OT1} \setfont\textsf\sfshape{10}{\mainmagstep}{OT1} \setfont\textsc\scshape{10}{\mainmagstep}{OT1} \setfont\textttsl\ttslshape{10}{\mainmagstep}{OT1TT} \font\texti=cmmi10 scaled \mainmagstep \font\textsy=cmsy10 scaled \mainmagstep \def\textecsize{1095} % A few fonts for @defun names and args. \setfont\defbf\bfshape{10}{\magstep1}{OT1} \setfont\deftt\ttshape{10}{\magstep1}{OT1TT} \setfont\defttsl\ttslshape{10}{\magstep1}{OT1TT} \def\df{\let\tentt=\deftt \let\tenbf = \defbf \let\tenttsl=\defttsl \bf} % Fonts for indices, footnotes, small examples (9pt). \def\smallnominalsize{9pt} \setfont\smallrm\rmshape{9}{1000}{OT1} \setfont\smalltt\ttshape{9}{1000}{OT1TT} \setfont\smallbf\bfshape{10}{900}{OT1} \setfont\smallit\itshape{9}{1000}{OT1IT} \setfont\smallsl\slshape{9}{1000}{OT1} \setfont\smallsf\sfshape{9}{1000}{OT1} \setfont\smallsc\scshape{10}{900}{OT1} \setfont\smallttsl\ttslshape{10}{900}{OT1TT} \font\smalli=cmmi9 \font\smallsy=cmsy9 \def\smallecsize{0900} % Fonts for small examples (8pt). \def\smallernominalsize{8pt} \setfont\smallerrm\rmshape{8}{1000}{OT1} \setfont\smallertt\ttshape{8}{1000}{OT1TT} \setfont\smallerbf\bfshape{10}{800}{OT1} \setfont\smallerit\itshape{8}{1000}{OT1IT} \setfont\smallersl\slshape{8}{1000}{OT1} \setfont\smallersf\sfshape{8}{1000}{OT1} \setfont\smallersc\scshape{10}{800}{OT1} \setfont\smallerttsl\ttslshape{10}{800}{OT1TT} \font\smalleri=cmmi8 \font\smallersy=cmsy8 \def\smallerecsize{0800} % Fonts for title page (20.4pt): \def\titlenominalsize{20pt} \setfont\titlerm\rmbshape{12}{\magstep3}{OT1} \setfont\titleit\itbshape{10}{\magstep4}{OT1IT} \setfont\titlesl\slbshape{10}{\magstep4}{OT1} \setfont\titlett\ttbshape{12}{\magstep3}{OT1TT} \setfont\titlettsl\ttslshape{10}{\magstep4}{OT1TT} \setfont\titlesf\sfbshape{17}{\magstep1}{OT1} \let\titlebf=\titlerm \setfont\titlesc\scbshape{10}{\magstep4}{OT1} \font\titlei=cmmi12 scaled \magstep3 \font\titlesy=cmsy10 scaled \magstep4 \def\authorrm{\secrm} \def\authortt{\sectt} \def\titleecsize{2074} % Chapter (and unnumbered) fonts (17.28pt). \def\chapnominalsize{17pt} \setfont\chaprm\rmbshape{12}{\magstep2}{OT1} \setfont\chapit\itbshape{10}{\magstep3}{OT1IT} \setfont\chapsl\slbshape{10}{\magstep3}{OT1} \setfont\chaptt\ttbshape{12}{\magstep2}{OT1TT} \setfont\chapttsl\ttslshape{10}{\magstep3}{OT1TT} \setfont\chapsf\sfbshape{17}{1000}{OT1} \let\chapbf=\chaprm \setfont\chapsc\scbshape{10}{\magstep3}{OT1} \font\chapi=cmmi12 scaled \magstep2 \font\chapsy=cmsy10 scaled \magstep3 \def\chapecsize{1728} % Section fonts (14.4pt). \def\secnominalsize{14pt} \setfont\secrm\rmbshape{12}{\magstep1}{OT1} \setfont\secit\itbshape{10}{\magstep2}{OT1IT} \setfont\secsl\slbshape{10}{\magstep2}{OT1} \setfont\sectt\ttbshape{12}{\magstep1}{OT1TT} \setfont\secttsl\ttslshape{10}{\magstep2}{OT1TT} \setfont\secsf\sfbshape{12}{\magstep1}{OT1} \let\secbf\secrm \setfont\secsc\scbshape{10}{\magstep2}{OT1} \font\seci=cmmi12 scaled \magstep1 \font\secsy=cmsy10 scaled \magstep2 \def\sececsize{1440} % Subsection fonts (13.15pt). \def\ssecnominalsize{13pt} \setfont\ssecrm\rmbshape{12}{\magstephalf}{OT1} \setfont\ssecit\itbshape{10}{1315}{OT1IT} \setfont\ssecsl\slbshape{10}{1315}{OT1} \setfont\ssectt\ttbshape{12}{\magstephalf}{OT1TT} \setfont\ssecttsl\ttslshape{10}{1315}{OT1TT} \setfont\ssecsf\sfbshape{12}{\magstephalf}{OT1} \let\ssecbf\ssecrm \setfont\ssecsc\scbshape{10}{1315}{OT1} \font\sseci=cmmi12 scaled \magstephalf \font\ssecsy=cmsy10 scaled 1315 \def\ssececsize{1200} % Reduced fonts for @acro in text (10pt). \def\reducednominalsize{10pt} \setfont\reducedrm\rmshape{10}{1000}{OT1} \setfont\reducedtt\ttshape{10}{1000}{OT1TT} \setfont\reducedbf\bfshape{10}{1000}{OT1} \setfont\reducedit\itshape{10}{1000}{OT1IT} \setfont\reducedsl\slshape{10}{1000}{OT1} \setfont\reducedsf\sfshape{10}{1000}{OT1} \setfont\reducedsc\scshape{10}{1000}{OT1} \setfont\reducedttsl\ttslshape{10}{1000}{OT1TT} \font\reducedi=cmmi10 \font\reducedsy=cmsy10 \def\reducedecsize{1000} % reset the current fonts \textfonts \rm } % end of 11pt text font size definitions % Definitions to make the main text be 10pt Computer Modern, with % section, chapter, etc., sizes following suit. This is for the GNU % Press printing of the Emacs 22 manual. Maybe other manuals in the % future. Used with @smallbook, which sets the leading to 12pt. % \def\definetextfontsizex{% % Text fonts (10pt). \def\textnominalsize{10pt} \edef\mainmagstep{1000} \setfont\textrm\rmshape{10}{\mainmagstep}{OT1} \setfont\texttt\ttshape{10}{\mainmagstep}{OT1TT} \setfont\textbf\bfshape{10}{\mainmagstep}{OT1} \setfont\textit\itshape{10}{\mainmagstep}{OT1IT} \setfont\textsl\slshape{10}{\mainmagstep}{OT1} \setfont\textsf\sfshape{10}{\mainmagstep}{OT1} \setfont\textsc\scshape{10}{\mainmagstep}{OT1} \setfont\textttsl\ttslshape{10}{\mainmagstep}{OT1TT} \font\texti=cmmi10 scaled \mainmagstep \font\textsy=cmsy10 scaled \mainmagstep \def\textecsize{1000} % A few fonts for @defun names and args. \setfont\defbf\bfshape{10}{\magstephalf}{OT1} \setfont\deftt\ttshape{10}{\magstephalf}{OT1TT} \setfont\defttsl\ttslshape{10}{\magstephalf}{OT1TT} \def\df{\let\tentt=\deftt \let\tenbf = \defbf \let\tenttsl=\defttsl \bf} % Fonts for indices, footnotes, small examples (9pt). \def\smallnominalsize{9pt} \setfont\smallrm\rmshape{9}{1000}{OT1} \setfont\smalltt\ttshape{9}{1000}{OT1TT} \setfont\smallbf\bfshape{10}{900}{OT1} \setfont\smallit\itshape{9}{1000}{OT1IT} \setfont\smallsl\slshape{9}{1000}{OT1} \setfont\smallsf\sfshape{9}{1000}{OT1} \setfont\smallsc\scshape{10}{900}{OT1} \setfont\smallttsl\ttslshape{10}{900}{OT1TT} \font\smalli=cmmi9 \font\smallsy=cmsy9 \def\smallecsize{0900} % Fonts for small examples (8pt). \def\smallernominalsize{8pt} \setfont\smallerrm\rmshape{8}{1000}{OT1} \setfont\smallertt\ttshape{8}{1000}{OT1TT} \setfont\smallerbf\bfshape{10}{800}{OT1} \setfont\smallerit\itshape{8}{1000}{OT1IT} \setfont\smallersl\slshape{8}{1000}{OT1} \setfont\smallersf\sfshape{8}{1000}{OT1} \setfont\smallersc\scshape{10}{800}{OT1} \setfont\smallerttsl\ttslshape{10}{800}{OT1TT} \font\smalleri=cmmi8 \font\smallersy=cmsy8 \def\smallerecsize{0800} % Fonts for title page (20.4pt): \def\titlenominalsize{20pt} \setfont\titlerm\rmbshape{12}{\magstep3}{OT1} \setfont\titleit\itbshape{10}{\magstep4}{OT1IT} \setfont\titlesl\slbshape{10}{\magstep4}{OT1} \setfont\titlett\ttbshape{12}{\magstep3}{OT1TT} \setfont\titlettsl\ttslshape{10}{\magstep4}{OT1TT} \setfont\titlesf\sfbshape{17}{\magstep1}{OT1} \let\titlebf=\titlerm \setfont\titlesc\scbshape{10}{\magstep4}{OT1} \font\titlei=cmmi12 scaled \magstep3 \font\titlesy=cmsy10 scaled \magstep4 \def\authorrm{\secrm} \def\authortt{\sectt} \def\titleecsize{2074} % Chapter fonts (14.4pt). \def\chapnominalsize{14pt} \setfont\chaprm\rmbshape{12}{\magstep1}{OT1} \setfont\chapit\itbshape{10}{\magstep2}{OT1IT} \setfont\chapsl\slbshape{10}{\magstep2}{OT1} \setfont\chaptt\ttbshape{12}{\magstep1}{OT1TT} \setfont\chapttsl\ttslshape{10}{\magstep2}{OT1TT} \setfont\chapsf\sfbshape{12}{\magstep1}{OT1} \let\chapbf\chaprm \setfont\chapsc\scbshape{10}{\magstep2}{OT1} \font\chapi=cmmi12 scaled \magstep1 \font\chapsy=cmsy10 scaled \magstep2 \def\chapecsize{1440} % Section fonts (12pt). \def\secnominalsize{12pt} \setfont\secrm\rmbshape{12}{1000}{OT1} \setfont\secit\itbshape{10}{\magstep1}{OT1IT} \setfont\secsl\slbshape{10}{\magstep1}{OT1} \setfont\sectt\ttbshape{12}{1000}{OT1TT} \setfont\secttsl\ttslshape{10}{\magstep1}{OT1TT} \setfont\secsf\sfbshape{12}{1000}{OT1} \let\secbf\secrm \setfont\secsc\scbshape{10}{\magstep1}{OT1} \font\seci=cmmi12 \font\secsy=cmsy10 scaled \magstep1 \def\sececsize{1200} % Subsection fonts (10pt). \def\ssecnominalsize{10pt} \setfont\ssecrm\rmbshape{10}{1000}{OT1} \setfont\ssecit\itbshape{10}{1000}{OT1IT} \setfont\ssecsl\slbshape{10}{1000}{OT1} \setfont\ssectt\ttbshape{10}{1000}{OT1TT} \setfont\ssecttsl\ttslshape{10}{1000}{OT1TT} \setfont\ssecsf\sfbshape{10}{1000}{OT1} \let\ssecbf\ssecrm \setfont\ssecsc\scbshape{10}{1000}{OT1} \font\sseci=cmmi10 \font\ssecsy=cmsy10 \def\ssececsize{1000} % Reduced fonts for @acro in text (9pt). \def\reducednominalsize{9pt} \setfont\reducedrm\rmshape{9}{1000}{OT1} \setfont\reducedtt\ttshape{9}{1000}{OT1TT} \setfont\reducedbf\bfshape{10}{900}{OT1} \setfont\reducedit\itshape{9}{1000}{OT1IT} \setfont\reducedsl\slshape{9}{1000}{OT1} \setfont\reducedsf\sfshape{9}{1000}{OT1} \setfont\reducedsc\scshape{10}{900}{OT1} \setfont\reducedttsl\ttslshape{10}{900}{OT1TT} \font\reducedi=cmmi9 \font\reducedsy=cmsy9 \def\reducedecsize{0900} % reduce space between paragraphs \divide\parskip by 2 % reset the current fonts \textfonts \rm } % end of 10pt text font size definitions % We provide the user-level command % @fonttextsize 10 % (or 11) to redefine the text font size. pt is assumed. % \def\xword{10} \def\xiword{11} % \parseargdef\fonttextsize{% \def\textsizearg{#1}% \wlog{doing @fonttextsize \textsizearg}% % % Set \globaldefs so that documents can use this inside @tex, since % makeinfo 4.8 does not support it, but we need it nonetheless. % \begingroup \globaldefs=1 \ifx\textsizearg\xword \definetextfontsizex \else \ifx\textsizearg\xiword \definetextfontsizexi \else \errhelp=\EMsimple \errmessage{@fonttextsize only supports `10' or `11', not `\textsizearg'} \fi\fi \endgroup } % In order for the font changes to affect most math symbols and letters, % we have to define the \textfont of the standard families. Since % texinfo doesn't allow for producing subscripts and superscripts except % in the main text, we don't bother to reset \scriptfont and % \scriptscriptfont (which would also require loading a lot more fonts). % \def\resetmathfonts{% \textfont0=\tenrm \textfont1=\teni \textfont2=\tensy \textfont\itfam=\tenit \textfont\slfam=\tensl \textfont\bffam=\tenbf \textfont\ttfam=\tentt \textfont\sffam=\tensf } % The font-changing commands redefine the meanings of \tenSTYLE, instead % of just \STYLE. We do this because \STYLE needs to also set the % current \fam for math mode. Our \STYLE (e.g., \rm) commands hardwire % \tenSTYLE to set the current font. % % Each font-changing command also sets the names \lsize (one size lower) % and \lllsize (three sizes lower). These relative commands are used in % the LaTeX logo and acronyms. % % This all needs generalizing, badly. % \def\textfonts{% \let\tenrm=\textrm \let\tenit=\textit \let\tensl=\textsl \let\tenbf=\textbf \let\tentt=\texttt \let\smallcaps=\textsc \let\tensf=\textsf \let\teni=\texti \let\tensy=\textsy \let\tenttsl=\textttsl \def\curfontsize{text}% \def\lsize{reduced}\def\lllsize{smaller}% \resetmathfonts \setleading{\textleading}} \def\titlefonts{% \let\tenrm=\titlerm \let\tenit=\titleit \let\tensl=\titlesl \let\tenbf=\titlebf \let\tentt=\titlett \let\smallcaps=\titlesc \let\tensf=\titlesf \let\teni=\titlei \let\tensy=\titlesy \let\tenttsl=\titlettsl \def\curfontsize{title}% \def\lsize{chap}\def\lllsize{subsec}% \resetmathfonts \setleading{25pt}} \def\titlefont#1{{\titlefonts\rm #1}} \def\chapfonts{% \let\tenrm=\chaprm \let\tenit=\chapit \let\tensl=\chapsl \let\tenbf=\chapbf \let\tentt=\chaptt \let\smallcaps=\chapsc \let\tensf=\chapsf \let\teni=\chapi \let\tensy=\chapsy \let\tenttsl=\chapttsl \def\curfontsize{chap}% \def\lsize{sec}\def\lllsize{text}% \resetmathfonts \setleading{19pt}} \def\secfonts{% \let\tenrm=\secrm \let\tenit=\secit \let\tensl=\secsl \let\tenbf=\secbf \let\tentt=\sectt \let\smallcaps=\secsc \let\tensf=\secsf \let\teni=\seci \let\tensy=\secsy \let\tenttsl=\secttsl \def\curfontsize{sec}% \def\lsize{subsec}\def\lllsize{reduced}% \resetmathfonts \setleading{16pt}} \def\subsecfonts{% \let\tenrm=\ssecrm \let\tenit=\ssecit \let\tensl=\ssecsl \let\tenbf=\ssecbf \let\tentt=\ssectt \let\smallcaps=\ssecsc \let\tensf=\ssecsf \let\teni=\sseci \let\tensy=\ssecsy \let\tenttsl=\ssecttsl \def\curfontsize{ssec}% \def\lsize{text}\def\lllsize{small}% \resetmathfonts \setleading{15pt}} \let\subsubsecfonts = \subsecfonts \def\reducedfonts{% \let\tenrm=\reducedrm \let\tenit=\reducedit \let\tensl=\reducedsl \let\tenbf=\reducedbf \let\tentt=\reducedtt \let\reducedcaps=\reducedsc \let\tensf=\reducedsf \let\teni=\reducedi \let\tensy=\reducedsy \let\tenttsl=\reducedttsl \def\curfontsize{reduced}% \def\lsize{small}\def\lllsize{smaller}% \resetmathfonts \setleading{10.5pt}} \def\smallfonts{% \let\tenrm=\smallrm \let\tenit=\smallit \let\tensl=\smallsl \let\tenbf=\smallbf \let\tentt=\smalltt \let\smallcaps=\smallsc \let\tensf=\smallsf \let\teni=\smalli \let\tensy=\smallsy \let\tenttsl=\smallttsl \def\curfontsize{small}% \def\lsize{smaller}\def\lllsize{smaller}% \resetmathfonts \setleading{10.5pt}} \def\smallerfonts{% \let\tenrm=\smallerrm \let\tenit=\smallerit \let\tensl=\smallersl \let\tenbf=\smallerbf \let\tentt=\smallertt \let\smallcaps=\smallersc \let\tensf=\smallersf \let\teni=\smalleri \let\tensy=\smallersy \let\tenttsl=\smallerttsl \def\curfontsize{smaller}% \def\lsize{smaller}\def\lllsize{smaller}% \resetmathfonts \setleading{9.5pt}} % Set the fonts to use with the @small... environments. \let\smallexamplefonts = \smallfonts % About \smallexamplefonts. If we use \smallfonts (9pt), @smallexample % can fit this many characters: % 8.5x11=86 smallbook=72 a4=90 a5=69 % If we use \scriptfonts (8pt), then we can fit this many characters: % 8.5x11=90+ smallbook=80 a4=90+ a5=77 % For me, subjectively, the few extra characters that fit aren't worth % the additional smallness of 8pt. So I'm making the default 9pt. % % By the way, for comparison, here's what fits with @example (10pt): % 8.5x11=71 smallbook=60 a4=75 a5=58 % % I wish the USA used A4 paper. % --karl, 24jan03. % Set up the default fonts, so we can use them for creating boxes. % \definetextfontsizexi % Define these so they can be easily changed for other fonts. \def\angleleft{$\langle$} \def\angleright{$\rangle$} % Count depth in font-changes, for error checks \newcount\fontdepth \fontdepth=0 % Fonts for short table of contents. \setfont\shortcontrm\rmshape{12}{1000}{OT1} \setfont\shortcontbf\bfshape{10}{\magstep1}{OT1} % no cmb12 \setfont\shortcontsl\slshape{12}{1000}{OT1} \setfont\shortconttt\ttshape{12}{1000}{OT1TT} %% Add scribe-like font environments, plus @l for inline lisp (usually sans %% serif) and @ii for TeX italic % \smartitalic{ARG} outputs arg in italics, followed by an italic correction % unless the following character is such as not to need one. \def\smartitalicx{\ifx\next,\else\ifx\next-\else\ifx\next.\else \ptexslash\fi\fi\fi} \def\smartslanted#1{{\ifusingtt\ttsl\sl #1}\futurelet\next\smartitalicx} \def\smartitalic#1{{\ifusingtt\ttsl\it #1}\futurelet\next\smartitalicx} % like \smartslanted except unconditionally uses \ttsl. % @var is set to this for defun arguments. \def\ttslanted#1{{\ttsl #1}\futurelet\next\smartitalicx} % like \smartslanted except unconditionally use \sl. We never want % ttsl for book titles, do we? \def\cite#1{{\sl #1}\futurelet\next\smartitalicx} \let\i=\smartitalic \let\slanted=\smartslanted \let\var=\smartslanted \let\dfn=\smartslanted \let\emph=\smartitalic % @b, explicit bold. \def\b#1{{\bf #1}} \let\strong=\b % @sansserif, explicit sans. \def\sansserif#1{{\sf #1}} % We can't just use \exhyphenpenalty, because that only has effect at % the end of a paragraph. Restore normal hyphenation at the end of the % group within which \nohyphenation is presumably called. % \def\nohyphenation{\hyphenchar\font = -1 \aftergroup\restorehyphenation} \def\restorehyphenation{\hyphenchar\font = `- } % Set sfcode to normal for the chars that usually have another value. % Can't use plain's \frenchspacing because it uses the `\x notation, and % sometimes \x has an active definition that messes things up. % \catcode`@=11 \def\plainfrenchspacing{% \sfcode\dotChar =\@m \sfcode\questChar=\@m \sfcode\exclamChar=\@m \sfcode\colonChar=\@m \sfcode\semiChar =\@m \sfcode\commaChar =\@m \def\endofsentencespacefactor{1000}% for @. and friends } \def\plainnonfrenchspacing{% \sfcode`\.3000\sfcode`\?3000\sfcode`\!3000 \sfcode`\:2000\sfcode`\;1500\sfcode`\,1250 \def\endofsentencespacefactor{3000}% for @. and friends } \catcode`@=\other \def\endofsentencespacefactor{3000}% default \def\t#1{% {\tt \rawbackslash \plainfrenchspacing #1}% \null } \def\samp#1{`\tclose{#1}'\null} \setfont\keyrm\rmshape{8}{1000}{OT1} \font\keysy=cmsy9 \def\key#1{{\keyrm\textfont2=\keysy \leavevmode\hbox{% \raise0.4pt\hbox{\angleleft}\kern-.08em\vtop{% \vbox{\hrule\kern-0.4pt \hbox{\raise0.4pt\hbox{\vphantom{\angleleft}}#1}}% \kern-0.4pt\hrule}% \kern-.06em\raise0.4pt\hbox{\angleright}}}} \def\key #1{{\nohyphenation \uppercase{#1}}\null} % The old definition, with no lozenge: %\def\key #1{{\ttsl \nohyphenation \uppercase{#1}}\null} \def\ctrl #1{{\tt \rawbackslash \hat}#1} % @file, @option are the same as @samp. \let\file=\samp \let\option=\samp % @code is a modification of @t, % which makes spaces the same size as normal in the surrounding text. \def\tclose#1{% {% % Change normal interword space to be same as for the current font. \spaceskip = \fontdimen2\font % % Switch to typewriter. \tt % % But `\ ' produces the large typewriter interword space. \def\ {{\spaceskip = 0pt{} }}% % % Turn off hyphenation. \nohyphenation % \rawbackslash \plainfrenchspacing #1% }% \null } % We *must* turn on hyphenation at `-' and `_' in @code. % Otherwise, it is too hard to avoid overfull hboxes % in the Emacs manual, the Library manual, etc. % Unfortunately, TeX uses one parameter (\hyphenchar) to control % both hyphenation at - and hyphenation within words. % We must therefore turn them both off (\tclose does that) % and arrange explicitly to hyphenate at a dash. % -- rms. { \catcode`\-=\active \catcode`\_=\active \catcode`\'=\active \catcode`\`=\active % \global\def\code{\begingroup \catcode\rquoteChar=\active \catcode\lquoteChar=\active \let'\codequoteright \let`\codequoteleft % \catcode\dashChar=\active \catcode\underChar=\active \ifallowcodebreaks \let-\codedash \let_\codeunder \else \let-\realdash \let_\realunder \fi \codex } } \def\realdash{-} \def\codedash{-\discretionary{}{}{}} \def\codeunder{% % this is all so @math{@code{var_name}+1} can work. In math mode, _ % is "active" (mathcode"8000) and \normalunderscore (or \char95, etc.) % will therefore expand the active definition of _, which is us % (inside @code that is), therefore an endless loop. \ifusingtt{\ifmmode \mathchar"075F % class 0=ordinary, family 7=ttfam, pos 0x5F=_. \else\normalunderscore \fi \discretionary{}{}{}}% {\_}% } \def\codex #1{\tclose{#1}\endgroup} % An additional complication: the above will allow breaks after, e.g., % each of the four underscores in __typeof__. This is undesirable in % some manuals, especially if they don't have long identifiers in % general. @allowcodebreaks provides a way to control this. % \newif\ifallowcodebreaks \allowcodebreakstrue \def\keywordtrue{true} \def\keywordfalse{false} \parseargdef\allowcodebreaks{% \def\txiarg{#1}% \ifx\txiarg\keywordtrue \allowcodebreakstrue \else\ifx\txiarg\keywordfalse \allowcodebreaksfalse \else \errhelp = \EMsimple \errmessage{Unknown @allowcodebreaks option `\txiarg'}% \fi\fi } % @kbd is like @code, except that if the argument is just one @key command, % then @kbd has no effect. % @kbdinputstyle -- arg is `distinct' (@kbd uses slanted tty font always), % `example' (@kbd uses ttsl only inside of @example and friends), % or `code' (@kbd uses normal tty font always). \parseargdef\kbdinputstyle{% \def\txiarg{#1}% \ifx\txiarg\worddistinct \gdef\kbdexamplefont{\ttsl}\gdef\kbdfont{\ttsl}% \else\ifx\txiarg\wordexample \gdef\kbdexamplefont{\ttsl}\gdef\kbdfont{\tt}% \else\ifx\txiarg\wordcode \gdef\kbdexamplefont{\tt}\gdef\kbdfont{\tt}% \else \errhelp = \EMsimple \errmessage{Unknown @kbdinputstyle option `\txiarg'}% \fi\fi\fi } \def\worddistinct{distinct} \def\wordexample{example} \def\wordcode{code} % Default is `distinct.' \kbdinputstyle distinct \def\xkey{\key} \def\kbdfoo#1#2#3\par{\def\one{#1}\def\three{#3}\def\threex{??}% \ifx\one\xkey\ifx\threex\three \key{#2}% \else{\tclose{\kbdfont\look}}\fi \else{\tclose{\kbdfont\look}}\fi} % For @indicateurl, @env, @command quotes seem unnecessary, so use \code. \let\indicateurl=\code \let\env=\code \let\command=\code % @clicksequence{File @click{} Open ...} \def\clicksequence#1{\begingroup #1\endgroup} % @clickstyle @arrow (by default) \parseargdef\clickstyle{\def\click{#1}} \def\click{\arrow} % @uref (abbreviation for `urlref') takes an optional (comma-separated) % second argument specifying the text to display and an optional third % arg as text to display instead of (rather than in addition to) the url % itself. First (mandatory) arg is the url. Perhaps eventually put in % a hypertex \special here. % \def\uref#1{\douref #1,,,\finish} \def\douref#1,#2,#3,#4\finish{\begingroup \unsepspaces \pdfurl{#1}% \setbox0 = \hbox{\ignorespaces #3}% \ifdim\wd0 > 0pt \unhbox0 % third arg given, show only that \else \setbox0 = \hbox{\ignorespaces #2}% \ifdim\wd0 > 0pt \ifpdf \unhbox0 % PDF: 2nd arg given, show only it \else \unhbox0\ (\code{#1})% DVI: 2nd arg given, show both it and url \fi \else \code{#1}% only url given, so show it \fi \fi \endlink \endgroup} % @url synonym for @uref, since that's how everyone uses it. % \let\url=\uref % rms does not like angle brackets --karl, 17may97. % So now @email is just like @uref, unless we are pdf. % %\def\email#1{\angleleft{\tt #1}\angleright} \ifpdf \def\email#1{\doemail#1,,\finish} \def\doemail#1,#2,#3\finish{\begingroup \unsepspaces \pdfurl{mailto:#1}% \setbox0 = \hbox{\ignorespaces #2}% \ifdim\wd0>0pt\unhbox0\else\code{#1}\fi \endlink \endgroup} \else \let\email=\uref \fi % Check if we are currently using a typewriter font. Since all the % Computer Modern typewriter fonts have zero interword stretch (and % shrink), and it is reasonable to expect all typewriter fonts to have % this property, we can check that font parameter. % \def\ifmonospace{\ifdim\fontdimen3\font=0pt } % Typeset a dimension, e.g., `in' or `pt'. The only reason for the % argument is to make the input look right: @dmn{pt} instead of @dmn{}pt. % \def\dmn#1{\thinspace #1} \def\kbd#1{\def\look{#1}\expandafter\kbdfoo\look??\par} % @l was never documented to mean ``switch to the Lisp font'', % and it is not used as such in any manual I can find. We need it for % Polish suppressed-l. --karl, 22sep96. %\def\l#1{{\li #1}\null} % Explicit font changes: @r, @sc, undocumented @ii. \def\r#1{{\rm #1}} % roman font \def\sc#1{{\smallcaps#1}} % smallcaps font \def\ii#1{{\it #1}} % italic font % @acronym for "FBI", "NATO", and the like. % We print this one point size smaller, since it's intended for % all-uppercase. % \def\acronym#1{\doacronym #1,,\finish} \def\doacronym#1,#2,#3\finish{% {\selectfonts\lsize #1}% \def\temp{#2}% \ifx\temp\empty \else \space ({\unsepspaces \ignorespaces \temp \unskip})% \fi } % @abbr for "Comput. J." and the like. % No font change, but don't do end-of-sentence spacing. % \def\abbr#1{\doabbr #1,,\finish} \def\doabbr#1,#2,#3\finish{% {\plainfrenchspacing #1}% \def\temp{#2}% \ifx\temp\empty \else \space ({\unsepspaces \ignorespaces \temp \unskip})% \fi } % @pounds{} is a sterling sign, which Knuth put in the CM italic font. % \def\pounds{{\it\$}} % @euro{} comes from a separate font, depending on the current style. % We use the free feym* fonts from the eurosym package by Henrik % Theiling, which support regular, slanted, bold and bold slanted (and % "outlined" (blackboard board, sort of) versions, which we don't need). % It is available from http://www.ctan.org/tex-archive/fonts/eurosym. % % Although only regular is the truly official Euro symbol, we ignore % that. The Euro is designed to be slightly taller than the regular % font height. % % feymr - regular % feymo - slanted % feybr - bold % feybo - bold slanted % % There is no good (free) typewriter version, to my knowledge. % A feymr10 euro is ~7.3pt wide, while a normal cmtt10 char is ~5.25pt wide. % Hmm. % % Also doesn't work in math. Do we need to do math with euro symbols? % Hope not. % % \def\euro{{\eurofont e}} \def\eurofont{% % We set the font at each command, rather than predefining it in % \textfonts and the other font-switching commands, so that % installations which never need the symbol don't have to have the % font installed. % % There is only one designed size (nominal 10pt), so we always scale % that to the current nominal size. % % By the way, simply using "at 1em" works for cmr10 and the like, but % does not work for cmbx10 and other extended/shrunken fonts. % \def\eurosize{\csname\curfontsize nominalsize\endcsname}% % \ifx\curfontstyle\bfstylename % bold: \font\thiseurofont = \ifusingit{feybo10}{feybr10} at \eurosize \else % regular: \font\thiseurofont = \ifusingit{feymo10}{feymr10} at \eurosize \fi \thiseurofont } % Hacks for glyphs from the EC fonts similar to \euro. We don't % use \let for the aliases, because sometimes we redefine the original % macro, and the alias should reflect the redefinition. \def\guillemetleft{{\ecfont \char"13}} \def\guillemotleft{\guillemetleft} \def\guillemetright{{\ecfont \char"14}} \def\guillemotright{\guillemetright} \def\guilsinglleft{{\ecfont \char"0E}} \def\guilsinglright{{\ecfont \char"0F}} \def\quotedblbase{{\ecfont \char"12}} \def\quotesinglbase{{\ecfont \char"0D}} % \def\ecfont{% % We can't distinguish serif/sanserif and italic/slanted, but this % is used for crude hacks anyway (like adding French and German % quotes to documents typeset with CM, where we lose kerning), so % hopefully nobody will notice/care. \edef\ecsize{\csname\curfontsize ecsize\endcsname}% \edef\nominalsize{\csname\curfontsize nominalsize\endcsname}% \ifx\curfontstyle\bfstylename % bold: \font\thisecfont = ecb\ifusingit{i}{x}\ecsize \space at \nominalsize \else % regular: \font\thisecfont = ec\ifusingit{ti}{rm}\ecsize \space at \nominalsize \fi \thisecfont } % @registeredsymbol - R in a circle. The font for the R should really % be smaller yet, but lllsize is the best we can do for now. % Adapted from the plain.tex definition of \copyright. % \def\registeredsymbol{% $^{{\ooalign{\hfil\raise.07ex\hbox{\selectfonts\lllsize R}% \hfil\crcr\Orb}}% }$% } % @textdegree - the normal degrees sign. % \def\textdegree{$^\circ$} % Laurent Siebenmann reports \Orb undefined with: % Textures 1.7.7 (preloaded format=plain 93.10.14) (68K) 16 APR 2004 02:38 % so we'll define it if necessary. % \ifx\Orb\undefined \def\Orb{\mathhexbox20D} \fi % Quotes. \chardef\quotedblleft="5C \chardef\quotedblright=`\" \chardef\quoteleft=`\` \chardef\quoteright=`\' \message{page headings,} \newskip\titlepagetopglue \titlepagetopglue = 1.5in \newskip\titlepagebottomglue \titlepagebottomglue = 2pc % First the title page. Must do @settitle before @titlepage. \newif\ifseenauthor \newif\iffinishedtitlepage % Do an implicit @contents or @shortcontents after @end titlepage if the % user says @setcontentsaftertitlepage or @setshortcontentsaftertitlepage. % \newif\ifsetcontentsaftertitlepage \let\setcontentsaftertitlepage = \setcontentsaftertitlepagetrue \newif\ifsetshortcontentsaftertitlepage \let\setshortcontentsaftertitlepage = \setshortcontentsaftertitlepagetrue \parseargdef\shorttitlepage{\begingroup\hbox{}\vskip 1.5in \chaprm \centerline{#1}% \endgroup\page\hbox{}\page} \envdef\titlepage{% % Open one extra group, as we want to close it in the middle of \Etitlepage. \begingroup \parindent=0pt \textfonts % Leave some space at the very top of the page. \vglue\titlepagetopglue % No rule at page bottom unless we print one at the top with @title. \finishedtitlepagetrue % % Most title ``pages'' are actually two pages long, with space % at the top of the second. We don't want the ragged left on the second. \let\oldpage = \page \def\page{% \iffinishedtitlepage\else \finishtitlepage \fi \let\page = \oldpage \page \null }% } \def\Etitlepage{% \iffinishedtitlepage\else \finishtitlepage \fi % It is important to do the page break before ending the group, % because the headline and footline are only empty inside the group. % If we use the new definition of \page, we always get a blank page % after the title page, which we certainly don't want. \oldpage \endgroup % % Need this before the \...aftertitlepage checks so that if they are % in effect the toc pages will come out with page numbers. \HEADINGSon % % If they want short, they certainly want long too. \ifsetshortcontentsaftertitlepage \shortcontents \contents \global\let\shortcontents = \relax \global\let\contents = \relax \fi % \ifsetcontentsaftertitlepage \contents \global\let\contents = \relax \global\let\shortcontents = \relax \fi } \def\finishtitlepage{% \vskip4pt \hrule height 2pt width \hsize \vskip\titlepagebottomglue \finishedtitlepagetrue } %%% Macros to be used within @titlepage: \let\subtitlerm=\tenrm \def\subtitlefont{\subtitlerm \normalbaselineskip = 13pt \normalbaselines} \def\authorfont{\authorrm \normalbaselineskip = 16pt \normalbaselines \let\tt=\authortt} \parseargdef\title{% \checkenv\titlepage \leftline{\titlefonts\rm #1} % print a rule at the page bottom also. \finishedtitlepagefalse \vskip4pt \hrule height 4pt width \hsize \vskip4pt } \parseargdef\subtitle{% \checkenv\titlepage {\subtitlefont \rightline{#1}}% } % @author should come last, but may come many times. % It can also be used inside @quotation. % \parseargdef\author{% \def\temp{\quotation}% \ifx\thisenv\temp \def\quotationauthor{#1}% printed in \Equotation. \else \checkenv\titlepage \ifseenauthor\else \vskip 0pt plus 1filll \seenauthortrue \fi {\authorfont \leftline{#1}}% \fi } %%% Set up page headings and footings. \let\thispage=\folio \newtoks\evenheadline % headline on even pages \newtoks\oddheadline % headline on odd pages \newtoks\evenfootline % footline on even pages \newtoks\oddfootline % footline on odd pages % Now make TeX use those variables \headline={{\textfonts\rm \ifodd\pageno \the\oddheadline \else \the\evenheadline \fi}} \footline={{\textfonts\rm \ifodd\pageno \the\oddfootline \else \the\evenfootline \fi}\HEADINGShook} \let\HEADINGShook=\relax % Commands to set those variables. % For example, this is what @headings on does % @evenheading @thistitle|@thispage|@thischapter % @oddheading @thischapter|@thispage|@thistitle % @evenfooting @thisfile|| % @oddfooting ||@thisfile \def\evenheading{\parsearg\evenheadingxxx} \def\evenheadingxxx #1{\evenheadingyyy #1\|\|\|\|\finish} \def\evenheadingyyy #1\|#2\|#3\|#4\finish{% \global\evenheadline={\rlap{\centerline{#2}}\line{#1\hfil#3}}} \def\oddheading{\parsearg\oddheadingxxx} \def\oddheadingxxx #1{\oddheadingyyy #1\|\|\|\|\finish} \def\oddheadingyyy #1\|#2\|#3\|#4\finish{% \global\oddheadline={\rlap{\centerline{#2}}\line{#1\hfil#3}}} \parseargdef\everyheading{\oddheadingxxx{#1}\evenheadingxxx{#1}}% \def\evenfooting{\parsearg\evenfootingxxx} \def\evenfootingxxx #1{\evenfootingyyy #1\|\|\|\|\finish} \def\evenfootingyyy #1\|#2\|#3\|#4\finish{% \global\evenfootline={\rlap{\centerline{#2}}\line{#1\hfil#3}}} \def\oddfooting{\parsearg\oddfootingxxx} \def\oddfootingxxx #1{\oddfootingyyy #1\|\|\|\|\finish} \def\oddfootingyyy #1\|#2\|#3\|#4\finish{% \global\oddfootline = {\rlap{\centerline{#2}}\line{#1\hfil#3}}% % % Leave some space for the footline. Hopefully ok to assume % @evenfooting will not be used by itself. \global\advance\pageheight by -12pt \global\advance\vsize by -12pt } \parseargdef\everyfooting{\oddfootingxxx{#1}\evenfootingxxx{#1}} % @evenheadingmarks top \thischapter <- chapter at the top of a page % @evenheadingmarks bottom \thischapter <- chapter at the bottom of a page % % The same set of arguments for: % % @oddheadingmarks % @evenfootingmarks % @oddfootingmarks % @everyheadingmarks % @everyfootingmarks \def\evenheadingmarks{\headingmarks{even}{heading}} \def\oddheadingmarks{\headingmarks{odd}{heading}} \def\evenfootingmarks{\headingmarks{even}{footing}} \def\oddfootingmarks{\headingmarks{odd}{footing}} \def\everyheadingmarks#1 {\headingmarks{even}{heading}{#1} \headingmarks{odd}{heading}{#1} } \def\everyfootingmarks#1 {\headingmarks{even}{footing}{#1} \headingmarks{odd}{footing}{#1} } % #1 = even/odd, #2 = heading/footing, #3 = top/bottom. \def\headingmarks#1#2#3 {% \expandafter\let\expandafter\temp \csname get#3headingmarks\endcsname \global\expandafter\let\csname get#1#2marks\endcsname \temp } \everyheadingmarks bottom \everyfootingmarks bottom % @headings double turns headings on for double-sided printing. % @headings single turns headings on for single-sided printing. % @headings off turns them off. % @headings on same as @headings double, retained for compatibility. % @headings after turns on double-sided headings after this page. % @headings doubleafter turns on double-sided headings after this page. % @headings singleafter turns on single-sided headings after this page. % By default, they are off at the start of a document, % and turned `on' after @end titlepage. \def\headings #1 {\csname HEADINGS#1\endcsname} \def\HEADINGSoff{% \global\evenheadline={\hfil} \global\evenfootline={\hfil} \global\oddheadline={\hfil} \global\oddfootline={\hfil}} \HEADINGSoff % When we turn headings on, set the page number to 1. % For double-sided printing, put current file name in lower left corner, % chapter name on inside top of right hand pages, document % title on inside top of left hand pages, and page numbers on outside top % edge of all pages. \def\HEADINGSdouble{% \global\pageno=1 \global\evenfootline={\hfil} \global\oddfootline={\hfil} \global\evenheadline={\line{\folio\hfil\thistitle}} \global\oddheadline={\line{\thischapter\hfil\folio}} \global\let\contentsalignmacro = \chapoddpage } \let\contentsalignmacro = \chappager % For single-sided printing, chapter title goes across top left of page, % page number on top right. \def\HEADINGSsingle{% \global\pageno=1 \global\evenfootline={\hfil} \global\oddfootline={\hfil} \global\evenheadline={\line{\thischapter\hfil\folio}} \global\oddheadline={\line{\thischapter\hfil\folio}} \global\let\contentsalignmacro = \chappager } \def\HEADINGSon{\HEADINGSdouble} \def\HEADINGSafter{\let\HEADINGShook=\HEADINGSdoublex} \let\HEADINGSdoubleafter=\HEADINGSafter \def\HEADINGSdoublex{% \global\evenfootline={\hfil} \global\oddfootline={\hfil} \global\evenheadline={\line{\folio\hfil\thistitle}} \global\oddheadline={\line{\thischapter\hfil\folio}} \global\let\contentsalignmacro = \chapoddpage } \def\HEADINGSsingleafter{\let\HEADINGShook=\HEADINGSsinglex} \def\HEADINGSsinglex{% \global\evenfootline={\hfil} \global\oddfootline={\hfil} \global\evenheadline={\line{\thischapter\hfil\folio}} \global\oddheadline={\line{\thischapter\hfil\folio}} \global\let\contentsalignmacro = \chappager } % Subroutines used in generating headings % This produces Day Month Year style of output. % Only define if not already defined, in case a txi-??.tex file has set % up a different format (e.g., txi-cs.tex does this). \ifx\today\undefined \def\today{% \number\day\space \ifcase\month \or\putwordMJan\or\putwordMFeb\or\putwordMMar\or\putwordMApr \or\putwordMMay\or\putwordMJun\or\putwordMJul\or\putwordMAug \or\putwordMSep\or\putwordMOct\or\putwordMNov\or\putwordMDec \fi \space\number\year} \fi % @settitle line... specifies the title of the document, for headings. % It generates no output of its own. \def\thistitle{\putwordNoTitle} \def\settitle{\parsearg{\gdef\thistitle}} \message{tables,} % Tables -- @table, @ftable, @vtable, @item(x). % default indentation of table text \newdimen\tableindent \tableindent=.8in % default indentation of @itemize and @enumerate text \newdimen\itemindent \itemindent=.3in % margin between end of table item and start of table text. \newdimen\itemmargin \itemmargin=.1in % used internally for \itemindent minus \itemmargin \newdimen\itemmax % Note @table, @ftable, and @vtable define @item, @itemx, etc., with % these defs. % They also define \itemindex % to index the item name in whatever manner is desired (perhaps none). \newif\ifitemxneedsnegativevskip \def\itemxpar{\par\ifitemxneedsnegativevskip\nobreak\vskip-\parskip\nobreak\fi} \def\internalBitem{\smallbreak \parsearg\itemzzz} \def\internalBitemx{\itemxpar \parsearg\itemzzz} \def\itemzzz #1{\begingroup % \advance\hsize by -\rightskip \advance\hsize by -\tableindent \setbox0=\hbox{\itemindicate{#1}}% \itemindex{#1}% \nobreak % This prevents a break before @itemx. % % If the item text does not fit in the space we have, put it on a line % by itself, and do not allow a page break either before or after that % line. We do not start a paragraph here because then if the next % command is, e.g., @kindex, the whatsit would get put into the % horizontal list on a line by itself, resulting in extra blank space. \ifdim \wd0>\itemmax % % Make this a paragraph so we get the \parskip glue and wrapping, % but leave it ragged-right. \begingroup \advance\leftskip by-\tableindent \advance\hsize by\tableindent \advance\rightskip by0pt plus1fil \leavevmode\unhbox0\par \endgroup % % We're going to be starting a paragraph, but we don't want the % \parskip glue -- logically it's part of the @item we just started. \nobreak \vskip-\parskip % % Stop a page break at the \parskip glue coming up. However, if % what follows is an environment such as @example, there will be no % \parskip glue; then the negative vskip we just inserted would % cause the example and the item to crash together. So we use this % bizarre value of 10001 as a signal to \aboveenvbreak to insert % \parskip glue after all. Section titles are handled this way also. % \penalty 10001 \endgroup \itemxneedsnegativevskipfalse \else % The item text fits into the space. Start a paragraph, so that the % following text (if any) will end up on the same line. \noindent % Do this with kerns and \unhbox so that if there is a footnote in % the item text, it can migrate to the main vertical list and % eventually be printed. \nobreak\kern-\tableindent \dimen0 = \itemmax \advance\dimen0 by \itemmargin \advance\dimen0 by -\wd0 \unhbox0 \nobreak\kern\dimen0 \endgroup \itemxneedsnegativevskiptrue \fi } \def\item{\errmessage{@item while not in a list environment}} \def\itemx{\errmessage{@itemx while not in a list environment}} % @table, @ftable, @vtable. \envdef\table{% \let\itemindex\gobble \tablecheck{table}% } \envdef\ftable{% \def\itemindex ##1{\doind {fn}{\code{##1}}}% \tablecheck{ftable}% } \envdef\vtable{% \def\itemindex ##1{\doind {vr}{\code{##1}}}% \tablecheck{vtable}% } \def\tablecheck#1{% \ifnum \the\catcode`\^^M=\active \endgroup \errmessage{This command won't work in this context; perhaps the problem is that we are \inenvironment\thisenv}% \def\next{\doignore{#1}}% \else \let\next\tablex \fi \next } \def\tablex#1{% \def\itemindicate{#1}% \parsearg\tabley } \def\tabley#1{% {% \makevalueexpandable \edef\temp{\noexpand\tablez #1\space\space\space}% \expandafter }\temp \endtablez } \def\tablez #1 #2 #3 #4\endtablez{% \aboveenvbreak \ifnum 0#1>0 \advance \leftskip by #1\mil \fi \ifnum 0#2>0 \tableindent=#2\mil \fi \ifnum 0#3>0 \advance \rightskip by #3\mil \fi \itemmax=\tableindent \advance \itemmax by -\itemmargin \advance \leftskip by \tableindent \exdentamount=\tableindent \parindent = 0pt \parskip = \smallskipamount \ifdim \parskip=0pt \parskip=2pt \fi \let\item = \internalBitem \let\itemx = \internalBitemx } \def\Etable{\endgraf\afterenvbreak} \let\Eftable\Etable \let\Evtable\Etable \let\Eitemize\Etable \let\Eenumerate\Etable % This is the counter used by @enumerate, which is really @itemize \newcount \itemno \envdef\itemize{\parsearg\doitemize} \def\doitemize#1{% \aboveenvbreak \itemmax=\itemindent \advance\itemmax by -\itemmargin \advance\leftskip by \itemindent \exdentamount=\itemindent \parindent=0pt \parskip=\smallskipamount \ifdim\parskip=0pt \parskip=2pt \fi \def\itemcontents{#1}% % @itemize with no arg is equivalent to @itemize @bullet. \ifx\itemcontents\empty\def\itemcontents{\bullet}\fi \let\item=\itemizeitem } % Definition of @item while inside @itemize and @enumerate. % \def\itemizeitem{% \advance\itemno by 1 % for enumerations {\let\par=\endgraf \smallbreak}% reasonable place to break {% % If the document has an @itemize directly after a section title, a % \nobreak will be last on the list, and \sectionheading will have % done a \vskip-\parskip. In that case, we don't want to zero % parskip, or the item text will crash with the heading. On the % other hand, when there is normal text preceding the item (as there % usually is), we do want to zero parskip, or there would be too much % space. In that case, we won't have a \nobreak before. At least % that's the theory. \ifnum\lastpenalty<10000 \parskip=0in \fi \noindent \hbox to 0pt{\hss \itemcontents \kern\itemmargin}% \vadjust{\penalty 1200}}% not good to break after first line of item. \flushcr } % \splitoff TOKENS\endmark defines \first to be the first token in % TOKENS, and \rest to be the remainder. % \def\splitoff#1#2\endmark{\def\first{#1}\def\rest{#2}}% % Allow an optional argument of an uppercase letter, lowercase letter, % or number, to specify the first label in the enumerated list. No % argument is the same as `1'. % \envparseargdef\enumerate{\enumeratey #1 \endenumeratey} \def\enumeratey #1 #2\endenumeratey{% % If we were given no argument, pretend we were given `1'. \def\thearg{#1}% \ifx\thearg\empty \def\thearg{1}\fi % % Detect if the argument is a single token. If so, it might be a % letter. Otherwise, the only valid thing it can be is a number. % (We will always have one token, because of the test we just made. % This is a good thing, since \splitoff doesn't work given nothing at % all -- the first parameter is undelimited.) \expandafter\splitoff\thearg\endmark \ifx\rest\empty % Only one token in the argument. It could still be anything. % A ``lowercase letter'' is one whose \lccode is nonzero. % An ``uppercase letter'' is one whose \lccode is both nonzero, and % not equal to itself. % Otherwise, we assume it's a number. % % We need the \relax at the end of the \ifnum lines to stop TeX from % continuing to look for a . % \ifnum\lccode\expandafter`\thearg=0\relax \numericenumerate % a number (we hope) \else % It's a letter. \ifnum\lccode\expandafter`\thearg=\expandafter`\thearg\relax \lowercaseenumerate % lowercase letter \else \uppercaseenumerate % uppercase letter \fi \fi \else % Multiple tokens in the argument. We hope it's a number. \numericenumerate \fi } % An @enumerate whose labels are integers. The starting integer is % given in \thearg. % \def\numericenumerate{% \itemno = \thearg \startenumeration{\the\itemno}% } % The starting (lowercase) letter is in \thearg. \def\lowercaseenumerate{% \itemno = \expandafter`\thearg \startenumeration{% % Be sure we're not beyond the end of the alphabet. \ifnum\itemno=0 \errmessage{No more lowercase letters in @enumerate; get a bigger alphabet}% \fi \char\lccode\itemno }% } % The starting (uppercase) letter is in \thearg. \def\uppercaseenumerate{% \itemno = \expandafter`\thearg \startenumeration{% % Be sure we're not beyond the end of the alphabet. \ifnum\itemno=0 \errmessage{No more uppercase letters in @enumerate; get a bigger alphabet} \fi \char\uccode\itemno }% } % Call \doitemize, adding a period to the first argument and supplying the % common last two arguments. Also subtract one from the initial value in % \itemno, since @item increments \itemno. % \def\startenumeration#1{% \advance\itemno by -1 \doitemize{#1.}\flushcr } % @alphaenumerate and @capsenumerate are abbreviations for giving an arg % to @enumerate. % \def\alphaenumerate{\enumerate{a}} \def\capsenumerate{\enumerate{A}} \def\Ealphaenumerate{\Eenumerate} \def\Ecapsenumerate{\Eenumerate} % @multitable macros % Amy Hendrickson, 8/18/94, 3/6/96 % % @multitable ... @end multitable will make as many columns as desired. % Contents of each column will wrap at width given in preamble. Width % can be specified either with sample text given in a template line, % or in percent of \hsize, the current width of text on page. % Table can continue over pages but will only break between lines. % To make preamble: % % Either define widths of columns in terms of percent of \hsize: % @multitable @columnfractions .25 .3 .45 % @item ... % % Numbers following @columnfractions are the percent of the total % current hsize to be used for each column. You may use as many % columns as desired. % Or use a template: % @multitable {Column 1 template} {Column 2 template} {Column 3 template} % @item ... % using the widest term desired in each column. % Each new table line starts with @item, each subsequent new column % starts with @tab. Empty columns may be produced by supplying @tab's % with nothing between them for as many times as empty columns are needed, % ie, @tab@tab@tab will produce two empty columns. % @item, @tab do not need to be on their own lines, but it will not hurt % if they are. % Sample multitable: % @multitable {Column 1 template} {Column 2 template} {Column 3 template} % @item first col stuff @tab second col stuff @tab third col % @item % first col stuff % @tab % second col stuff % @tab % third col % @item first col stuff @tab second col stuff % @tab Many paragraphs of text may be used in any column. % % They will wrap at the width determined by the template. % @item@tab@tab This will be in third column. % @end multitable % Default dimensions may be reset by user. % @multitableparskip is vertical space between paragraphs in table. % @multitableparindent is paragraph indent in table. % @multitablecolmargin is horizontal space to be left between columns. % @multitablelinespace is space to leave between table items, baseline % to baseline. % 0pt means it depends on current normal line spacing. % \newskip\multitableparskip \newskip\multitableparindent \newdimen\multitablecolspace \newskip\multitablelinespace \multitableparskip=0pt \multitableparindent=6pt \multitablecolspace=12pt \multitablelinespace=0pt % Macros used to set up halign preamble: % \let\endsetuptable\relax \def\xendsetuptable{\endsetuptable} \let\columnfractions\relax \def\xcolumnfractions{\columnfractions} \newif\ifsetpercent % #1 is the @columnfraction, usually a decimal number like .5, but might % be just 1. We just use it, whatever it is. % \def\pickupwholefraction#1 {% \global\advance\colcount by 1 \expandafter\xdef\csname col\the\colcount\endcsname{#1\hsize}% \setuptable } \newcount\colcount \def\setuptable#1{% \def\firstarg{#1}% \ifx\firstarg\xendsetuptable \let\go = \relax \else \ifx\firstarg\xcolumnfractions \global\setpercenttrue \else \ifsetpercent \let\go\pickupwholefraction \else \global\advance\colcount by 1 \setbox0=\hbox{#1\unskip\space}% Add a normal word space as a % separator; typically that is always in the input, anyway. \expandafter\xdef\csname col\the\colcount\endcsname{\the\wd0}% \fi \fi \ifx\go\pickupwholefraction % Put the argument back for the \pickupwholefraction call, so % we'll always have a period there to be parsed. \def\go{\pickupwholefraction#1}% \else \let\go = \setuptable \fi% \fi \go } % multitable-only commands. % % @headitem starts a heading row, which we typeset in bold. % Assignments have to be global since we are inside the implicit group % of an alignment entry. Note that \everycr resets \everytab. \def\headitem{\checkenv\multitable \crcr \global\everytab={\bf}\the\everytab}% % % A \tab used to include \hskip1sp. But then the space in a template % line is not enough. That is bad. So let's go back to just `&' until % we encounter the problem it was intended to solve again. % --karl, nathan@acm.org, 20apr99. \def\tab{\checkenv\multitable &\the\everytab}% % @multitable ... @end multitable definitions: % \newtoks\everytab % insert after every tab. % \envdef\multitable{% \vskip\parskip \startsavinginserts % % @item within a multitable starts a normal row. % We use \def instead of \let so that if one of the multitable entries % contains an @itemize, we don't choke on the \item (seen as \crcr aka % \endtemplate) expanding \doitemize. \def\item{\crcr}% % \tolerance=9500 \hbadness=9500 \setmultitablespacing \parskip=\multitableparskip \parindent=\multitableparindent \overfullrule=0pt \global\colcount=0 % \everycr = {% \noalign{% \global\everytab={}% \global\colcount=0 % Reset the column counter. % Check for saved footnotes, etc. \checkinserts % Keeps underfull box messages off when table breaks over pages. %\filbreak % Maybe so, but it also creates really weird page breaks when the % table breaks over pages. Wouldn't \vfil be better? Wait until the % problem manifests itself, so it can be fixed for real --karl. }% }% % \parsearg\domultitable } \def\domultitable#1{% % To parse everything between @multitable and @item: \setuptable#1 \endsetuptable % % This preamble sets up a generic column definition, which will % be used as many times as user calls for columns. % \vtop will set a single line and will also let text wrap and % continue for many paragraphs if desired. \halign\bgroup &% \global\advance\colcount by 1 \multistrut \vtop{% % Use the current \colcount to find the correct column width: \hsize=\expandafter\csname col\the\colcount\endcsname % % In order to keep entries from bumping into each other % we will add a \leftskip of \multitablecolspace to all columns after % the first one. % % If a template has been used, we will add \multitablecolspace % to the width of each template entry. % % If the user has set preamble in terms of percent of \hsize we will % use that dimension as the width of the column, and the \leftskip % will keep entries from bumping into each other. Table will start at % left margin and final column will justify at right margin. % % Make sure we don't inherit \rightskip from the outer environment. \rightskip=0pt \ifnum\colcount=1 % The first column will be indented with the surrounding text. \advance\hsize by\leftskip \else \ifsetpercent \else % If user has not set preamble in terms of percent of \hsize % we will advance \hsize by \multitablecolspace. \advance\hsize by \multitablecolspace \fi % In either case we will make \leftskip=\multitablecolspace: \leftskip=\multitablecolspace \fi % Ignoring space at the beginning and end avoids an occasional spurious % blank line, when TeX decides to break the line at the space before the % box from the multistrut, so the strut ends up on a line by itself. % For example: % @multitable @columnfractions .11 .89 % @item @code{#} % @tab Legal holiday which is valid in major parts of the whole country. % Is automatically provided with highlighting sequences respectively % marking characters. \noindent\ignorespaces##\unskip\multistrut }\cr } \def\Emultitable{% \crcr \egroup % end the \halign \global\setpercentfalse } \def\setmultitablespacing{% \def\multistrut{\strut}% just use the standard line spacing % % Compute \multitablelinespace (if not defined by user) for use in % \multitableparskip calculation. We used define \multistrut based on % this, but (ironically) that caused the spacing to be off. % See bug-texinfo report from Werner Lemberg, 31 Oct 2004 12:52:20 +0100. \ifdim\multitablelinespace=0pt \setbox0=\vbox{X}\global\multitablelinespace=\the\baselineskip \global\advance\multitablelinespace by-\ht0 \fi %% Test to see if parskip is larger than space between lines of %% table. If not, do nothing. %% If so, set to same dimension as multitablelinespace. \ifdim\multitableparskip>\multitablelinespace \global\multitableparskip=\multitablelinespace \global\advance\multitableparskip-7pt %% to keep parskip somewhat smaller %% than skip between lines in the table. \fi% \ifdim\multitableparskip=0pt \global\multitableparskip=\multitablelinespace \global\advance\multitableparskip-7pt %% to keep parskip somewhat smaller %% than skip between lines in the table. \fi} \message{conditionals,} % @iftex, @ifnotdocbook, @ifnothtml, @ifnotinfo, @ifnotplaintext, % @ifnotxml always succeed. They currently do nothing; we don't % attempt to check whether the conditionals are properly nested. But we % have to remember that they are conditionals, so that @end doesn't % attempt to close an environment group. % \def\makecond#1{% \expandafter\let\csname #1\endcsname = \relax \expandafter\let\csname iscond.#1\endcsname = 1 } \makecond{iftex} \makecond{ifnotdocbook} \makecond{ifnothtml} \makecond{ifnotinfo} \makecond{ifnotplaintext} \makecond{ifnotxml} % Ignore @ignore, @ifhtml, @ifinfo, and the like. % \def\direntry{\doignore{direntry}} \def\documentdescription{\doignore{documentdescription}} \def\docbook{\doignore{docbook}} \def\html{\doignore{html}} \def\ifdocbook{\doignore{ifdocbook}} \def\ifhtml{\doignore{ifhtml}} \def\ifinfo{\doignore{ifinfo}} \def\ifnottex{\doignore{ifnottex}} \def\ifplaintext{\doignore{ifplaintext}} \def\ifxml{\doignore{ifxml}} \def\ignore{\doignore{ignore}} \def\menu{\doignore{menu}} \def\xml{\doignore{xml}} % Ignore text until a line `@end #1', keeping track of nested conditionals. % % A count to remember the depth of nesting. \newcount\doignorecount \def\doignore#1{\begingroup % Scan in ``verbatim'' mode: \obeylines \catcode`\@ = \other \catcode`\{ = \other \catcode`\} = \other % % Make sure that spaces turn into tokens that match what \doignoretext wants. \spaceisspace % % Count number of #1's that we've seen. \doignorecount = 0 % % Swallow text until we reach the matching `@end #1'. \dodoignore{#1}% } { \catcode`_=11 % We want to use \_STOP_ which cannot appear in texinfo source. \obeylines % % \gdef\dodoignore#1{% % #1 contains the command name as a string, e.g., `ifinfo'. % % Define a command to find the next `@end #1'. \long\def\doignoretext##1^^M@end #1{% \doignoretextyyy##1^^M@#1\_STOP_}% % % And this command to find another #1 command, at the beginning of a % line. (Otherwise, we would consider a line `@c @ifset', for % example, to count as an @ifset for nesting.) \long\def\doignoretextyyy##1^^M@#1##2\_STOP_{\doignoreyyy{##2}\_STOP_}% % % And now expand that command. \doignoretext ^^M% }% } \def\doignoreyyy#1{% \def\temp{#1}% \ifx\temp\empty % Nothing found. \let\next\doignoretextzzz \else % Found a nested condition, ... \advance\doignorecount by 1 \let\next\doignoretextyyy % ..., look for another. % If we're here, #1 ends with ^^M\ifinfo (for example). \fi \next #1% the token \_STOP_ is present just after this macro. } % We have to swallow the remaining "\_STOP_". % \def\doignoretextzzz#1{% \ifnum\doignorecount = 0 % We have just found the outermost @end. \let\next\enddoignore \else % Still inside a nested condition. \advance\doignorecount by -1 \let\next\doignoretext % Look for the next @end. \fi \next } % Finish off ignored text. { \obeylines% % Ignore anything after the last `@end #1'; this matters in verbatim % environments, where otherwise the newline after an ignored conditional % would result in a blank line in the output. \gdef\enddoignore#1^^M{\endgroup\ignorespaces}% } % @set VAR sets the variable VAR to an empty value. % @set VAR REST-OF-LINE sets VAR to the value REST-OF-LINE. % % Since we want to separate VAR from REST-OF-LINE (which might be % empty), we can't just use \parsearg; we have to insert a space of our % own to delimit the rest of the line, and then take it out again if we % didn't need it. % We rely on the fact that \parsearg sets \catcode`\ =10. % \parseargdef\set{\setyyy#1 \endsetyyy} \def\setyyy#1 #2\endsetyyy{% {% \makevalueexpandable \def\temp{#2}% \edef\next{\gdef\makecsname{SET#1}}% \ifx\temp\empty \next{}% \else \setzzz#2\endsetzzz \fi }% } % Remove the trailing space \setxxx inserted. \def\setzzz#1 \endsetzzz{\next{#1}} % @clear VAR clears (i.e., unsets) the variable VAR. % \parseargdef\clear{% {% \makevalueexpandable \global\expandafter\let\csname SET#1\endcsname=\relax }% } % @value{foo} gets the text saved in variable foo. \def\value{\begingroup\makevalueexpandable\valuexxx} \def\valuexxx#1{\expandablevalue{#1}\endgroup} { \catcode`\- = \active \catcode`\_ = \active % \gdef\makevalueexpandable{% \let\value = \expandablevalue % We don't want these characters active, ... \catcode`\-=\other \catcode`\_=\other % ..., but we might end up with active ones in the argument if % we're called from @code, as @code{@value{foo-bar_}}, though. % So \let them to their normal equivalents. \let-\realdash \let_\normalunderscore } } % We have this subroutine so that we can handle at least some @value's % properly in indexes (we call \makevalueexpandable in \indexdummies). % The command has to be fully expandable (if the variable is set), since % the result winds up in the index file. This means that if the % variable's value contains other Texinfo commands, it's almost certain % it will fail (although perhaps we could fix that with sufficient work % to do a one-level expansion on the result, instead of complete). % \def\expandablevalue#1{% \expandafter\ifx\csname SET#1\endcsname\relax {[No value for ``#1'']}% \message{Variable `#1', used in @value, is not set.}% \else \csname SET#1\endcsname \fi } % @ifset VAR ... @end ifset reads the `...' iff VAR has been defined % with @set. % % To get special treatment of `@end ifset,' call \makeond and the redefine. % \makecond{ifset} \def\ifset{\parsearg{\doifset{\let\next=\ifsetfail}}} \def\doifset#1#2{% {% \makevalueexpandable \let\next=\empty \expandafter\ifx\csname SET#2\endcsname\relax #1% If not set, redefine \next. \fi \expandafter }\next } \def\ifsetfail{\doignore{ifset}} % @ifclear VAR ... @end ifclear reads the `...' iff VAR has never been % defined with @set, or has been undefined with @clear. % % The `\else' inside the `\doifset' parameter is a trick to reuse the % above code: if the variable is not set, do nothing, if it is set, % then redefine \next to \ifclearfail. % \makecond{ifclear} \def\ifclear{\parsearg{\doifset{\else \let\next=\ifclearfail}}} \def\ifclearfail{\doignore{ifclear}} % @dircategory CATEGORY -- specify a category of the dir file % which this file should belong to. Ignore this in TeX. \let\dircategory=\comment % @defininfoenclose. \let\definfoenclose=\comment \message{indexing,} % Index generation facilities % Define \newwrite to be identical to plain tex's \newwrite % except not \outer, so it can be used within macros and \if's. \edef\newwrite{\makecsname{ptexnewwrite}} % \newindex {foo} defines an index named foo. % It automatically defines \fooindex such that % \fooindex ...rest of line... puts an entry in the index foo. % It also defines \fooindfile to be the number of the output channel for % the file that accumulates this index. The file's extension is foo. % The name of an index should be no more than 2 characters long % for the sake of vms. % \def\newindex#1{% \iflinks \expandafter\newwrite \csname#1indfile\endcsname \openout \csname#1indfile\endcsname \jobname.#1 % Open the file \fi \expandafter\xdef\csname#1index\endcsname{% % Define @#1index \noexpand\doindex{#1}} } % @defindex foo == \newindex{foo} % \def\defindex{\parsearg\newindex} % Define @defcodeindex, like @defindex except put all entries in @code. % \def\defcodeindex{\parsearg\newcodeindex} % \def\newcodeindex#1{% \iflinks \expandafter\newwrite \csname#1indfile\endcsname \openout \csname#1indfile\endcsname \jobname.#1 \fi \expandafter\xdef\csname#1index\endcsname{% \noexpand\docodeindex{#1}}% } % @synindex foo bar makes index foo feed into index bar. % Do this instead of @defindex foo if you don't want it as a separate index. % % @syncodeindex foo bar similar, but put all entries made for index foo % inside @code. % \def\synindex#1 #2 {\dosynindex\doindex{#1}{#2}} \def\syncodeindex#1 #2 {\dosynindex\docodeindex{#1}{#2}} % #1 is \doindex or \docodeindex, #2 the index getting redefined (foo), % #3 the target index (bar). \def\dosynindex#1#2#3{% % Only do \closeout if we haven't already done it, else we'll end up % closing the target index. \expandafter \ifx\csname donesynindex#2\endcsname \undefined % The \closeout helps reduce unnecessary open files; the limit on the % Acorn RISC OS is a mere 16 files. \expandafter\closeout\csname#2indfile\endcsname \expandafter\let\csname\donesynindex#2\endcsname = 1 \fi % redefine \fooindfile: \expandafter\let\expandafter\temp\expandafter=\csname#3indfile\endcsname \expandafter\let\csname#2indfile\endcsname=\temp % redefine \fooindex: \expandafter\xdef\csname#2index\endcsname{\noexpand#1{#3}}% } % Define \doindex, the driver for all \fooindex macros. % Argument #1 is generated by the calling \fooindex macro, % and it is "foo", the name of the index. % \doindex just uses \parsearg; it calls \doind for the actual work. % This is because \doind is more useful to call from other macros. % There is also \dosubind {index}{topic}{subtopic} % which makes an entry in a two-level index such as the operation index. \def\doindex#1{\edef\indexname{#1}\parsearg\singleindexer} \def\singleindexer #1{\doind{\indexname}{#1}} % like the previous two, but they put @code around the argument. \def\docodeindex#1{\edef\indexname{#1}\parsearg\singlecodeindexer} \def\singlecodeindexer #1{\doind{\indexname}{\code{#1}}} % Take care of Texinfo commands that can appear in an index entry. % Since there are some commands we want to expand, and others we don't, % we have to laboriously prevent expansion for those that we don't. % \def\indexdummies{% \escapechar = `\\ % use backslash in output files. \def\@{@}% change to @@ when we switch to @ as escape char in index files. \def\ {\realbackslash\space }% % % Need these in case \tex is in effect and \{ is a \delimiter again. % But can't use \lbracecmd and \rbracecmd because texindex assumes % braces and backslashes are used only as delimiters. \let\{ = \mylbrace \let\} = \myrbrace % % I don't entirely understand this, but when an index entry is % generated from a macro call, the \endinput which \scanmacro inserts % causes processing to be prematurely terminated. This is, % apparently, because \indexsorttmp is fully expanded, and \endinput % is an expandable command. The redefinition below makes \endinput % disappear altogether for that purpose -- although logging shows that % processing continues to some further point. On the other hand, it % seems \endinput does not hurt in the printed index arg, since that % is still getting written without apparent harm. % % Sample source (mac-idx3.tex, reported by Graham Percival to % help-texinfo, 22may06): % @macro funindex {WORD} % @findex xyz % @end macro % ... % @funindex commtest % % The above is not enough to reproduce the bug, but it gives the flavor. % % Sample whatsit resulting: % .@write3{\entry{xyz}{@folio }{@code {xyz@endinput }}} % % So: \let\endinput = \empty % % Do the redefinitions. \commondummies } % For the aux and toc files, @ is the escape character. So we want to % redefine everything using @ as the escape character (instead of % \realbackslash, still used for index files). When everything uses @, % this will be simpler. % \def\atdummies{% \def\@{@@}% \def\ {@ }% \let\{ = \lbraceatcmd \let\} = \rbraceatcmd % % Do the redefinitions. \commondummies \otherbackslash } % Called from \indexdummies and \atdummies. % \def\commondummies{% % % \definedummyword defines \#1 as \string\#1\space, thus effectively % preventing its expansion. This is used only for control% words, % not control letters, because the \space would be incorrect for % control characters, but is needed to separate the control word % from whatever follows. % % For control letters, we have \definedummyletter, which omits the % space. % % These can be used both for control words that take an argument and % those that do not. If it is followed by {arg} in the input, then % that will dutifully get written to the index (or wherever). % \def\definedummyword ##1{\def##1{\string##1\space}}% \def\definedummyletter##1{\def##1{\string##1}}% \let\definedummyaccent\definedummyletter % \commondummiesnofonts % \definedummyletter\_% % % Non-English letters. \definedummyword\AA \definedummyword\AE \definedummyword\L \definedummyword\OE \definedummyword\O \definedummyword\aa \definedummyword\ae \definedummyword\l \definedummyword\oe \definedummyword\o \definedummyword\ss \definedummyword\exclamdown \definedummyword\questiondown \definedummyword\ordf \definedummyword\ordm % % Although these internal commands shouldn't show up, sometimes they do. \definedummyword\bf \definedummyword\gtr \definedummyword\hat \definedummyword\less \definedummyword\sf \definedummyword\sl \definedummyword\tclose \definedummyword\tt % \definedummyword\LaTeX \definedummyword\TeX % % Assorted special characters. \definedummyword\bullet \definedummyword\comma \definedummyword\copyright \definedummyword\registeredsymbol \definedummyword\dots \definedummyword\enddots \definedummyword\equiv \definedummyword\error \definedummyword\euro \definedummyword\guillemetleft \definedummyword\guillemetright \definedummyword\guilsinglleft \definedummyword\guilsinglright \definedummyword\expansion \definedummyword\minus \definedummyword\pounds \definedummyword\point \definedummyword\print \definedummyword\quotedblbase \definedummyword\quotedblleft \definedummyword\quotedblright \definedummyword\quoteleft \definedummyword\quoteright \definedummyword\quotesinglbase \definedummyword\result \definedummyword\textdegree % % We want to disable all macros so that they are not expanded by \write. \macrolist % \normalturnoffactive % % Handle some cases of @value -- where it does not contain any % (non-fully-expandable) commands. \makevalueexpandable } % \commondummiesnofonts: common to \commondummies and \indexnofonts. % \def\commondummiesnofonts{% % Control letters and accents. \definedummyletter\!% \definedummyaccent\"% \definedummyaccent\'% \definedummyletter\*% \definedummyaccent\,% \definedummyletter\.% \definedummyletter\/% \definedummyletter\:% \definedummyaccent\=% \definedummyletter\?% \definedummyaccent\^% \definedummyaccent\`% \definedummyaccent\~% \definedummyword\u \definedummyword\v \definedummyword\H \definedummyword\dotaccent \definedummyword\ringaccent \definedummyword\tieaccent \definedummyword\ubaraccent \definedummyword\udotaccent \definedummyword\dotless % % Texinfo font commands. \definedummyword\b \definedummyword\i \definedummyword\r \definedummyword\sc \definedummyword\t % % Commands that take arguments. \definedummyword\acronym \definedummyword\cite \definedummyword\code \definedummyword\command \definedummyword\dfn \definedummyword\emph \definedummyword\env \definedummyword\file \definedummyword\kbd \definedummyword\key \definedummyword\math \definedummyword\option \definedummyword\pxref \definedummyword\ref \definedummyword\samp \definedummyword\strong \definedummyword\tie \definedummyword\uref \definedummyword\url \definedummyword\var \definedummyword\verb \definedummyword\w \definedummyword\xref } % \indexnofonts is used when outputting the strings to sort the index % by, and when constructing control sequence names. It eliminates all % control sequences and just writes whatever the best ASCII sort string % would be for a given command (usually its argument). % \def\indexnofonts{% % Accent commands should become @asis. \def\definedummyaccent##1{\let##1\asis}% % We can just ignore other control letters. \def\definedummyletter##1{\let##1\empty}% % Hopefully, all control words can become @asis. \let\definedummyword\definedummyaccent % \commondummiesnofonts % % Don't no-op \tt, since it isn't a user-level command % and is used in the definitions of the active chars like <, >, |, etc. % Likewise with the other plain tex font commands. %\let\tt=\asis % \def\ { }% \def\@{@}% % how to handle braces? \def\_{\normalunderscore}% % % Non-English letters. \def\AA{AA}% \def\AE{AE}% \def\L{L}% \def\OE{OE}% \def\O{O}% \def\aa{aa}% \def\ae{ae}% \def\l{l}% \def\oe{oe}% \def\o{o}% \def\ss{ss}% \def\exclamdown{!}% \def\questiondown{?}% \def\ordf{a}% \def\ordm{o}% % \def\LaTeX{LaTeX}% \def\TeX{TeX}% % % Assorted special characters. % (The following {} will end up in the sort string, but that's ok.) \def\bullet{bullet}% \def\comma{,}% \def\copyright{copyright}% \def\registeredsymbol{R}% \def\dots{...}% \def\enddots{...}% \def\equiv{==}% \def\error{error}% \def\euro{euro}% \def\guillemetleft{<<}% \def\guillemetright{>>}% \def\guilsinglleft{<}% \def\guilsinglright{>}% \def\expansion{==>}% \def\minus{-}% \def\pounds{pounds}% \def\point{.}% \def\print{-|}% \def\quotedblbase{"}% \def\quotedblleft{"}% \def\quotedblright{"}% \def\quoteleft{`}% \def\quoteright{'}% \def\quotesinglbase{,}% \def\result{=>}% \def\textdegree{degrees}% % % We need to get rid of all macros, leaving only the arguments (if present). % Of course this is not nearly correct, but it is the best we can do for now. % makeinfo does not expand macros in the argument to @deffn, which ends up % writing an index entry, and texindex isn't prepared for an index sort entry % that starts with \. % % Since macro invocations are followed by braces, we can just redefine them % to take a single TeX argument. The case of a macro invocation that % goes to end-of-line is not handled. % \macrolist } \let\indexbackslash=0 %overridden during \printindex. \let\SETmarginindex=\relax % put index entries in margin (undocumented)? % Most index entries go through here, but \dosubind is the general case. % #1 is the index name, #2 is the entry text. \def\doind#1#2{\dosubind{#1}{#2}{}} % Workhorse for all \fooindexes. % #1 is name of index, #2 is stuff to put there, #3 is subentry -- % empty if called from \doind, as we usually are (the main exception % is with most defuns, which call us directly). % \def\dosubind#1#2#3{% \iflinks {% % Store the main index entry text (including the third arg). \toks0 = {#2}% % If third arg is present, precede it with a space. \def\thirdarg{#3}% \ifx\thirdarg\empty \else \toks0 = \expandafter{\the\toks0 \space #3}% \fi % \edef\writeto{\csname#1indfile\endcsname}% % \safewhatsit\dosubindwrite }% \fi } % Write the entry in \toks0 to the index file: % \def\dosubindwrite{% % Put the index entry in the margin if desired. \ifx\SETmarginindex\relax\else \insert\margin{\hbox{\vrule height8pt depth3pt width0pt \the\toks0}}% \fi % % Remember, we are within a group. \indexdummies % Must do this here, since \bf, etc expand at this stage \def\backslashcurfont{\indexbackslash}% \indexbackslash isn't defined now % so it will be output as is; and it will print as backslash. % % Process the index entry with all font commands turned off, to % get the string to sort by. {\indexnofonts \edef\temp{\the\toks0}% need full expansion \xdef\indexsorttmp{\temp}% }% % % Set up the complete index entry, with both the sort key and % the original text, including any font commands. We write % three arguments to \entry to the .?? file (four in the % subentry case), texindex reduces to two when writing the .??s % sorted result. \edef\temp{% \write\writeto{% \string\entry{\indexsorttmp}{\noexpand\folio}{\the\toks0}}% }% \temp } % Take care of unwanted page breaks/skips around a whatsit: % % If a skip is the last thing on the list now, preserve it % by backing up by \lastskip, doing the \write, then inserting % the skip again. Otherwise, the whatsit generated by the % \write or \pdfdest will make \lastskip zero. The result is that % sequences like this: % @end defun % @tindex whatever % @defun ... % will have extra space inserted, because the \medbreak in the % start of the @defun won't see the skip inserted by the @end of % the previous defun. % % But don't do any of this if we're not in vertical mode. We % don't want to do a \vskip and prematurely end a paragraph. % % Avoid page breaks due to these extra skips, too. % % But wait, there is a catch there: % We'll have to check whether \lastskip is zero skip. \ifdim is not % sufficient for this purpose, as it ignores stretch and shrink parts % of the skip. The only way seems to be to check the textual % representation of the skip. % % The following is almost like \def\zeroskipmacro{0.0pt} except that % the ``p'' and ``t'' characters have catcode \other, not 11 (letter). % \edef\zeroskipmacro{\expandafter\the\csname z@skip\endcsname} % \newskip\whatsitskip \newcount\whatsitpenalty % % ..., ready, GO: % \def\safewhatsit#1{% \ifhmode #1% \else % \lastskip and \lastpenalty cannot both be nonzero simultaneously. \whatsitskip = \lastskip \edef\lastskipmacro{\the\lastskip}% \whatsitpenalty = \lastpenalty % % If \lastskip is nonzero, that means the last item was a % skip. And since a skip is discardable, that means this % -\whatsitskip glue we're inserting is preceded by a % non-discardable item, therefore it is not a potential % breakpoint, therefore no \nobreak needed. \ifx\lastskipmacro\zeroskipmacro \else \vskip-\whatsitskip \fi % #1% % \ifx\lastskipmacro\zeroskipmacro % If \lastskip was zero, perhaps the last item was a penalty, and % perhaps it was >=10000, e.g., a \nobreak. In that case, we want % to re-insert the same penalty (values >10000 are used for various % signals); since we just inserted a non-discardable item, any % following glue (such as a \parskip) would be a breakpoint. For example: % % @deffn deffn-whatever % @vindex index-whatever % Description. % would allow a break between the index-whatever whatsit % and the "Description." paragraph. \ifnum\whatsitpenalty>9999 \penalty\whatsitpenalty \fi \else % On the other hand, if we had a nonzero \lastskip, % this make-up glue would be preceded by a non-discardable item % (the whatsit from the \write), so we must insert a \nobreak. \nobreak\vskip\whatsitskip \fi \fi } % The index entry written in the file actually looks like % \entry {sortstring}{page}{topic} % or % \entry {sortstring}{page}{topic}{subtopic} % The texindex program reads in these files and writes files % containing these kinds of lines: % \initial {c} % before the first topic whose initial is c % \entry {topic}{pagelist} % for a topic that is used without subtopics % \primary {topic} % for the beginning of a topic that is used with subtopics % \secondary {subtopic}{pagelist} % for each subtopic. % Define the user-accessible indexing commands % @findex, @vindex, @kindex, @cindex. \def\findex {\fnindex} \def\kindex {\kyindex} \def\cindex {\cpindex} \def\vindex {\vrindex} \def\tindex {\tpindex} \def\pindex {\pgindex} \def\cindexsub {\begingroup\obeylines\cindexsub} {\obeylines % \gdef\cindexsub "#1" #2^^M{\endgroup % \dosubind{cp}{#2}{#1}}} % Define the macros used in formatting output of the sorted index material. % @printindex causes a particular index (the ??s file) to get printed. % It does not print any chapter heading (usually an @unnumbered). % \parseargdef\printindex{\begingroup \dobreak \chapheadingskip{10000}% % \smallfonts \rm \tolerance = 9500 \plainfrenchspacing \everypar = {}% don't want the \kern\-parindent from indentation suppression. % % See if the index file exists and is nonempty. % Change catcode of @ here so that if the index file contains % \initial {@} % as its first line, TeX doesn't complain about mismatched braces % (because it thinks @} is a control sequence). \catcode`\@ = 11 \openin 1 \jobname.#1s \ifeof 1 % \enddoublecolumns gets confused if there is no text in the index, % and it loses the chapter title and the aux file entries for the % index. The easiest way to prevent this problem is to make sure % there is some text. \putwordIndexNonexistent \else % % If the index file exists but is empty, then \openin leaves \ifeof % false. We have to make TeX try to read something from the file, so % it can discover if there is anything in it. \read 1 to \temp \ifeof 1 \putwordIndexIsEmpty \else % Index files are almost Texinfo source, but we use \ as the escape % character. It would be better to use @, but that's too big a change % to make right now. \def\indexbackslash{\backslashcurfont}% \catcode`\\ = 0 \escapechar = `\\ \begindoublecolumns \input \jobname.#1s \enddoublecolumns \fi \fi \closein 1 \endgroup} % These macros are used by the sorted index file itself. % Change them to control the appearance of the index. \def\initial#1{{% % Some minor font changes for the special characters. \let\tentt=\sectt \let\tt=\sectt \let\sf=\sectt % % Remove any glue we may have, we'll be inserting our own. \removelastskip % % We like breaks before the index initials, so insert a bonus. \nobreak \vskip 0pt plus 3\baselineskip \penalty 0 \vskip 0pt plus -3\baselineskip % % Typeset the initial. Making this add up to a whole number of % baselineskips increases the chance of the dots lining up from column % to column. It still won't often be perfect, because of the stretch % we need before each entry, but it's better. % % No shrink because it confuses \balancecolumns. \vskip 1.67\baselineskip plus .5\baselineskip \leftline{\secbf #1}% % Do our best not to break after the initial. \nobreak \vskip .33\baselineskip plus .1\baselineskip }} % \entry typesets a paragraph consisting of the text (#1), dot leaders, and % then page number (#2) flushed to the right margin. It is used for index % and table of contents entries. The paragraph is indented by \leftskip. % % A straightforward implementation would start like this: % \def\entry#1#2{... % But this freezes the catcodes in the argument, and can cause problems to % @code, which sets - active. This problem was fixed by a kludge--- % ``-'' was active throughout whole index, but this isn't really right. % % The right solution is to prevent \entry from swallowing the whole text. % --kasal, 21nov03 \def\entry{% \begingroup % % Start a new paragraph if necessary, so our assignments below can't % affect previous text. \par % % Do not fill out the last line with white space. \parfillskip = 0in % % No extra space above this paragraph. \parskip = 0in % % Do not prefer a separate line ending with a hyphen to fewer lines. \finalhyphendemerits = 0 % % \hangindent is only relevant when the entry text and page number % don't both fit on one line. In that case, bob suggests starting the % dots pretty far over on the line. Unfortunately, a large % indentation looks wrong when the entry text itself is broken across % lines. So we use a small indentation and put up with long leaders. % % \hangafter is reset to 1 (which is the value we want) at the start % of each paragraph, so we need not do anything with that. \hangindent = 2em % % When the entry text needs to be broken, just fill out the first line % with blank space. \rightskip = 0pt plus1fil % % A bit of stretch before each entry for the benefit of balancing % columns. \vskip 0pt plus1pt % % Swallow the left brace of the text (first parameter): \afterassignment\doentry \let\temp = } \def\doentry{% \bgroup % Instead of the swallowed brace. \noindent \aftergroup\finishentry % And now comes the text of the entry. } \def\finishentry#1{% % #1 is the page number. % % The following is kludged to not output a line of dots in the index if % there are no page numbers. The next person who breaks this will be % cursed by a Unix daemon. \setbox\boxA = \hbox{#1}% \ifdim\wd\boxA = 0pt \ % \else % % If we must, put the page number on a line of its own, and fill out % this line with blank space. (The \hfil is overwhelmed with the % fill leaders glue in \indexdotfill if the page number does fit.) \hfil\penalty50 \null\nobreak\indexdotfill % Have leaders before the page number. % % The `\ ' here is removed by the implicit \unskip that TeX does as % part of (the primitive) \par. Without it, a spurious underfull % \hbox ensues. \ifpdf \pdfgettoks#1.% \ \the\toksA \else \ #1% \fi \fi \par \endgroup } % Like plain.tex's \dotfill, except uses up at least 1 em. \def\indexdotfill{\cleaders \hbox{$\mathsurround=0pt \mkern1.5mu.\mkern1.5mu$}\hskip 1em plus 1fill} \def\primary #1{\line{#1\hfil}} \newskip\secondaryindent \secondaryindent=0.5cm \def\secondary#1#2{{% \parfillskip=0in \parskip=0in \hangindent=1in \hangafter=1 \noindent\hskip\secondaryindent\hbox{#1}\indexdotfill \ifpdf \pdfgettoks#2.\ \the\toksA % The page number ends the paragraph. \else #2 \fi \par }} % Define two-column mode, which we use to typeset indexes. % Adapted from the TeXbook, page 416, which is to say, % the manmac.tex format used to print the TeXbook itself. \catcode`\@=11 \newbox\partialpage \newdimen\doublecolumnhsize \def\begindoublecolumns{\begingroup % ended by \enddoublecolumns % Grab any single-column material above us. \output = {% % % Here is a possibility not foreseen in manmac: if we accumulate a % whole lot of material, we might end up calling this \output % routine twice in a row (see the doublecol-lose test, which is % essentially a couple of indexes with @setchapternewpage off). In % that case we just ship out what is in \partialpage with the normal % output routine. Generally, \partialpage will be empty when this % runs and this will be a no-op. See the indexspread.tex test case. \ifvoid\partialpage \else \onepageout{\pagecontents\partialpage}% \fi % \global\setbox\partialpage = \vbox{% % Unvbox the main output page. \unvbox\PAGE \kern-\topskip \kern\baselineskip }% }% \eject % run that output routine to set \partialpage % % Use the double-column output routine for subsequent pages. \output = {\doublecolumnout}% % % Change the page size parameters. We could do this once outside this % routine, in each of @smallbook, @afourpaper, and the default 8.5x11 % format, but then we repeat the same computation. Repeating a couple % of assignments once per index is clearly meaningless for the % execution time, so we may as well do it in one place. % % First we halve the line length, less a little for the gutter between % the columns. We compute the gutter based on the line length, so it % changes automatically with the paper format. The magic constant % below is chosen so that the gutter has the same value (well, +-<1pt) % as it did when we hard-coded it. % % We put the result in a separate register, \doublecolumhsize, so we % can restore it in \pagesofar, after \hsize itself has (potentially) % been clobbered. % \doublecolumnhsize = \hsize \advance\doublecolumnhsize by -.04154\hsize \divide\doublecolumnhsize by 2 \hsize = \doublecolumnhsize % % Double the \vsize as well. (We don't need a separate register here, % since nobody clobbers \vsize.) \vsize = 2\vsize } % The double-column output routine for all double-column pages except % the last. % \def\doublecolumnout{% \splittopskip=\topskip \splitmaxdepth=\maxdepth % Get the available space for the double columns -- the normal % (undoubled) page height minus any material left over from the % previous page. \dimen@ = \vsize \divide\dimen@ by 2 \advance\dimen@ by -\ht\partialpage % % box0 will be the left-hand column, box2 the right. \setbox0=\vsplit255 to\dimen@ \setbox2=\vsplit255 to\dimen@ \onepageout\pagesofar \unvbox255 \penalty\outputpenalty } % % Re-output the contents of the output page -- any previous material, % followed by the two boxes we just split, in box0 and box2. \def\pagesofar{% \unvbox\partialpage % \hsize = \doublecolumnhsize \wd0=\hsize \wd2=\hsize \hbox to\pagewidth{\box0\hfil\box2}% } % % All done with double columns. \def\enddoublecolumns{% % The following penalty ensures that the page builder is exercised % _before_ we change the output routine. This is necessary in the % following situation: % % The last section of the index consists only of a single entry. % Before this section, \pagetotal is less than \pagegoal, so no % break occurs before the last section starts. However, the last % section, consisting of \initial and the single \entry, does not % fit on the page and has to be broken off. Without the following % penalty the page builder will not be exercised until \eject % below, and by that time we'll already have changed the output % routine to the \balancecolumns version, so the next-to-last % double-column page will be processed with \balancecolumns, which % is wrong: The two columns will go to the main vertical list, with % the broken-off section in the recent contributions. As soon as % the output routine finishes, TeX starts reconsidering the page % break. The two columns and the broken-off section both fit on the % page, because the two columns now take up only half of the page % goal. When TeX sees \eject from below which follows the final % section, it invokes the new output routine that we've set after % \balancecolumns below; \onepageout will try to fit the two columns % and the final section into the vbox of \pageheight (see % \pagebody), causing an overfull box. % % Note that glue won't work here, because glue does not exercise the % page builder, unlike penalties (see The TeXbook, pp. 280-281). \penalty0 % \output = {% % Split the last of the double-column material. Leave it on the % current page, no automatic page break. \balancecolumns % % If we end up splitting too much material for the current page, % though, there will be another page break right after this \output % invocation ends. Having called \balancecolumns once, we do not % want to call it again. Therefore, reset \output to its normal % definition right away. (We hope \balancecolumns will never be % called on to balance too much material, but if it is, this makes % the output somewhat more palatable.) \global\output = {\onepageout{\pagecontents\PAGE}}% }% \eject \endgroup % started in \begindoublecolumns % % \pagegoal was set to the doubled \vsize above, since we restarted % the current page. We're now back to normal single-column % typesetting, so reset \pagegoal to the normal \vsize (after the % \endgroup where \vsize got restored). \pagegoal = \vsize } % % Called at the end of the double column material. \def\balancecolumns{% \setbox0 = \vbox{\unvbox255}% like \box255 but more efficient, see p.120. \dimen@ = \ht0 \advance\dimen@ by \topskip \advance\dimen@ by-\baselineskip \divide\dimen@ by 2 % target to split to %debug\message{final 2-column material height=\the\ht0, target=\the\dimen@.}% \splittopskip = \topskip % Loop until we get a decent breakpoint. {% \vbadness = 10000 \loop \global\setbox3 = \copy0 \global\setbox1 = \vsplit3 to \dimen@ \ifdim\ht3>\dimen@ \global\advance\dimen@ by 1pt \repeat }% %debug\message{split to \the\dimen@, column heights: \the\ht1, \the\ht3.}% \setbox0=\vbox to\dimen@{\unvbox1}% \setbox2=\vbox to\dimen@{\unvbox3}% % \pagesofar } \catcode`\@ = \other \message{sectioning,} % Chapters, sections, etc. % \unnumberedno is an oxymoron, of course. But we count the unnumbered % sections so that we can refer to them unambiguously in the pdf % outlines by their "section number". We avoid collisions with chapter % numbers by starting them at 10000. (If a document ever has 10000 % chapters, we're in trouble anyway, I'm sure.) \newcount\unnumberedno \unnumberedno = 10000 \newcount\chapno \newcount\secno \secno=0 \newcount\subsecno \subsecno=0 \newcount\subsubsecno \subsubsecno=0 % This counter is funny since it counts through charcodes of letters A, B, ... \newcount\appendixno \appendixno = `\@ % % \def\appendixletter{\char\the\appendixno} % We do the following ugly conditional instead of the above simple % construct for the sake of pdftex, which needs the actual % letter in the expansion, not just typeset. % \def\appendixletter{% \ifnum\appendixno=`A A% \else\ifnum\appendixno=`B B% \else\ifnum\appendixno=`C C% \else\ifnum\appendixno=`D D% \else\ifnum\appendixno=`E E% \else\ifnum\appendixno=`F F% \else\ifnum\appendixno=`G G% \else\ifnum\appendixno=`H H% \else\ifnum\appendixno=`I I% \else\ifnum\appendixno=`J J% \else\ifnum\appendixno=`K K% \else\ifnum\appendixno=`L L% \else\ifnum\appendixno=`M M% \else\ifnum\appendixno=`N N% \else\ifnum\appendixno=`O O% \else\ifnum\appendixno=`P P% \else\ifnum\appendixno=`Q Q% \else\ifnum\appendixno=`R R% \else\ifnum\appendixno=`S S% \else\ifnum\appendixno=`T T% \else\ifnum\appendixno=`U U% \else\ifnum\appendixno=`V V% \else\ifnum\appendixno=`W W% \else\ifnum\appendixno=`X X% \else\ifnum\appendixno=`Y Y% \else\ifnum\appendixno=`Z Z% % The \the is necessary, despite appearances, because \appendixletter is % expanded while writing the .toc file. \char\appendixno is not % expandable, thus it is written literally, thus all appendixes come out % with the same letter (or @) in the toc without it. \else\char\the\appendixno \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi} % Each @chapter defines these (using marks) as the number+name, number % and name of the chapter. Page headings and footings can use % these. @section does likewise. \def\thischapter{} \def\thischapternum{} \def\thischaptername{} \def\thissection{} \def\thissectionnum{} \def\thissectionname{} \newcount\absseclevel % used to calculate proper heading level \newcount\secbase\secbase=0 % @raisesections/@lowersections modify this count % @raisesections: treat @section as chapter, @subsection as section, etc. \def\raisesections{\global\advance\secbase by -1} \let\up=\raisesections % original BFox name % @lowersections: treat @chapter as section, @section as subsection, etc. \def\lowersections{\global\advance\secbase by 1} \let\down=\lowersections % original BFox name % we only have subsub. \chardef\maxseclevel = 3 % % A numbered section within an unnumbered changes to unnumbered too. % To achieve this, remember the "biggest" unnum. sec. we are currently in: \chardef\unmlevel = \maxseclevel % % Trace whether the current chapter is an appendix or not: % \chapheadtype is "N" or "A", unnumbered chapters are ignored. \def\chapheadtype{N} % Choose a heading macro % #1 is heading type % #2 is heading level % #3 is text for heading \def\genhead#1#2#3{% % Compute the abs. sec. level: \absseclevel=#2 \advance\absseclevel by \secbase % Make sure \absseclevel doesn't fall outside the range: \ifnum \absseclevel < 0 \absseclevel = 0 \else \ifnum \absseclevel > 3 \absseclevel = 3 \fi \fi % The heading type: \def\headtype{#1}% \if \headtype U% \ifnum \absseclevel < \unmlevel \chardef\unmlevel = \absseclevel \fi \else % Check for appendix sections: \ifnum \absseclevel = 0 \edef\chapheadtype{\headtype}% \else \if \headtype A\if \chapheadtype N% \errmessage{@appendix... within a non-appendix chapter}% \fi\fi \fi % Check for numbered within unnumbered: \ifnum \absseclevel > \unmlevel \def\headtype{U}% \else \chardef\unmlevel = 3 \fi \fi % Now print the heading: \if \headtype U% \ifcase\absseclevel \unnumberedzzz{#3}% \or \unnumberedseczzz{#3}% \or \unnumberedsubseczzz{#3}% \or \unnumberedsubsubseczzz{#3}% \fi \else \if \headtype A% \ifcase\absseclevel \appendixzzz{#3}% \or \appendixsectionzzz{#3}% \or \appendixsubseczzz{#3}% \or \appendixsubsubseczzz{#3}% \fi \else \ifcase\absseclevel \chapterzzz{#3}% \or \seczzz{#3}% \or \numberedsubseczzz{#3}% \or \numberedsubsubseczzz{#3}% \fi \fi \fi \suppressfirstparagraphindent } % an interface: \def\numhead{\genhead N} \def\apphead{\genhead A} \def\unnmhead{\genhead U} % @chapter, @appendix, @unnumbered. Increment top-level counter, reset % all lower-level sectioning counters to zero. % % Also set \chaplevelprefix, which we prepend to @float sequence numbers % (e.g., figures), q.v. By default (before any chapter), that is empty. \let\chaplevelprefix = \empty % \outer\parseargdef\chapter{\numhead0{#1}} % normally numhead0 calls chapterzzz \def\chapterzzz#1{% % section resetting is \global in case the chapter is in a group, such % as an @include file. \global\secno=0 \global\subsecno=0 \global\subsubsecno=0 \global\advance\chapno by 1 % % Used for \float. \gdef\chaplevelprefix{\the\chapno.}% \resetallfloatnos % \message{\putwordChapter\space \the\chapno}% % % Write the actual heading. \chapmacro{#1}{Ynumbered}{\the\chapno}% % % So @section and the like are numbered underneath this chapter. \global\let\section = \numberedsec \global\let\subsection = \numberedsubsec \global\let\subsubsection = \numberedsubsubsec } \outer\parseargdef\appendix{\apphead0{#1}} % normally apphead0 calls appendixzzz \def\appendixzzz#1{% \global\secno=0 \global\subsecno=0 \global\subsubsecno=0 \global\advance\appendixno by 1 \gdef\chaplevelprefix{\appendixletter.}% \resetallfloatnos % \def\appendixnum{\putwordAppendix\space \appendixletter}% \message{\appendixnum}% % \chapmacro{#1}{Yappendix}{\appendixletter}% % \global\let\section = \appendixsec \global\let\subsection = \appendixsubsec \global\let\subsubsection = \appendixsubsubsec } \outer\parseargdef\unnumbered{\unnmhead0{#1}} % normally unnmhead0 calls unnumberedzzz \def\unnumberedzzz#1{% \global\secno=0 \global\subsecno=0 \global\subsubsecno=0 \global\advance\unnumberedno by 1 % % Since an unnumbered has no number, no prefix for figures. \global\let\chaplevelprefix = \empty \resetallfloatnos % % This used to be simply \message{#1}, but TeX fully expands the % argument to \message. Therefore, if #1 contained @-commands, TeX % expanded them. For example, in `@unnumbered The @cite{Book}', TeX % expanded @cite (which turns out to cause errors because \cite is meant % to be executed, not expanded). % % Anyway, we don't want the fully-expanded definition of @cite to appear % as a result of the \message, we just want `@cite' itself. We use % \the to achieve this: TeX expands \the only once, % simply yielding the contents of . (We also do this for % the toc entries.) \toks0 = {#1}% \message{(\the\toks0)}% % \chapmacro{#1}{Ynothing}{\the\unnumberedno}% % \global\let\section = \unnumberedsec \global\let\subsection = \unnumberedsubsec \global\let\subsubsection = \unnumberedsubsubsec } % @centerchap is like @unnumbered, but the heading is centered. \outer\parseargdef\centerchap{% % Well, we could do the following in a group, but that would break % an assumption that \chapmacro is called at the outermost level. % Thus we are safer this way: --kasal, 24feb04 \let\centerparametersmaybe = \centerparameters \unnmhead0{#1}% \let\centerparametersmaybe = \relax } % @top is like @unnumbered. \let\top\unnumbered % Sections. \outer\parseargdef\numberedsec{\numhead1{#1}} % normally calls seczzz \def\seczzz#1{% \global\subsecno=0 \global\subsubsecno=0 \global\advance\secno by 1 \sectionheading{#1}{sec}{Ynumbered}{\the\chapno.\the\secno}% } \outer\parseargdef\appendixsection{\apphead1{#1}} % normally calls appendixsectionzzz \def\appendixsectionzzz#1{% \global\subsecno=0 \global\subsubsecno=0 \global\advance\secno by 1 \sectionheading{#1}{sec}{Yappendix}{\appendixletter.\the\secno}% } \let\appendixsec\appendixsection \outer\parseargdef\unnumberedsec{\unnmhead1{#1}} % normally calls unnumberedseczzz \def\unnumberedseczzz#1{% \global\subsecno=0 \global\subsubsecno=0 \global\advance\secno by 1 \sectionheading{#1}{sec}{Ynothing}{\the\unnumberedno.\the\secno}% } % Subsections. \outer\parseargdef\numberedsubsec{\numhead2{#1}} % normally calls numberedsubseczzz \def\numberedsubseczzz#1{% \global\subsubsecno=0 \global\advance\subsecno by 1 \sectionheading{#1}{subsec}{Ynumbered}{\the\chapno.\the\secno.\the\subsecno}% } \outer\parseargdef\appendixsubsec{\apphead2{#1}} % normally calls appendixsubseczzz \def\appendixsubseczzz#1{% \global\subsubsecno=0 \global\advance\subsecno by 1 \sectionheading{#1}{subsec}{Yappendix}% {\appendixletter.\the\secno.\the\subsecno}% } \outer\parseargdef\unnumberedsubsec{\unnmhead2{#1}} %normally calls unnumberedsubseczzz \def\unnumberedsubseczzz#1{% \global\subsubsecno=0 \global\advance\subsecno by 1 \sectionheading{#1}{subsec}{Ynothing}% {\the\unnumberedno.\the\secno.\the\subsecno}% } % Subsubsections. \outer\parseargdef\numberedsubsubsec{\numhead3{#1}} % normally numberedsubsubseczzz \def\numberedsubsubseczzz#1{% \global\advance\subsubsecno by 1 \sectionheading{#1}{subsubsec}{Ynumbered}% {\the\chapno.\the\secno.\the\subsecno.\the\subsubsecno}% } \outer\parseargdef\appendixsubsubsec{\apphead3{#1}} % normally appendixsubsubseczzz \def\appendixsubsubseczzz#1{% \global\advance\subsubsecno by 1 \sectionheading{#1}{subsubsec}{Yappendix}% {\appendixletter.\the\secno.\the\subsecno.\the\subsubsecno}% } \outer\parseargdef\unnumberedsubsubsec{\unnmhead3{#1}} %normally unnumberedsubsubseczzz \def\unnumberedsubsubseczzz#1{% \global\advance\subsubsecno by 1 \sectionheading{#1}{subsubsec}{Ynothing}% {\the\unnumberedno.\the\secno.\the\subsecno.\the\subsubsecno}% } % These macros control what the section commands do, according % to what kind of chapter we are in (ordinary, appendix, or unnumbered). % Define them by default for a numbered chapter. \let\section = \numberedsec \let\subsection = \numberedsubsec \let\subsubsection = \numberedsubsubsec % Define @majorheading, @heading and @subheading % NOTE on use of \vbox for chapter headings, section headings, and such: % 1) We use \vbox rather than the earlier \line to permit % overlong headings to fold. % 2) \hyphenpenalty is set to 10000 because hyphenation in a % heading is obnoxious; this forbids it. % 3) Likewise, headings look best if no \parindent is used, and % if justification is not attempted. Hence \raggedright. \def\majorheading{% {\advance\chapheadingskip by 10pt \chapbreak }% \parsearg\chapheadingzzz } \def\chapheading{\chapbreak \parsearg\chapheadingzzz} \def\chapheadingzzz#1{% {\chapfonts \vbox{\hyphenpenalty=10000\tolerance=5000 \parindent=0pt\raggedright \rm #1\hfill}}% \bigskip \par\penalty 200\relax \suppressfirstparagraphindent } % @heading, @subheading, @subsubheading. \parseargdef\heading{\sectionheading{#1}{sec}{Yomitfromtoc}{} \suppressfirstparagraphindent} \parseargdef\subheading{\sectionheading{#1}{subsec}{Yomitfromtoc}{} \suppressfirstparagraphindent} \parseargdef\subsubheading{\sectionheading{#1}{subsubsec}{Yomitfromtoc}{} \suppressfirstparagraphindent} % These macros generate a chapter, section, etc. heading only % (including whitespace, linebreaking, etc. around it), % given all the information in convenient, parsed form. %%% Args are the skip and penalty (usually negative) \def\dobreak#1#2{\par\ifdim\lastskip<#1\removelastskip\penalty#2\vskip#1\fi} %%% Define plain chapter starts, and page on/off switching for it % Parameter controlling skip before chapter headings (if needed) \newskip\chapheadingskip \def\chapbreak{\dobreak \chapheadingskip {-4000}} \def\chappager{\par\vfill\supereject} % Because \domark is called before \chapoddpage, the filler page will % get the headings for the next chapter, which is wrong. But we don't % care -- we just disable all headings on the filler page. \def\chapoddpage{% \chappager \ifodd\pageno \else \begingroup \evenheadline={\hfil}\evenfootline={\hfil}% \oddheadline={\hfil}\oddfootline={\hfil}% \hbox to 0pt{}% \chappager \endgroup \fi } \def\setchapternewpage #1 {\csname CHAPPAG#1\endcsname} \def\CHAPPAGoff{% \global\let\contentsalignmacro = \chappager \global\let\pchapsepmacro=\chapbreak \global\let\pagealignmacro=\chappager} \def\CHAPPAGon{% \global\let\contentsalignmacro = \chappager \global\let\pchapsepmacro=\chappager \global\let\pagealignmacro=\chappager \global\def\HEADINGSon{\HEADINGSsingle}} \def\CHAPPAGodd{% \global\let\contentsalignmacro = \chapoddpage \global\let\pchapsepmacro=\chapoddpage \global\let\pagealignmacro=\chapoddpage \global\def\HEADINGSon{\HEADINGSdouble}} \CHAPPAGon % Chapter opening. % % #1 is the text, #2 is the section type (Ynumbered, Ynothing, % Yappendix, Yomitfromtoc), #3 the chapter number. % % To test against our argument. \def\Ynothingkeyword{Ynothing} \def\Yomitfromtockeyword{Yomitfromtoc} \def\Yappendixkeyword{Yappendix} % \def\chapmacro#1#2#3{% % Insert the first mark before the heading break (see notes for \domark). \let\prevchapterdefs=\lastchapterdefs \let\prevsectiondefs=\lastsectiondefs \gdef\lastsectiondefs{\gdef\thissectionname{}\gdef\thissectionnum{}% \gdef\thissection{}}% % \def\temptype{#2}% \ifx\temptype\Ynothingkeyword \gdef\lastchapterdefs{\gdef\thischaptername{#1}\gdef\thischapternum{}% \gdef\thischapter{\thischaptername}}% \else\ifx\temptype\Yomitfromtockeyword \gdef\lastchapterdefs{\gdef\thischaptername{#1}\gdef\thischapternum{}% \gdef\thischapter{}}% \else\ifx\temptype\Yappendixkeyword \toks0={#1}% \xdef\lastchapterdefs{% \gdef\noexpand\thischaptername{\the\toks0}% \gdef\noexpand\thischapternum{\appendixletter}% \gdef\noexpand\thischapter{\putwordAppendix{} \noexpand\thischapternum: \noexpand\thischaptername}% }% \else \toks0={#1}% \xdef\lastchapterdefs{% \gdef\noexpand\thischaptername{\the\toks0}% \gdef\noexpand\thischapternum{\the\chapno}% \gdef\noexpand\thischapter{\putwordChapter{} \noexpand\thischapternum: \noexpand\thischaptername}% }% \fi\fi\fi % % Output the mark. Pass it through \safewhatsit, to take care of % the preceding space. \safewhatsit\domark % % Insert the chapter heading break. \pchapsepmacro % % Now the second mark, after the heading break. No break points % between here and the heading. \let\prevchapterdefs=\lastchapterdefs \let\prevsectiondefs=\lastsectiondefs \domark % {% \chapfonts \rm % % Have to define \lastsection before calling \donoderef, because the % xref code eventually uses it. On the other hand, it has to be called % after \pchapsepmacro, or the headline will change too soon. \gdef\lastsection{#1}% % % Only insert the separating space if we have a chapter/appendix % number, and don't print the unnumbered ``number''. \ifx\temptype\Ynothingkeyword \setbox0 = \hbox{}% \def\toctype{unnchap}% \else\ifx\temptype\Yomitfromtockeyword \setbox0 = \hbox{}% contents like unnumbered, but no toc entry \def\toctype{omit}% \else\ifx\temptype\Yappendixkeyword \setbox0 = \hbox{\putwordAppendix{} #3\enspace}% \def\toctype{app}% \else \setbox0 = \hbox{#3\enspace}% \def\toctype{numchap}% \fi\fi\fi % % Write the toc entry for this chapter. Must come before the % \donoderef, because we include the current node name in the toc % entry, and \donoderef resets it to empty. \writetocentry{\toctype}{#1}{#3}% % % For pdftex, we have to write out the node definition (aka, make % the pdfdest) after any page break, but before the actual text has % been typeset. If the destination for the pdf outline is after the % text, then jumping from the outline may wind up with the text not % being visible, for instance under high magnification. \donoderef{#2}% % % Typeset the actual heading. \nobreak % Avoid page breaks at the interline glue. \vbox{\hyphenpenalty=10000 \tolerance=5000 \parindent=0pt \raggedright \hangindent=\wd0 \centerparametersmaybe \unhbox0 #1\par}% }% \nobreak\bigskip % no page break after a chapter title \nobreak } % @centerchap -- centered and unnumbered. \let\centerparametersmaybe = \relax \def\centerparameters{% \advance\rightskip by 3\rightskip \leftskip = \rightskip \parfillskip = 0pt } % I don't think this chapter style is supported any more, so I'm not % updating it with the new noderef stuff. We'll see. --karl, 11aug03. % \def\setchapterstyle #1 {\csname CHAPF#1\endcsname} % \def\unnchfopen #1{% \chapoddpage {\chapfonts \vbox{\hyphenpenalty=10000\tolerance=5000 \parindent=0pt\raggedright \rm #1\hfill}}\bigskip \par\nobreak } \def\chfopen #1#2{\chapoddpage {\chapfonts \vbox to 3in{\vfil \hbox to\hsize{\hfil #2} \hbox to\hsize{\hfil #1} \vfil}}% \par\penalty 5000 % } \def\centerchfopen #1{% \chapoddpage {\chapfonts \vbox{\hyphenpenalty=10000\tolerance=5000 \parindent=0pt \hfill {\rm #1}\hfill}}\bigskip \par\nobreak } \def\CHAPFopen{% \global\let\chapmacro=\chfopen \global\let\centerchapmacro=\centerchfopen} % Section titles. These macros combine the section number parts and % call the generic \sectionheading to do the printing. % \newskip\secheadingskip \def\secheadingbreak{\dobreak \secheadingskip{-1000}} % Subsection titles. \newskip\subsecheadingskip \def\subsecheadingbreak{\dobreak \subsecheadingskip{-500}} % Subsubsection titles. \def\subsubsecheadingskip{\subsecheadingskip} \def\subsubsecheadingbreak{\subsecheadingbreak} % Print any size, any type, section title. % % #1 is the text, #2 is the section level (sec/subsec/subsubsec), #3 is % the section type for xrefs (Ynumbered, Ynothing, Yappendix), #4 is the % section number. % \def\seckeyword{sec} % \def\sectionheading#1#2#3#4{% {% % Switch to the right set of fonts. \csname #2fonts\endcsname \rm % \def\sectionlevel{#2}% \def\temptype{#3}% % % Insert first mark before the heading break (see notes for \domark). \let\prevsectiondefs=\lastsectiondefs \ifx\temptype\Ynothingkeyword \ifx\sectionlevel\seckeyword \gdef\lastsectiondefs{\gdef\thissectionname{#1}\gdef\thissectionnum{}% \gdef\thissection{\thissectionname}}% \fi \else\ifx\temptype\Yomitfromtockeyword % Don't redefine \thissection. \else\ifx\temptype\Yappendixkeyword \ifx\sectionlevel\seckeyword \toks0={#1}% \xdef\lastsectiondefs{% \gdef\noexpand\thissectionname{\the\toks0}% \gdef\noexpand\thissectionnum{#4}% \gdef\noexpand\thissection{\putwordSection{} \noexpand\thissectionnum: \noexpand\thissectionname}% }% \fi \else \ifx\sectionlevel\seckeyword \toks0={#1}% \xdef\lastsectiondefs{% \gdef\noexpand\thissectionname{\the\toks0}% \gdef\noexpand\thissectionnum{#4}% \gdef\noexpand\thissection{\putwordSection{} \noexpand\thissectionnum: \noexpand\thissectionname}% }% \fi \fi\fi\fi % % Output the mark. Pass it through \safewhatsit, to take care of % the preceding space. \safewhatsit\domark % % Insert space above the heading. \csname #2headingbreak\endcsname % % Now the second mark, after the heading break. No break points % between here and the heading. \let\prevsectiondefs=\lastsectiondefs \domark % % Only insert the space after the number if we have a section number. \ifx\temptype\Ynothingkeyword \setbox0 = \hbox{}% \def\toctype{unn}% \gdef\lastsection{#1}% \else\ifx\temptype\Yomitfromtockeyword % for @headings -- no section number, don't include in toc, % and don't redefine \lastsection. \setbox0 = \hbox{}% \def\toctype{omit}% \let\sectionlevel=\empty \else\ifx\temptype\Yappendixkeyword \setbox0 = \hbox{#4\enspace}% \def\toctype{app}% \gdef\lastsection{#1}% \else \setbox0 = \hbox{#4\enspace}% \def\toctype{num}% \gdef\lastsection{#1}% \fi\fi\fi % % Write the toc entry (before \donoderef). See comments in \chapmacro. \writetocentry{\toctype\sectionlevel}{#1}{#4}% % % Write the node reference (= pdf destination for pdftex). % Again, see comments in \chapmacro. \donoderef{#3}% % % Interline glue will be inserted when the vbox is completed. % That glue will be a valid breakpoint for the page, since it'll be % preceded by a whatsit (usually from the \donoderef, or from the % \writetocentry if there was no node). We don't want to allow that % break, since then the whatsits could end up on page n while the % section is on page n+1, thus toc/etc. are wrong. Debian bug 276000. \nobreak % % Output the actual section heading. \vbox{\hyphenpenalty=10000 \tolerance=5000 \parindent=0pt \raggedright \hangindent=\wd0 % zero if no section number \unhbox0 #1}% }% % Add extra space after the heading -- half of whatever came above it. % Don't allow stretch, though. \kern .5 \csname #2headingskip\endcsname % % Do not let the kern be a potential breakpoint, as it would be if it % was followed by glue. \nobreak % % We'll almost certainly start a paragraph next, so don't let that % glue accumulate. (Not a breakpoint because it's preceded by a % discardable item.) \vskip-\parskip % % This is purely so the last item on the list is a known \penalty > % 10000. This is so \startdefun can avoid allowing breakpoints after % section headings. Otherwise, it would insert a valid breakpoint between: % % @section sec-whatever % @deffn def-whatever \penalty 10001 } \message{toc,} % Table of contents. \newwrite\tocfile % Write an entry to the toc file, opening it if necessary. % Called from @chapter, etc. % % Example usage: \writetocentry{sec}{Section Name}{\the\chapno.\the\secno} % We append the current node name (if any) and page number as additional % arguments for the \{chap,sec,...}entry macros which will eventually % read this. The node name is used in the pdf outlines as the % destination to jump to. % % We open the .toc file for writing here instead of at @setfilename (or % any other fixed time) so that @contents can be anywhere in the document. % But if #1 is `omit', then we don't do anything. This is used for the % table of contents chapter openings themselves. % \newif\iftocfileopened \def\omitkeyword{omit}% % \def\writetocentry#1#2#3{% \edef\writetoctype{#1}% \ifx\writetoctype\omitkeyword \else \iftocfileopened\else \immediate\openout\tocfile = \jobname.toc \global\tocfileopenedtrue \fi % \iflinks {\atdummies \edef\temp{% \write\tocfile{@#1entry{#2}{#3}{\lastnode}{\noexpand\folio}}}% \temp }% \fi \fi % % Tell \shipout to create a pdf destination on each page, if we're % writing pdf. These are used in the table of contents. We can't % just write one on every page because the title pages are numbered % 1 and 2 (the page numbers aren't printed), and so are the first % two pages of the document. Thus, we'd have two destinations named % `1', and two named `2'. \ifpdf \global\pdfmakepagedesttrue \fi } % These characters do not print properly in the Computer Modern roman % fonts, so we must take special care. This is more or less redundant % with the Texinfo input format setup at the end of this file. % \def\activecatcodes{% \catcode`\"=\active \catcode`\$=\active \catcode`\<=\active \catcode`\>=\active \catcode`\\=\active \catcode`\^=\active \catcode`\_=\active \catcode`\|=\active \catcode`\~=\active } % Read the toc file, which is essentially Texinfo input. \def\readtocfile{% \setupdatafile \activecatcodes \input \tocreadfilename } \newskip\contentsrightmargin \contentsrightmargin=1in \newcount\savepageno \newcount\lastnegativepageno \lastnegativepageno = -1 % Prepare to read what we've written to \tocfile. % \def\startcontents#1{% % If @setchapternewpage on, and @headings double, the contents should % start on an odd page, unlike chapters. Thus, we maintain % \contentsalignmacro in parallel with \pagealignmacro. % From: Torbjorn Granlund \contentsalignmacro \immediate\closeout\tocfile % % Don't need to put `Contents' or `Short Contents' in the headline. % It is abundantly clear what they are. \chapmacro{#1}{Yomitfromtoc}{}% % \savepageno = \pageno \begingroup % Set up to handle contents files properly. \raggedbottom % Worry more about breakpoints than the bottom. \advance\hsize by -\contentsrightmargin % Don't use the full line length. % % Roman numerals for page numbers. \ifnum \pageno>0 \global\pageno = \lastnegativepageno \fi } % redefined for the two-volume lispref. We always output on % \jobname.toc even if this is redefined. % \def\tocreadfilename{\jobname.toc} % Normal (long) toc. % \def\contents{% \startcontents{\putwordTOC}% \openin 1 \tocreadfilename\space \ifeof 1 \else \readtocfile \fi \vfill \eject \contentsalignmacro % in case @setchapternewpage odd is in effect \ifeof 1 \else \pdfmakeoutlines \fi \closein 1 \endgroup \lastnegativepageno = \pageno \global\pageno = \savepageno } % And just the chapters. \def\summarycontents{% \startcontents{\putwordShortTOC}% % \let\numchapentry = \shortchapentry \let\appentry = \shortchapentry \let\unnchapentry = \shortunnchapentry % We want a true roman here for the page numbers. \secfonts \let\rm=\shortcontrm \let\bf=\shortcontbf \let\sl=\shortcontsl \let\tt=\shortconttt \rm \hyphenpenalty = 10000 \advance\baselineskip by 1pt % Open it up a little. \def\numsecentry##1##2##3##4{} \let\appsecentry = \numsecentry \let\unnsecentry = \numsecentry \let\numsubsecentry = \numsecentry \let\appsubsecentry = \numsecentry \let\unnsubsecentry = \numsecentry \let\numsubsubsecentry = \numsecentry \let\appsubsubsecentry = \numsecentry \let\unnsubsubsecentry = \numsecentry \openin 1 \tocreadfilename\space \ifeof 1 \else \readtocfile \fi \closein 1 \vfill \eject \contentsalignmacro % in case @setchapternewpage odd is in effect \endgroup \lastnegativepageno = \pageno \global\pageno = \savepageno } \let\shortcontents = \summarycontents % Typeset the label for a chapter or appendix for the short contents. % The arg is, e.g., `A' for an appendix, or `3' for a chapter. % \def\shortchaplabel#1{% % This space should be enough, since a single number is .5em, and the % widest letter (M) is 1em, at least in the Computer Modern fonts. % But use \hss just in case. % (This space doesn't include the extra space that gets added after % the label; that gets put in by \shortchapentry above.) % % We'd like to right-justify chapter numbers, but that looks strange % with appendix letters. And right-justifying numbers and % left-justifying letters looks strange when there is less than 10 % chapters. Have to read the whole toc once to know how many chapters % there are before deciding ... \hbox to 1em{#1\hss}% } % These macros generate individual entries in the table of contents. % The first argument is the chapter or section name. % The last argument is the page number. % The arguments in between are the chapter number, section number, ... % Chapters, in the main contents. \def\numchapentry#1#2#3#4{\dochapentry{#2\labelspace#1}{#4}} % % Chapters, in the short toc. % See comments in \dochapentry re vbox and related settings. \def\shortchapentry#1#2#3#4{% \tocentry{\shortchaplabel{#2}\labelspace #1}{\doshortpageno\bgroup#4\egroup}% } % Appendices, in the main contents. % Need the word Appendix, and a fixed-size box. % \def\appendixbox#1{% % We use M since it's probably the widest letter. \setbox0 = \hbox{\putwordAppendix{} M}% \hbox to \wd0{\putwordAppendix{} #1\hss}} % \def\appentry#1#2#3#4{\dochapentry{\appendixbox{#2}\labelspace#1}{#4}} % Unnumbered chapters. \def\unnchapentry#1#2#3#4{\dochapentry{#1}{#4}} \def\shortunnchapentry#1#2#3#4{\tocentry{#1}{\doshortpageno\bgroup#4\egroup}} % Sections. \def\numsecentry#1#2#3#4{\dosecentry{#2\labelspace#1}{#4}} \let\appsecentry=\numsecentry \def\unnsecentry#1#2#3#4{\dosecentry{#1}{#4}} % Subsections. \def\numsubsecentry#1#2#3#4{\dosubsecentry{#2\labelspace#1}{#4}} \let\appsubsecentry=\numsubsecentry \def\unnsubsecentry#1#2#3#4{\dosubsecentry{#1}{#4}} % And subsubsections. \def\numsubsubsecentry#1#2#3#4{\dosubsubsecentry{#2\labelspace#1}{#4}} \let\appsubsubsecentry=\numsubsubsecentry \def\unnsubsubsecentry#1#2#3#4{\dosubsubsecentry{#1}{#4}} % This parameter controls the indentation of the various levels. % Same as \defaultparindent. \newdimen\tocindent \tocindent = 15pt % Now for the actual typesetting. In all these, #1 is the text and #2 is the % page number. % % If the toc has to be broken over pages, we want it to be at chapters % if at all possible; hence the \penalty. \def\dochapentry#1#2{% \penalty-300 \vskip1\baselineskip plus.33\baselineskip minus.25\baselineskip \begingroup \chapentryfonts \tocentry{#1}{\dopageno\bgroup#2\egroup}% \endgroup \nobreak\vskip .25\baselineskip plus.1\baselineskip } \def\dosecentry#1#2{\begingroup \secentryfonts \leftskip=\tocindent \tocentry{#1}{\dopageno\bgroup#2\egroup}% \endgroup} \def\dosubsecentry#1#2{\begingroup \subsecentryfonts \leftskip=2\tocindent \tocentry{#1}{\dopageno\bgroup#2\egroup}% \endgroup} \def\dosubsubsecentry#1#2{\begingroup \subsubsecentryfonts \leftskip=3\tocindent \tocentry{#1}{\dopageno\bgroup#2\egroup}% \endgroup} % We use the same \entry macro as for the index entries. \let\tocentry = \entry % Space between chapter (or whatever) number and the title. \def\labelspace{\hskip1em \relax} \def\dopageno#1{{\rm #1}} \def\doshortpageno#1{{\rm #1}} \def\chapentryfonts{\secfonts \rm} \def\secentryfonts{\textfonts} \def\subsecentryfonts{\textfonts} \def\subsubsecentryfonts{\textfonts} \message{environments,} % @foo ... @end foo. % @point{}, @result{}, @expansion{}, @print{}, @equiv{}. % % Since these characters are used in examples, they should be an even number of % \tt widths. Each \tt character is 1en, so two makes it 1em. % \def\point{$\star$} \def\arrow{\leavevmode\raise.05ex\hbox to 1em{\hfil$\rightarrow$\hfil}} \def\result{\leavevmode\raise.05ex\hbox to 1em{\hfil$\Rightarrow$\hfil}} \def\expansion{\leavevmode\hbox to 1em{\hfil$\mapsto$\hfil}} \def\print{\leavevmode\lower.1ex\hbox to 1em{\hfil$\dashv$\hfil}} \def\equiv{\leavevmode\hbox to 1em{\hfil$\ptexequiv$\hfil}} % The @error{} command. % Adapted from the TeXbook's \boxit. % \newbox\errorbox % {\tentt \global\dimen0 = 3em}% Width of the box. \dimen2 = .55pt % Thickness of rules % The text. (`r' is open on the right, `e' somewhat less so on the left.) \setbox0 = \hbox{\kern-.75pt \reducedsf error\kern-1.5pt} % \setbox\errorbox=\hbox to \dimen0{\hfil \hsize = \dimen0 \advance\hsize by -5.8pt % Space to left+right. \advance\hsize by -2\dimen2 % Rules. \vbox{% \hrule height\dimen2 \hbox{\vrule width\dimen2 \kern3pt % Space to left of text. \vtop{\kern2.4pt \box0 \kern2.4pt}% Space above/below. \kern3pt\vrule width\dimen2}% Space to right. \hrule height\dimen2} \hfil} % \def\error{\leavevmode\lower.7ex\copy\errorbox} % @tex ... @end tex escapes into raw Tex temporarily. % One exception: @ is still an escape character, so that @end tex works. % But \@ or @@ will get a plain tex @ character. \envdef\tex{% \catcode `\\=0 \catcode `\{=1 \catcode `\}=2 \catcode `\$=3 \catcode `\&=4 \catcode `\#=6 \catcode `\^=7 \catcode `\_=8 \catcode `\~=\active \let~=\tie \catcode `\%=14 \catcode `\+=\other \catcode `\"=\other \catcode `\|=\other \catcode `\<=\other \catcode `\>=\other \escapechar=`\\ % \let\b=\ptexb \let\bullet=\ptexbullet \let\c=\ptexc \let\,=\ptexcomma \let\.=\ptexdot \let\dots=\ptexdots \let\equiv=\ptexequiv \let\!=\ptexexclam \let\i=\ptexi \let\indent=\ptexindent \let\noindent=\ptexnoindent \let\{=\ptexlbrace \let\+=\tabalign \let\}=\ptexrbrace \let\/=\ptexslash \let\*=\ptexstar \let\t=\ptext \expandafter \let\csname top\endcsname=\ptextop % outer \let\frenchspacing=\plainfrenchspacing % \def\endldots{\mathinner{\ldots\ldots\ldots\ldots}}% \def\enddots{\relax\ifmmode\endldots\else$\mathsurround=0pt \endldots\,$\fi}% \def\@{@}% } % There is no need to define \Etex. % Define @lisp ... @end lisp. % @lisp environment forms a group so it can rebind things, % including the definition of @end lisp (which normally is erroneous). % Amount to narrow the margins by for @lisp. \newskip\lispnarrowing \lispnarrowing=0.4in % This is the definition that ^^M gets inside @lisp, @example, and other % such environments. \null is better than a space, since it doesn't % have any width. \def\lisppar{\null\endgraf} % This space is always present above and below environments. \newskip\envskipamount \envskipamount = 0pt % Make spacing and below environment symmetrical. We use \parskip here % to help in doing that, since in @example-like environments \parskip % is reset to zero; thus the \afterenvbreak inserts no space -- but the % start of the next paragraph will insert \parskip. % \def\aboveenvbreak{{% % =10000 instead of <10000 because of a special case in \itemzzz and % \sectionheading, q.v. \ifnum \lastpenalty=10000 \else \advance\envskipamount by \parskip \endgraf \ifdim\lastskip<\envskipamount \removelastskip % it's not a good place to break if the last penalty was \nobreak % or better ... \ifnum\lastpenalty<10000 \penalty-50 \fi \vskip\envskipamount \fi \fi }} \let\afterenvbreak = \aboveenvbreak % \nonarrowing is a flag. If "set", @lisp etc don't narrow margins; it will % also clear it, so that its embedded environments do the narrowing again. \let\nonarrowing=\relax % @cartouche ... @end cartouche: draw rectangle w/rounded corners around % environment contents. \font\circle=lcircle10 \newdimen\circthick \newdimen\cartouter\newdimen\cartinner \newskip\normbskip\newskip\normpskip\newskip\normlskip \circthick=\fontdimen8\circle % \def\ctl{{\circle\char'013\hskip -6pt}}% 6pt from pl file: 1/2charwidth \def\ctr{{\hskip 6pt\circle\char'010}} \def\cbl{{\circle\char'012\hskip -6pt}} \def\cbr{{\hskip 6pt\circle\char'011}} \def\carttop{\hbox to \cartouter{\hskip\lskip \ctl\leaders\hrule height\circthick\hfil\ctr \hskip\rskip}} \def\cartbot{\hbox to \cartouter{\hskip\lskip \cbl\leaders\hrule height\circthick\hfil\cbr \hskip\rskip}} % \newskip\lskip\newskip\rskip \envdef\cartouche{% \ifhmode\par\fi % can't be in the midst of a paragraph. \startsavinginserts \lskip=\leftskip \rskip=\rightskip \leftskip=0pt\rightskip=0pt % we want these *outside*. \cartinner=\hsize \advance\cartinner by-\lskip \advance\cartinner by-\rskip \cartouter=\hsize \advance\cartouter by 18.4pt % allow for 3pt kerns on either % side, and for 6pt waste from % each corner char, and rule thickness \normbskip=\baselineskip \normpskip=\parskip \normlskip=\lineskip % Flag to tell @lisp, etc., not to narrow margin. \let\nonarrowing = t% \vbox\bgroup \baselineskip=0pt\parskip=0pt\lineskip=0pt \carttop \hbox\bgroup \hskip\lskip \vrule\kern3pt \vbox\bgroup \kern3pt \hsize=\cartinner \baselineskip=\normbskip \lineskip=\normlskip \parskip=\normpskip \vskip -\parskip \comment % For explanation, see the end of \def\group. } \def\Ecartouche{% \ifhmode\par\fi \kern3pt \egroup \kern3pt\vrule \hskip\rskip \egroup \cartbot \egroup \checkinserts } % This macro is called at the beginning of all the @example variants, % inside a group. \def\nonfillstart{% \aboveenvbreak \hfuzz = 12pt % Don't be fussy \sepspaces % Make spaces be word-separators rather than space tokens. \let\par = \lisppar % don't ignore blank lines \obeylines % each line of input is a line of output \parskip = 0pt \parindent = 0pt \emergencystretch = 0pt % don't try to avoid overfull boxes \ifx\nonarrowing\relax \advance \leftskip by \lispnarrowing \exdentamount=\lispnarrowing \else \let\nonarrowing = \relax \fi \let\exdent=\nofillexdent } % If you want all examples etc. small: @set dispenvsize small. % If you want even small examples the full size: @set dispenvsize nosmall. % This affects the following displayed environments: % @example, @display, @format, @lisp % \def\smallword{small} \def\nosmallword{nosmall} \let\SETdispenvsize\relax \def\setnormaldispenv{% \ifx\SETdispenvsize\smallword % end paragraph for sake of leading, in case document has no blank % line. This is redundant with what happens in \aboveenvbreak, but % we need to do it before changing the fonts, and it's inconvenient % to change the fonts afterward. \ifnum \lastpenalty=10000 \else \endgraf \fi \smallexamplefonts \rm \fi } \def\setsmalldispenv{% \ifx\SETdispenvsize\nosmallword \else \ifnum \lastpenalty=10000 \else \endgraf \fi \smallexamplefonts \rm \fi } % We often define two environments, @foo and @smallfoo. % Let's do it by one command: \def\makedispenv #1#2{ \expandafter\envdef\csname#1\endcsname {\setnormaldispenv #2} \expandafter\envdef\csname small#1\endcsname {\setsmalldispenv #2} \expandafter\let\csname E#1\endcsname \afterenvbreak \expandafter\let\csname Esmall#1\endcsname \afterenvbreak } % Define two synonyms: \def\maketwodispenvs #1#2#3{ \makedispenv{#1}{#3} \makedispenv{#2}{#3} } % @lisp: indented, narrowed, typewriter font; @example: same as @lisp. % % @smallexample and @smalllisp: use smaller fonts. % Originally contributed by Pavel@xerox. % \maketwodispenvs {lisp}{example}{% \nonfillstart \tt\quoteexpand \let\kbdfont = \kbdexamplefont % Allow @kbd to do something special. \gobble % eat return } % @display/@smalldisplay: same as @lisp except keep current font. % \makedispenv {display}{% \nonfillstart \gobble } % @format/@smallformat: same as @display except don't narrow margins. % \makedispenv{format}{% \let\nonarrowing = t% \nonfillstart \gobble } % @flushleft: same as @format, but doesn't obey \SETdispenvsize. \envdef\flushleft{% \let\nonarrowing = t% \nonfillstart \gobble } \let\Eflushleft = \afterenvbreak % @flushright. % \envdef\flushright{% \let\nonarrowing = t% \nonfillstart \advance\leftskip by 0pt plus 1fill \gobble } \let\Eflushright = \afterenvbreak % @quotation does normal linebreaking (hence we can't use \nonfillstart) % and narrows the margins. We keep \parskip nonzero in general, since % we're doing normal filling. So, when using \aboveenvbreak and % \afterenvbreak, temporarily make \parskip 0. % \envdef\quotation{% {\parskip=0pt \aboveenvbreak}% because \aboveenvbreak inserts \parskip \parindent=0pt % % @cartouche defines \nonarrowing to inhibit narrowing at next level down. \ifx\nonarrowing\relax \advance\leftskip by \lispnarrowing \advance\rightskip by \lispnarrowing \exdentamount = \lispnarrowing \else \let\nonarrowing = \relax \fi \parsearg\quotationlabel } % We have retained a nonzero parskip for the environment, since we're % doing normal filling. % \def\Equotation{% \par \ifx\quotationauthor\undefined\else % indent a bit. \leftline{\kern 2\leftskip \sl ---\quotationauthor}% \fi {\parskip=0pt \afterenvbreak}% } % If we're given an argument, typeset it in bold with a colon after. \def\quotationlabel#1{% \def\temp{#1}% \ifx\temp\empty \else {\bf #1: }% \fi } % LaTeX-like @verbatim...@end verbatim and @verb{...} % If we want to allow any as delimiter, % we need the curly braces so that makeinfo sees the @verb command, eg: % `@verbx...x' would look like the '@verbx' command. --janneke@gnu.org % % [Knuth]: Donald Ervin Knuth, 1996. The TeXbook. % % [Knuth] p.344; only we need to do the other characters Texinfo sets % active too. Otherwise, they get lost as the first character on a % verbatim line. \def\dospecials{% \do\ \do\\\do\{\do\}\do\$\do\&% \do\#\do\^\do\^^K\do\_\do\^^A\do\%\do\~% \do\<\do\>\do\|\do\@\do+\do\"% } % % [Knuth] p. 380 \def\uncatcodespecials{% \def\do##1{\catcode`##1=\other}\dospecials} % % [Knuth] pp. 380,381,391 % Disable Spanish ligatures ?` and !` of \tt font \begingroup \catcode`\`=\active\gdef`{\relax\lq} \endgroup % % Setup for the @verb command. % % Eight spaces for a tab \begingroup \catcode`\^^I=\active \gdef\tabeightspaces{\catcode`\^^I=\active\def^^I{\ \ \ \ \ \ \ \ }} \endgroup % \def\setupverb{% \tt % easiest (and conventionally used) font for verbatim \def\par{\leavevmode\endgraf}% \catcode`\`=\active \tabeightspaces % Respect line breaks, % print special symbols as themselves, and % make each space count % must do in this order: \obeylines \uncatcodespecials \sepspaces } % Setup for the @verbatim environment % % Real tab expansion \newdimen\tabw \setbox0=\hbox{\tt\space} \tabw=8\wd0 % tab amount % \def\starttabbox{\setbox0=\hbox\bgroup} % Allow an option to not replace quotes with a regular directed right % quote/apostrophe (char 0x27), but instead use the undirected quote % from cmtt (char 0x0d). The undirected quote is ugly, so don't make it % the default, but it works for pasting with more pdf viewers (at least % evince), the lilypond developers report. xpdf does work with the % regular 0x27. % \def\codequoteright{% \expandafter\ifx\csname SETtxicodequoteundirected\endcsname\relax \expandafter\ifx\csname SETcodequoteundirected\endcsname\relax '% \else \char'15 \fi \else \char'15 \fi } % % and a similar option for the left quote char vs. a grave accent. % Modern fonts display ASCII 0x60 as a grave accent, so some people like % the code environments to do likewise. % \def\codequoteleft{% \expandafter\ifx\csname SETtxicodequotebacktick\endcsname\relax \expandafter\ifx\csname SETcodequotebacktick\endcsname\relax `% \else \char'22 \fi \else \char'22 \fi } % \begingroup \catcode`\^^I=\active \gdef\tabexpand{% \catcode`\^^I=\active \def^^I{\leavevmode\egroup \dimen0=\wd0 % the width so far, or since the previous tab \divide\dimen0 by\tabw \multiply\dimen0 by\tabw % compute previous multiple of \tabw \advance\dimen0 by\tabw % advance to next multiple of \tabw \wd0=\dimen0 \box0 \starttabbox }% } \catcode`\'=\active \gdef\rquoteexpand{\catcode\rquoteChar=\active \def'{\codequoteright}}% % \catcode`\`=\active \gdef\lquoteexpand{\catcode\lquoteChar=\active \def`{\codequoteleft}}% % \gdef\quoteexpand{\rquoteexpand \lquoteexpand}% \endgroup % start the verbatim environment. \def\setupverbatim{% \let\nonarrowing = t% \nonfillstart % Easiest (and conventionally used) font for verbatim \tt \def\par{\leavevmode\egroup\box0\endgraf}% \catcode`\`=\active \tabexpand \quoteexpand % Respect line breaks, % print special symbols as themselves, and % make each space count % must do in this order: \obeylines \uncatcodespecials \sepspaces \everypar{\starttabbox}% } % Do the @verb magic: verbatim text is quoted by unique % delimiter characters. Before first delimiter expect a % right brace, after last delimiter expect closing brace: % % \def\doverb'{'#1'}'{#1} % % [Knuth] p. 382; only eat outer {} \begingroup \catcode`[=1\catcode`]=2\catcode`\{=\other\catcode`\}=\other \gdef\doverb{#1[\def\next##1#1}[##1\endgroup]\next] \endgroup % \def\verb{\begingroup\setupverb\doverb} % % % Do the @verbatim magic: define the macro \doverbatim so that % the (first) argument ends when '@end verbatim' is reached, ie: % % \def\doverbatim#1@end verbatim{#1} % % For Texinfo it's a lot easier than for LaTeX, % because texinfo's \verbatim doesn't stop at '\end{verbatim}': % we need not redefine '\', '{' and '}'. % % Inspired by LaTeX's verbatim command set [latex.ltx] % \begingroup \catcode`\ =\active \obeylines % % ignore everything up to the first ^^M, that's the newline at the end % of the @verbatim input line itself. Otherwise we get an extra blank % line in the output. \xdef\doverbatim#1^^M#2@end verbatim{#2\noexpand\end\gobble verbatim}% % We really want {...\end verbatim} in the body of the macro, but % without the active space; thus we have to use \xdef and \gobble. \endgroup % \envdef\verbatim{% \setupverbatim\doverbatim } \let\Everbatim = \afterenvbreak % @verbatiminclude FILE - insert text of file in verbatim environment. % \def\verbatiminclude{\parseargusing\filenamecatcodes\doverbatiminclude} % \def\doverbatiminclude#1{% {% \makevalueexpandable \setupverbatim \input #1 \afterenvbreak }% } % @copying ... @end copying. % Save the text away for @insertcopying later. % % We save the uninterpreted tokens, rather than creating a box. % Saving the text in a box would be much easier, but then all the % typesetting commands (@smallbook, font changes, etc.) have to be done % beforehand -- and a) we want @copying to be done first in the source % file; b) letting users define the frontmatter in as flexible order as % possible is very desirable. % \def\copying{\checkenv{}\begingroup\scanargctxt\docopying} \def\docopying#1@end copying{\endgroup\def\copyingtext{#1}} % \def\insertcopying{% \begingroup \parindent = 0pt % paragraph indentation looks wrong on title page \scanexp\copyingtext \endgroup } \message{defuns,} % @defun etc. \newskip\defbodyindent \defbodyindent=.4in \newskip\defargsindent \defargsindent=50pt \newskip\deflastargmargin \deflastargmargin=18pt \newcount\defunpenalty % Start the processing of @deffn: \def\startdefun{% \ifnum\lastpenalty<10000 \medbreak \defunpenalty=10003 % Will keep this @deffn together with the % following @def command, see below. \else % If there are two @def commands in a row, we'll have a \nobreak, % which is there to keep the function description together with its % header. But if there's nothing but headers, we need to allow a % break somewhere. Check specifically for penalty 10002, inserted % by \printdefunline, instead of 10000, since the sectioning % commands also insert a nobreak penalty, and we don't want to allow % a break between a section heading and a defun. % % As a minor refinement, we avoid "club" headers by signalling % with penalty of 10003 after the very first @deffn in the % sequence (see above), and penalty of 10002 after any following % @def command. \ifnum\lastpenalty=10002 \penalty2000 \else \defunpenalty=10002 \fi % % Similarly, after a section heading, do not allow a break. % But do insert the glue. \medskip % preceded by discardable penalty, so not a breakpoint \fi % \parindent=0in \advance\leftskip by \defbodyindent \exdentamount=\defbodyindent } \def\dodefunx#1{% % First, check whether we are in the right environment: \checkenv#1% % % As above, allow line break if we have multiple x headers in a row. % It's not a great place, though. \ifnum\lastpenalty=10002 \penalty3000 \else \defunpenalty=10002 \fi % % And now, it's time to reuse the body of the original defun: \expandafter\gobbledefun#1% } \def\gobbledefun#1\startdefun{} % \printdefunline \deffnheader{text} % \def\printdefunline#1#2{% \begingroup % call \deffnheader: #1#2 \endheader % common ending: \interlinepenalty = 10000 \advance\rightskip by 0pt plus 1fil \endgraf \nobreak\vskip -\parskip \penalty\defunpenalty % signal to \startdefun and \dodefunx % Some of the @defun-type tags do not enable magic parentheses, % rendering the following check redundant. But we don't optimize. \checkparencounts \endgroup } \def\Edefun{\endgraf\medbreak} % \makedefun{deffn} creates \deffn, \deffnx and \Edeffn; % the only thing remaining is to define \deffnheader. % \def\makedefun#1{% \expandafter\let\csname E#1\endcsname = \Edefun \edef\temp{\noexpand\domakedefun \makecsname{#1}\makecsname{#1x}\makecsname{#1header}}% \temp } % \domakedefun \deffn \deffnx \deffnheader % % Define \deffn and \deffnx, without parameters. % \deffnheader has to be defined explicitly. % \def\domakedefun#1#2#3{% \envdef#1{% \startdefun \parseargusing\activeparens{\printdefunline#3}% }% \def#2{\dodefunx#1}% \def#3% } %%% Untyped functions: % @deffn category name args \makedefun{deffn}{\deffngeneral{}} % @deffn category class name args \makedefun{defop}#1 {\defopon{#1\ \putwordon}} % \defopon {category on}class name args \def\defopon#1#2 {\deffngeneral{\putwordon\ \code{#2}}{#1\ \code{#2}} } % \deffngeneral {subind}category name args % \def\deffngeneral#1#2 #3 #4\endheader{% % Remember that \dosubind{fn}{foo}{} is equivalent to \doind{fn}{foo}. \dosubind{fn}{\code{#3}}{#1}% \defname{#2}{}{#3}\magicamp\defunargs{#4\unskip}% } %%% Typed functions: % @deftypefn category type name args \makedefun{deftypefn}{\deftypefngeneral{}} % @deftypeop category class type name args \makedefun{deftypeop}#1 {\deftypeopon{#1\ \putwordon}} % \deftypeopon {category on}class type name args \def\deftypeopon#1#2 {\deftypefngeneral{\putwordon\ \code{#2}}{#1\ \code{#2}} } % \deftypefngeneral {subind}category type name args % \def\deftypefngeneral#1#2 #3 #4 #5\endheader{% \dosubind{fn}{\code{#4}}{#1}% \defname{#2}{#3}{#4}\defunargs{#5\unskip}% } %%% Typed variables: % @deftypevr category type var args \makedefun{deftypevr}{\deftypecvgeneral{}} % @deftypecv category class type var args \makedefun{deftypecv}#1 {\deftypecvof{#1\ \putwordof}} % \deftypecvof {category of}class type var args \def\deftypecvof#1#2 {\deftypecvgeneral{\putwordof\ \code{#2}}{#1\ \code{#2}} } % \deftypecvgeneral {subind}category type var args % \def\deftypecvgeneral#1#2 #3 #4 #5\endheader{% \dosubind{vr}{\code{#4}}{#1}% \defname{#2}{#3}{#4}\defunargs{#5\unskip}% } %%% Untyped variables: % @defvr category var args \makedefun{defvr}#1 {\deftypevrheader{#1} {} } % @defcv category class var args \makedefun{defcv}#1 {\defcvof{#1\ \putwordof}} % \defcvof {category of}class var args \def\defcvof#1#2 {\deftypecvof{#1}#2 {} } %%% Type: % @deftp category name args \makedefun{deftp}#1 #2 #3\endheader{% \doind{tp}{\code{#2}}% \defname{#1}{}{#2}\defunargs{#3\unskip}% } % Remaining @defun-like shortcuts: \makedefun{defun}{\deffnheader{\putwordDeffunc} } \makedefun{defmac}{\deffnheader{\putwordDefmac} } \makedefun{defspec}{\deffnheader{\putwordDefspec} } \makedefun{deftypefun}{\deftypefnheader{\putwordDeffunc} } \makedefun{defvar}{\defvrheader{\putwordDefvar} } \makedefun{defopt}{\defvrheader{\putwordDefopt} } \makedefun{deftypevar}{\deftypevrheader{\putwordDefvar} } \makedefun{defmethod}{\defopon\putwordMethodon} \makedefun{deftypemethod}{\deftypeopon\putwordMethodon} \makedefun{defivar}{\defcvof\putwordInstanceVariableof} \makedefun{deftypeivar}{\deftypecvof\putwordInstanceVariableof} % \defname, which formats the name of the @def (not the args). % #1 is the category, such as "Function". % #2 is the return type, if any. % #3 is the function name. % % We are followed by (but not passed) the arguments, if any. % \def\defname#1#2#3{% % Get the values of \leftskip and \rightskip as they were outside the @def... \advance\leftskip by -\defbodyindent % % How we'll format the type name. Putting it in brackets helps % distinguish it from the body text that may end up on the next line % just below it. \def\temp{#1}% \setbox0=\hbox{\kern\deflastargmargin \ifx\temp\empty\else [\rm\temp]\fi} % % Figure out line sizes for the paragraph shape. % The first line needs space for \box0; but if \rightskip is nonzero, % we need only space for the part of \box0 which exceeds it: \dimen0=\hsize \advance\dimen0 by -\wd0 \advance\dimen0 by \rightskip % The continuations: \dimen2=\hsize \advance\dimen2 by -\defargsindent % (plain.tex says that \dimen1 should be used only as global.) \parshape 2 0in \dimen0 \defargsindent \dimen2 % % Put the type name to the right margin. \noindent \hbox to 0pt{% \hfil\box0 \kern-\hsize % \hsize has to be shortened this way: \kern\leftskip % Intentionally do not respect \rightskip, since we need the space. }% % % Allow all lines to be underfull without complaint: \tolerance=10000 \hbadness=10000 \exdentamount=\defbodyindent {% % defun fonts. We use typewriter by default (used to be bold) because: % . we're printing identifiers, they should be in tt in principle. % . in languages with many accents, such as Czech or French, it's % common to leave accents off identifiers. The result looks ok in % tt, but exceedingly strange in rm. % . we don't want -- and --- to be treated as ligatures. % . this still does not fix the ?` and !` ligatures, but so far no % one has made identifiers using them :). \df \tt \def\temp{#2}% return value type \ifx\temp\empty\else \tclose{\temp} \fi #3% output function name }% {\rm\enskip}% hskip 0.5 em of \tenrm % \boldbrax % arguments will be output next, if any. } % Print arguments in slanted roman (not ttsl), inconsistently with using % tt for the name. This is because literal text is sometimes needed in % the argument list (groff manual), and ttsl and tt are not very % distinguishable. Prevent hyphenation at `-' chars. % \def\defunargs#1{% % use sl by default (not ttsl), % tt for the names. \df \sl \hyphenchar\font=0 % % On the other hand, if an argument has two dashes (for instance), we % want a way to get ttsl. Let's try @var for that. \let\var=\ttslanted #1% \sl\hyphenchar\font=45 } % We want ()&[] to print specially on the defun line. % \def\activeparens{% \catcode`\(=\active \catcode`\)=\active \catcode`\[=\active \catcode`\]=\active \catcode`\&=\active } % Make control sequences which act like normal parenthesis chars. \let\lparen = ( \let\rparen = ) % Be sure that we always have a definition for `(', etc. For example, % if the fn name has parens in it, \boldbrax will not be in effect yet, % so TeX would otherwise complain about undefined control sequence. { \activeparens \global\let(=\lparen \global\let)=\rparen \global\let[=\lbrack \global\let]=\rbrack \global\let& = \& \gdef\boldbrax{\let(=\opnr\let)=\clnr\let[=\lbrb\let]=\rbrb} \gdef\magicamp{\let&=\amprm} } \newcount\parencount % If we encounter &foo, then turn on ()-hacking afterwards \newif\ifampseen \def\amprm#1 {\ampseentrue{\bf\ }} \def\parenfont{% \ifampseen % At the first level, print parens in roman, % otherwise use the default font. \ifnum \parencount=1 \rm \fi \else % The \sf parens (in \boldbrax) actually are a little bolder than % the contained text. This is especially needed for [ and ] . \sf \fi } \def\infirstlevel#1{% \ifampseen \ifnum\parencount=1 #1% \fi \fi } \def\bfafterword#1 {#1 \bf} \def\opnr{% \global\advance\parencount by 1 {\parenfont(}% \infirstlevel \bfafterword } \def\clnr{% {\parenfont)}% \infirstlevel \sl \global\advance\parencount by -1 } \newcount\brackcount \def\lbrb{% \global\advance\brackcount by 1 {\bf[}% } \def\rbrb{% {\bf]}% \global\advance\brackcount by -1 } \def\checkparencounts{% \ifnum\parencount=0 \else \badparencount \fi \ifnum\brackcount=0 \else \badbrackcount \fi } % these should not use \errmessage; the glibc manual, at least, actually % has such constructs (when documenting function pointers). \def\badparencount{% \message{Warning: unbalanced parentheses in @def...}% \global\parencount=0 } \def\badbrackcount{% \message{Warning: unbalanced square brackets in @def...}% \global\brackcount=0 } \message{macros,} % @macro. % To do this right we need a feature of e-TeX, \scantokens, % which we arrange to emulate with a temporary file in ordinary TeX. \ifx\eTeXversion\undefined \newwrite\macscribble \def\scantokens#1{% \toks0={#1}% \immediate\openout\macscribble=\jobname.tmp \immediate\write\macscribble{\the\toks0}% \immediate\closeout\macscribble \input \jobname.tmp } \fi \def\scanmacro#1{% \begingroup \newlinechar`\^^M \let\xeatspaces\eatspaces % Undo catcode changes of \startcontents and \doprintindex % When called from @insertcopying or (short)caption, we need active % backslash to get it printed correctly. Previously, we had % \catcode`\\=\other instead. We'll see whether a problem appears % with macro expansion. --kasal, 19aug04 \catcode`\@=0 \catcode`\\=\active \escapechar=`\@ % ... and \example \spaceisspace % % Append \endinput to make sure that TeX does not see the ending newline. % I've verified that it is necessary both for e-TeX and for ordinary TeX % --kasal, 29nov03 \scantokens{#1\endinput}% \endgroup } \def\scanexp#1{% \edef\temp{\noexpand\scanmacro{#1}}% \temp } \newcount\paramno % Count of parameters \newtoks\macname % Macro name \newif\ifrecursive % Is it recursive? % List of all defined macros in the form % \definedummyword\macro1\definedummyword\macro2... % Currently is also contains all @aliases; the list can be split % if there is a need. \def\macrolist{} % Add the macro to \macrolist \def\addtomacrolist#1{\expandafter \addtomacrolistxxx \csname#1\endcsname} \def\addtomacrolistxxx#1{% \toks0 = \expandafter{\macrolist\definedummyword#1}% \xdef\macrolist{\the\toks0}% } % Utility routines. % This does \let #1 = #2, with \csnames; that is, % \let \csname#1\endcsname = \csname#2\endcsname % (except of course we have to play expansion games). % \def\cslet#1#2{% \expandafter\let \csname#1\expandafter\endcsname \csname#2\endcsname } % Trim leading and trailing spaces off a string. % Concepts from aro-bend problem 15 (see CTAN). {\catcode`\@=11 \gdef\eatspaces #1{\expandafter\trim@\expandafter{#1 }} \gdef\trim@ #1{\trim@@ @#1 @ #1 @ @@} \gdef\trim@@ #1@ #2@ #3@@{\trim@@@\empty #2 @} \def\unbrace#1{#1} \unbrace{\gdef\trim@@@ #1 } #2@{#1} } % Trim a single trailing ^^M off a string. {\catcode`\^^M=\other \catcode`\Q=3% \gdef\eatcr #1{\eatcra #1Q^^MQ}% \gdef\eatcra#1^^MQ{\eatcrb#1Q}% \gdef\eatcrb#1Q#2Q{#1}% } % Macro bodies are absorbed as an argument in a context where % all characters are catcode 10, 11 or 12, except \ which is active % (as in normal texinfo). It is necessary to change the definition of \. % Non-ASCII encodings make 8-bit characters active, so un-activate % them to avoid their expansion. Must do this non-globally, to % confine the change to the current group. % It's necessary to have hard CRs when the macro is executed. This is % done by making ^^M (\endlinechar) catcode 12 when reading the macro % body, and then making it the \newlinechar in \scanmacro. \def\scanctxt{% \catcode`\"=\other \catcode`\+=\other \catcode`\<=\other \catcode`\>=\other \catcode`\@=\other \catcode`\^=\other \catcode`\_=\other \catcode`\|=\other \catcode`\~=\other \ifx\declaredencoding\ascii \else \setnonasciicharscatcodenonglobal\other \fi } \def\scanargctxt{% \scanctxt \catcode`\\=\other \catcode`\^^M=\other } \def\macrobodyctxt{% \scanctxt \catcode`\{=\other \catcode`\}=\other \catcode`\^^M=\other \usembodybackslash } \def\macroargctxt{% \scanctxt \catcode`\\=\other } % \mbodybackslash is the definition of \ in @macro bodies. % It maps \foo\ => \csname macarg.foo\endcsname => #N % where N is the macro parameter number. % We define \csname macarg.\endcsname to be \realbackslash, so % \\ in macro replacement text gets you a backslash. {\catcode`@=0 @catcode`@\=@active @gdef@usembodybackslash{@let\=@mbodybackslash} @gdef@mbodybackslash#1\{@csname macarg.#1@endcsname} } \expandafter\def\csname macarg.\endcsname{\realbackslash} \def\macro{\recursivefalse\parsearg\macroxxx} \def\rmacro{\recursivetrue\parsearg\macroxxx} \def\macroxxx#1{% \getargs{#1}% now \macname is the macname and \argl the arglist \ifx\argl\empty % no arguments \paramno=0% \else \expandafter\parsemargdef \argl;% \fi \if1\csname ismacro.\the\macname\endcsname \message{Warning: redefining \the\macname}% \else \expandafter\ifx\csname \the\macname\endcsname \relax \else \errmessage{Macro name \the\macname\space already defined}\fi \global\cslet{macsave.\the\macname}{\the\macname}% \global\expandafter\let\csname ismacro.\the\macname\endcsname=1% \addtomacrolist{\the\macname}% \fi \begingroup \macrobodyctxt \ifrecursive \expandafter\parsermacbody \else \expandafter\parsemacbody \fi} \parseargdef\unmacro{% \if1\csname ismacro.#1\endcsname \global\cslet{#1}{macsave.#1}% \global\expandafter\let \csname ismacro.#1\endcsname=0% % Remove the macro name from \macrolist: \begingroup \expandafter\let\csname#1\endcsname \relax \let\definedummyword\unmacrodo \xdef\macrolist{\macrolist}% \endgroup \else \errmessage{Macro #1 not defined}% \fi } % Called by \do from \dounmacro on each macro. The idea is to omit any % macro definitions that have been changed to \relax. % \def\unmacrodo#1{% \ifx #1\relax % remove this \else \noexpand\definedummyword \noexpand#1% \fi } % This makes use of the obscure feature that if the last token of a % is #, then the preceding argument is delimited by % an opening brace, and that opening brace is not consumed. \def\getargs#1{\getargsxxx#1{}} \def\getargsxxx#1#{\getmacname #1 \relax\getmacargs} \def\getmacname #1 #2\relax{\macname={#1}} \def\getmacargs#1{\def\argl{#1}} % Parse the optional {params} list. Set up \paramno and \paramlist % so \defmacro knows what to do. Define \macarg.blah for each blah % in the params list, to be ##N where N is the position in that list. % That gets used by \mbodybackslash (above). % We need to get `macro parameter char #' into several definitions. % The technique used is stolen from LaTeX: let \hash be something % unexpandable, insert that wherever you need a #, and then redefine % it to # just before using the token list produced. % % The same technique is used to protect \eatspaces till just before % the macro is used. \def\parsemargdef#1;{\paramno=0\def\paramlist{}% \let\hash\relax\let\xeatspaces\relax\parsemargdefxxx#1,;,} \def\parsemargdefxxx#1,{% \if#1;\let\next=\relax \else \let\next=\parsemargdefxxx \advance\paramno by 1% \expandafter\edef\csname macarg.\eatspaces{#1}\endcsname {\xeatspaces{\hash\the\paramno}}% \edef\paramlist{\paramlist\hash\the\paramno,}% \fi\next} % These two commands read recursive and nonrecursive macro bodies. % (They're different since rec and nonrec macros end differently.) \long\def\parsemacbody#1@end macro% {\xdef\temp{\eatcr{#1}}\endgroup\defmacro}% \long\def\parsermacbody#1@end rmacro% {\xdef\temp{\eatcr{#1}}\endgroup\defmacro}% % This defines the macro itself. There are six cases: recursive and % nonrecursive macros of zero, one, and many arguments. % Much magic with \expandafter here. % \xdef is used so that macro definitions will survive the file % they're defined in; @include reads the file inside a group. \def\defmacro{% \let\hash=##% convert placeholders to macro parameter chars \ifrecursive \ifcase\paramno % 0 \expandafter\xdef\csname\the\macname\endcsname{% \noexpand\scanmacro{\temp}}% \or % 1 \expandafter\xdef\csname\the\macname\endcsname{% \bgroup\noexpand\macroargctxt \noexpand\braceorline \expandafter\noexpand\csname\the\macname xxx\endcsname}% \expandafter\xdef\csname\the\macname xxx\endcsname##1{% \egroup\noexpand\scanmacro{\temp}}% \else % many \expandafter\xdef\csname\the\macname\endcsname{% \bgroup\noexpand\macroargctxt \noexpand\csname\the\macname xx\endcsname}% \expandafter\xdef\csname\the\macname xx\endcsname##1{% \expandafter\noexpand\csname\the\macname xxx\endcsname ##1,}% \expandafter\expandafter \expandafter\xdef \expandafter\expandafter \csname\the\macname xxx\endcsname \paramlist{\egroup\noexpand\scanmacro{\temp}}% \fi \else \ifcase\paramno % 0 \expandafter\xdef\csname\the\macname\endcsname{% \noexpand\norecurse{\the\macname}% \noexpand\scanmacro{\temp}\egroup}% \or % 1 \expandafter\xdef\csname\the\macname\endcsname{% \bgroup\noexpand\macroargctxt \noexpand\braceorline \expandafter\noexpand\csname\the\macname xxx\endcsname}% \expandafter\xdef\csname\the\macname xxx\endcsname##1{% \egroup \noexpand\norecurse{\the\macname}% \noexpand\scanmacro{\temp}\egroup}% \else % many \expandafter\xdef\csname\the\macname\endcsname{% \bgroup\noexpand\macroargctxt \expandafter\noexpand\csname\the\macname xx\endcsname}% \expandafter\xdef\csname\the\macname xx\endcsname##1{% \expandafter\noexpand\csname\the\macname xxx\endcsname ##1,}% \expandafter\expandafter \expandafter\xdef \expandafter\expandafter \csname\the\macname xxx\endcsname \paramlist{% \egroup \noexpand\norecurse{\the\macname}% \noexpand\scanmacro{\temp}\egroup}% \fi \fi} \def\norecurse#1{\bgroup\cslet{#1}{macsave.#1}} % \braceorline decides whether the next nonwhitespace character is a % {. If so it reads up to the closing }, if not, it reads the whole % line. Whatever was read is then fed to the next control sequence % as an argument (by \parsebrace or \parsearg) \def\braceorline#1{\let\macnamexxx=#1\futurelet\nchar\braceorlinexxx} \def\braceorlinexxx{% \ifx\nchar\bgroup\else \expandafter\parsearg \fi \macnamexxx} % @alias. % We need some trickery to remove the optional spaces around the equal % sign. Just make them active and then expand them all to nothing. \def\alias{\parseargusing\obeyspaces\aliasxxx} \def\aliasxxx #1{\aliasyyy#1\relax} \def\aliasyyy #1=#2\relax{% {% \expandafter\let\obeyedspace=\empty \addtomacrolist{#1}% \xdef\next{\global\let\makecsname{#1}=\makecsname{#2}}% }% \next } \message{cross references,} \newwrite\auxfile \newif\ifhavexrefs % True if xref values are known. \newif\ifwarnedxrefs % True if we warned once that they aren't known. % @inforef is relatively simple. \def\inforef #1{\inforefzzz #1,,,,**} \def\inforefzzz #1,#2,#3,#4**{\putwordSee{} \putwordInfo{} \putwordfile{} \file{\ignorespaces #3{}}, node \samp{\ignorespaces#1{}}} % @node's only job in TeX is to define \lastnode, which is used in % cross-references. The @node line might or might not have commas, and % might or might not have spaces before the first comma, like: % @node foo , bar , ... % We don't want such trailing spaces in the node name. % \parseargdef\node{\checkenv{}\donode #1 ,\finishnodeparse} % % also remove a trailing comma, in case of something like this: % @node Help-Cross, , , Cross-refs \def\donode#1 ,#2\finishnodeparse{\dodonode #1,\finishnodeparse} \def\dodonode#1,#2\finishnodeparse{\gdef\lastnode{#1}} \let\nwnode=\node \let\lastnode=\empty % Write a cross-reference definition for the current node. #1 is the % type (Ynumbered, Yappendix, Ynothing). % \def\donoderef#1{% \ifx\lastnode\empty\else \setref{\lastnode}{#1}% \global\let\lastnode=\empty \fi } % @anchor{NAME} -- define xref target at arbitrary point. % \newcount\savesfregister % \def\savesf{\relax \ifhmode \savesfregister=\spacefactor \fi} \def\restoresf{\relax \ifhmode \spacefactor=\savesfregister \fi} \def\anchor#1{\savesf \setref{#1}{Ynothing}\restoresf \ignorespaces} % \setref{NAME}{SNT} defines a cross-reference point NAME (a node or an % anchor), which consists of three parts: % 1) NAME-title - the current sectioning name taken from \lastsection, % or the anchor name. % 2) NAME-snt - section number and type, passed as the SNT arg, or % empty for anchors. % 3) NAME-pg - the page number. % % This is called from \donoderef, \anchor, and \dofloat. In the case of % floats, there is an additional part, which is not written here: % 4) NAME-lof - the text as it should appear in a @listoffloats. % \def\setref#1#2{% \pdfmkdest{#1}% \iflinks {% \atdummies % preserve commands, but don't expand them \edef\writexrdef##1##2{% \write\auxfile{@xrdef{#1-% #1 of \setref, expanded by the \edef ##1}{##2}}% these are parameters of \writexrdef }% \toks0 = \expandafter{\lastsection}% \immediate \writexrdef{title}{\the\toks0 }% \immediate \writexrdef{snt}{\csname #2\endcsname}% \Ynumbered etc. \safewhatsit{\writexrdef{pg}{\folio}}% will be written later, during \shipout }% \fi } % @xref, @pxref, and @ref generate cross-references. For \xrefX, #1 is % the node name, #2 the name of the Info cross-reference, #3 the printed % node name, #4 the name of the Info file, #5 the name of the printed % manual. All but the node name can be omitted. % \def\pxref#1{\putwordsee{} \xrefX[#1,,,,,,,]} \def\xref#1{\putwordSee{} \xrefX[#1,,,,,,,]} \def\ref#1{\xrefX[#1,,,,,,,]} \def\xrefX[#1,#2,#3,#4,#5,#6]{\begingroup \unsepspaces \def\printedmanual{\ignorespaces #5}% \def\printedrefname{\ignorespaces #3}% \setbox1=\hbox{\printedmanual\unskip}% \setbox0=\hbox{\printedrefname\unskip}% \ifdim \wd0 = 0pt % No printed node name was explicitly given. \expandafter\ifx\csname SETxref-automatic-section-title\endcsname\relax % Use the node name inside the square brackets. \def\printedrefname{\ignorespaces #1}% \else % Use the actual chapter/section title appear inside % the square brackets. Use the real section title if we have it. \ifdim \wd1 > 0pt % It is in another manual, so we don't have it. \def\printedrefname{\ignorespaces #1}% \else \ifhavexrefs % We know the real title if we have the xref values. \def\printedrefname{\refx{#1-title}{}}% \else % Otherwise just copy the Info node name. \def\printedrefname{\ignorespaces #1}% \fi% \fi \fi \fi % % Make link in pdf output. \ifpdf {\indexnofonts \turnoffactive % This expands tokens, so do it after making catcode changes, so _ % etc. don't get their TeX definitions. \getfilename{#4}% % % See comments at \activebackslashdouble. {\activebackslashdouble \xdef\pdfxrefdest{#1}% \backslashparens\pdfxrefdest}% % \leavevmode \startlink attr{/Border [0 0 0]}% \ifnum\filenamelength>0 goto file{\the\filename.pdf} name{\pdfxrefdest}% \else goto name{\pdfmkpgn{\pdfxrefdest}}% \fi }% \setcolor{\linkcolor}% \fi % % Float references are printed completely differently: "Figure 1.2" % instead of "[somenode], p.3". We distinguish them by the % LABEL-title being set to a magic string. {% % Have to otherify everything special to allow the \csname to % include an _ in the xref name, etc. \indexnofonts \turnoffactive \expandafter\global\expandafter\let\expandafter\Xthisreftitle \csname XR#1-title\endcsname }% \iffloat\Xthisreftitle % If the user specified the print name (third arg) to the ref, % print it instead of our usual "Figure 1.2". \ifdim\wd0 = 0pt \refx{#1-snt}{}% \else \printedrefname \fi % % if the user also gave the printed manual name (fifth arg), append % "in MANUALNAME". \ifdim \wd1 > 0pt \space \putwordin{} \cite{\printedmanual}% \fi \else % node/anchor (non-float) references. % % If we use \unhbox0 and \unhbox1 to print the node names, TeX does not % insert empty discretionaries after hyphens, which means that it will % not find a line break at a hyphen in a node names. Since some manuals % are best written with fairly long node names, containing hyphens, this % is a loss. Therefore, we give the text of the node name again, so it % is as if TeX is seeing it for the first time. \ifdim \wd1 > 0pt \putwordSection{} ``\printedrefname'' \putwordin{} \cite{\printedmanual}% \else % _ (for example) has to be the character _ for the purposes of the % control sequence corresponding to the node, but it has to expand % into the usual \leavevmode...\vrule stuff for purposes of % printing. So we \turnoffactive for the \refx-snt, back on for the % printing, back off for the \refx-pg. {\turnoffactive % Only output a following space if the -snt ref is nonempty; for % @unnumbered and @anchor, it won't be. \setbox2 = \hbox{\ignorespaces \refx{#1-snt}{}}% \ifdim \wd2 > 0pt \refx{#1-snt}\space\fi }% % output the `[mynode]' via a macro so it can be overridden. \xrefprintnodename\printedrefname % % But we always want a comma and a space: ,\space % % output the `page 3'. \turnoffactive \putwordpage\tie\refx{#1-pg}{}% \fi \fi \endlink \endgroup} % This macro is called from \xrefX for the `[nodename]' part of xref % output. It's a separate macro only so it can be changed more easily, % since square brackets don't work well in some documents. Particularly % one that Bob is working on :). % \def\xrefprintnodename#1{[#1]} % Things referred to by \setref. % \def\Ynothing{} \def\Yomitfromtoc{} \def\Ynumbered{% \ifnum\secno=0 \putwordChapter@tie \the\chapno \else \ifnum\subsecno=0 \putwordSection@tie \the\chapno.\the\secno \else \ifnum\subsubsecno=0 \putwordSection@tie \the\chapno.\the\secno.\the\subsecno \else \putwordSection@tie \the\chapno.\the\secno.\the\subsecno.\the\subsubsecno \fi\fi\fi } \def\Yappendix{% \ifnum\secno=0 \putwordAppendix@tie @char\the\appendixno{}% \else \ifnum\subsecno=0 \putwordSection@tie @char\the\appendixno.\the\secno \else \ifnum\subsubsecno=0 \putwordSection@tie @char\the\appendixno.\the\secno.\the\subsecno \else \putwordSection@tie @char\the\appendixno.\the\secno.\the\subsecno.\the\subsubsecno \fi\fi\fi } % Define \refx{NAME}{SUFFIX} to reference a cross-reference string named NAME. % If its value is nonempty, SUFFIX is output afterward. % \def\refx#1#2{% {% \indexnofonts \otherbackslash \expandafter\global\expandafter\let\expandafter\thisrefX \csname XR#1\endcsname }% \ifx\thisrefX\relax % If not defined, say something at least. \angleleft un\-de\-fined\angleright \iflinks \ifhavexrefs \message{\linenumber Undefined cross reference `#1'.}% \else \ifwarnedxrefs\else \global\warnedxrefstrue \message{Cross reference values unknown; you must run TeX again.}% \fi \fi \fi \else % It's defined, so just use it. \thisrefX \fi #2% Output the suffix in any case. } % This is the macro invoked by entries in the aux file. Usually it's % just a \def (we prepend XR to the control sequence name to avoid % collisions). But if this is a float type, we have more work to do. % \def\xrdef#1#2{% {% The node name might contain 8-bit characters, which in our current % implementation are changed to commands like @'e. Don't let these % mess up the control sequence name. \indexnofonts \turnoffactive \xdef\safexrefname{#1}% }% % \expandafter\gdef\csname XR\safexrefname\endcsname{#2}% remember this xref % % Was that xref control sequence that we just defined for a float? \expandafter\iffloat\csname XR\safexrefname\endcsname % it was a float, and we have the (safe) float type in \iffloattype. \expandafter\let\expandafter\floatlist \csname floatlist\iffloattype\endcsname % % Is this the first time we've seen this float type? \expandafter\ifx\floatlist\relax \toks0 = {\do}% yes, so just \do \else % had it before, so preserve previous elements in list. \toks0 = \expandafter{\floatlist\do}% \fi % % Remember this xref in the control sequence \floatlistFLOATTYPE, % for later use in \listoffloats. \expandafter\xdef\csname floatlist\iffloattype\endcsname{\the\toks0 {\safexrefname}}% \fi } % Read the last existing aux file, if any. No error if none exists. % \def\tryauxfile{% \openin 1 \jobname.aux \ifeof 1 \else \readdatafile{aux}% \global\havexrefstrue \fi \closein 1 } \def\setupdatafile{% \catcode`\^^@=\other \catcode`\^^A=\other \catcode`\^^B=\other \catcode`\^^C=\other \catcode`\^^D=\other \catcode`\^^E=\other \catcode`\^^F=\other \catcode`\^^G=\other \catcode`\^^H=\other \catcode`\^^K=\other \catcode`\^^L=\other \catcode`\^^N=\other \catcode`\^^P=\other \catcode`\^^Q=\other \catcode`\^^R=\other \catcode`\^^S=\other \catcode`\^^T=\other \catcode`\^^U=\other \catcode`\^^V=\other \catcode`\^^W=\other \catcode`\^^X=\other \catcode`\^^Z=\other \catcode`\^^[=\other \catcode`\^^\=\other \catcode`\^^]=\other \catcode`\^^^=\other \catcode`\^^_=\other % It was suggested to set the catcode of ^ to 7, which would allow ^^e4 etc. % in xref tags, i.e., node names. But since ^^e4 notation isn't % supported in the main text, it doesn't seem desirable. Furthermore, % that is not enough: for node names that actually contain a ^ % character, we would end up writing a line like this: 'xrdef {'hat % b-title}{'hat b} and \xrdef does a \csname...\endcsname on the first % argument, and \hat is not an expandable control sequence. It could % all be worked out, but why? Either we support ^^ or we don't. % % The other change necessary for this was to define \auxhat: % \def\auxhat{\def^{'hat }}% extra space so ok if followed by letter % and then to call \auxhat in \setq. % \catcode`\^=\other % % Special characters. Should be turned off anyway, but... \catcode`\~=\other \catcode`\[=\other \catcode`\]=\other \catcode`\"=\other \catcode`\_=\other \catcode`\|=\other \catcode`\<=\other \catcode`\>=\other \catcode`\$=\other \catcode`\#=\other \catcode`\&=\other \catcode`\%=\other \catcode`+=\other % avoid \+ for paranoia even though we've turned it off % % This is to support \ in node names and titles, since the \ % characters end up in a \csname. It's easier than % leaving it active and making its active definition an actual \ % character. What I don't understand is why it works in the *value* % of the xrdef. Seems like it should be a catcode12 \, and that % should not typeset properly. But it works, so I'm moving on for % now. --karl, 15jan04. \catcode`\\=\other % % Make the characters 128-255 be printing characters. {% \count1=128 \def\loop{% \catcode\count1=\other \advance\count1 by 1 \ifnum \count1<256 \loop \fi }% }% % % @ is our escape character in .aux files, and we need braces. \catcode`\{=1 \catcode`\}=2 \catcode`\@=0 } \def\readdatafile#1{% \begingroup \setupdatafile \input\jobname.#1 \endgroup} \message{insertions,} % including footnotes. \newcount \footnoteno % The trailing space in the following definition for supereject is % vital for proper filling; pages come out unaligned when you do a % pagealignmacro call if that space before the closing brace is % removed. (Generally, numeric constants should always be followed by a % space to prevent strange expansion errors.) \def\supereject{\par\penalty -20000\footnoteno =0 } % @footnotestyle is meaningful for info output only. \let\footnotestyle=\comment {\catcode `\@=11 % % Auto-number footnotes. Otherwise like plain. \gdef\footnote{% \let\indent=\ptexindent \let\noindent=\ptexnoindent \global\advance\footnoteno by \@ne \edef\thisfootno{$^{\the\footnoteno}$}% % % In case the footnote comes at the end of a sentence, preserve the % extra spacing after we do the footnote number. \let\@sf\empty \ifhmode\edef\@sf{\spacefactor\the\spacefactor}\ptexslash\fi % % Remove inadvertent blank space before typesetting the footnote number. \unskip \thisfootno\@sf \dofootnote }% % Don't bother with the trickery in plain.tex to not require the % footnote text as a parameter. Our footnotes don't need to be so general. % % Oh yes, they do; otherwise, @ifset (and anything else that uses % \parseargline) fails inside footnotes because the tokens are fixed when % the footnote is read. --karl, 16nov96. % \gdef\dofootnote{% \insert\footins\bgroup % We want to typeset this text as a normal paragraph, even if the % footnote reference occurs in (for example) a display environment. % So reset some parameters. \hsize=\pagewidth \interlinepenalty\interfootnotelinepenalty \splittopskip\ht\strutbox % top baseline for broken footnotes \splitmaxdepth\dp\strutbox \floatingpenalty\@MM \leftskip\z@skip \rightskip\z@skip \spaceskip\z@skip \xspaceskip\z@skip \parindent\defaultparindent % \smallfonts \rm % % Because we use hanging indentation in footnotes, a @noindent appears % to exdent this text, so make it be a no-op. makeinfo does not use % hanging indentation so @noindent can still be needed within footnote % text after an @example or the like (not that this is good style). \let\noindent = \relax % % Hang the footnote text off the number. Use \everypar in case the % footnote extends for more than one paragraph. \everypar = {\hang}% \textindent{\thisfootno}% % % Don't crash into the line above the footnote text. Since this % expands into a box, it must come within the paragraph, lest it % provide a place where TeX can split the footnote. \footstrut \futurelet\next\fo@t } }%end \catcode `\@=11 % In case a @footnote appears in a vbox, save the footnote text and create % the real \insert just after the vbox finished. Otherwise, the insertion % would be lost. % Similarly, if a @footnote appears inside an alignment, save the footnote % text to a box and make the \insert when a row of the table is finished. % And the same can be done for other insert classes. --kasal, 16nov03. % Replace the \insert primitive by a cheating macro. % Deeper inside, just make sure that the saved insertions are not spilled % out prematurely. % \def\startsavinginserts{% \ifx \insert\ptexinsert \let\insert\saveinsert \else \let\checkinserts\relax \fi } % This \insert replacement works for both \insert\footins{foo} and % \insert\footins\bgroup foo\egroup, but it doesn't work for \insert27{foo}. % \def\saveinsert#1{% \edef\next{\noexpand\savetobox \makeSAVEname#1}% \afterassignment\next % swallow the left brace \let\temp = } \def\makeSAVEname#1{\makecsname{SAVE\expandafter\gobble\string#1}} \def\savetobox#1{\global\setbox#1 = \vbox\bgroup \unvbox#1} \def\checksaveins#1{\ifvoid#1\else \placesaveins#1\fi} \def\placesaveins#1{% \ptexinsert \csname\expandafter\gobblesave\string#1\endcsname {\box#1}% } % eat @SAVE -- beware, all of them have catcode \other: { \def\dospecials{\do S\do A\do V\do E} \uncatcodespecials % ;-) \gdef\gobblesave @SAVE{} } % initialization: \def\newsaveins #1{% \edef\next{\noexpand\newsaveinsX \makeSAVEname#1}% \next } \def\newsaveinsX #1{% \csname newbox\endcsname #1% \expandafter\def\expandafter\checkinserts\expandafter{\checkinserts \checksaveins #1}% } % initialize: \let\checkinserts\empty \newsaveins\footins \newsaveins\margin % @image. We use the macros from epsf.tex to support this. % If epsf.tex is not installed and @image is used, we complain. % % Check for and read epsf.tex up front. If we read it only at @image % time, we might be inside a group, and then its definitions would get % undone and the next image would fail. \openin 1 = epsf.tex \ifeof 1 \else % Do not bother showing banner with epsf.tex v2.7k (available in % doc/epsf.tex and on ctan). \def\epsfannounce{\toks0 = }% \input epsf.tex \fi \closein 1 % % We will only complain once about lack of epsf.tex. \newif\ifwarnednoepsf \newhelp\noepsfhelp{epsf.tex must be installed for images to work. It is also included in the Texinfo distribution, or you can get it from ftp://tug.org/tex/epsf.tex.} % \def\image#1{% \ifx\epsfbox\undefined \ifwarnednoepsf \else \errhelp = \noepsfhelp \errmessage{epsf.tex not found, images will be ignored}% \global\warnednoepsftrue \fi \else \imagexxx #1,,,,,\finish \fi } % % Arguments to @image: % #1 is (mandatory) image filename; we tack on .eps extension. % #2 is (optional) width, #3 is (optional) height. % #4 is (ignored optional) html alt text. % #5 is (ignored optional) extension. % #6 is just the usual extra ignored arg for parsing this stuff. \newif\ifimagevmode \def\imagexxx#1,#2,#3,#4,#5,#6\finish{\begingroup \catcode`\^^M = 5 % in case we're inside an example \normalturnoffactive % allow _ et al. in names % If the image is by itself, center it. \ifvmode \imagevmodetrue \nobreak\medskip % Usually we'll have text after the image which will insert % \parskip glue, so insert it here too to equalize the space % above and below. \nobreak\vskip\parskip \nobreak \fi % % Leave vertical mode so that indentation from an enclosing % environment such as @quotation is respected. On the other hand, if % it's at the top level, we don't want the normal paragraph indentation. \noindent % % Output the image. \ifpdf \dopdfimage{#1}{#2}{#3}% \else % \epsfbox itself resets \epsf?size at each figure. \setbox0 = \hbox{\ignorespaces #2}\ifdim\wd0 > 0pt \epsfxsize=#2\relax \fi \setbox0 = \hbox{\ignorespaces #3}\ifdim\wd0 > 0pt \epsfysize=#3\relax \fi \epsfbox{#1.eps}% \fi % \ifimagevmode \medskip \fi % space after the standalone image \endgroup} % @float FLOATTYPE,LABEL,LOC ... @end float for displayed figures, tables, % etc. We don't actually implement floating yet, we always include the % float "here". But it seemed the best name for the future. % \envparseargdef\float{\eatcommaspace\eatcommaspace\dofloat#1, , ,\finish} % There may be a space before second and/or third parameter; delete it. \def\eatcommaspace#1, {#1,} % #1 is the optional FLOATTYPE, the text label for this float, typically % "Figure", "Table", "Example", etc. Can't contain commas. If omitted, % this float will not be numbered and cannot be referred to. % % #2 is the optional xref label. Also must be present for the float to % be referable. % % #3 is the optional positioning argument; for now, it is ignored. It % will somehow specify the positions allowed to float to (here, top, bottom). % % We keep a separate counter for each FLOATTYPE, which we reset at each % chapter-level command. \let\resetallfloatnos=\empty % \def\dofloat#1,#2,#3,#4\finish{% \let\thiscaption=\empty \let\thisshortcaption=\empty % % don't lose footnotes inside @float. % % BEWARE: when the floats start float, we have to issue warning whenever an % insert appears inside a float which could possibly float. --kasal, 26may04 % \startsavinginserts % % We can't be used inside a paragraph. \par % \vtop\bgroup \def\floattype{#1}% \def\floatlabel{#2}% \def\floatloc{#3}% we do nothing with this yet. % \ifx\floattype\empty \let\safefloattype=\empty \else {% % the floattype might have accents or other special characters, % but we need to use it in a control sequence name. \indexnofonts \turnoffactive \xdef\safefloattype{\floattype}% }% \fi % % If label is given but no type, we handle that as the empty type. \ifx\floatlabel\empty \else % We want each FLOATTYPE to be numbered separately (Figure 1, % Table 1, Figure 2, ...). (And if no label, no number.) % \expandafter\getfloatno\csname\safefloattype floatno\endcsname \global\advance\floatno by 1 % {% % This magic value for \lastsection is output by \setref as the % XREFLABEL-title value. \xrefX uses it to distinguish float % labels (which have a completely different output format) from % node and anchor labels. And \xrdef uses it to construct the % lists of floats. % \edef\lastsection{\floatmagic=\safefloattype}% \setref{\floatlabel}{Yfloat}% }% \fi % % start with \parskip glue, I guess. \vskip\parskip % % Don't suppress indentation if a float happens to start a section. \restorefirstparagraphindent } % we have these possibilities: % @float Foo,lbl & @caption{Cap}: Foo 1.1: Cap % @float Foo,lbl & no caption: Foo 1.1 % @float Foo & @caption{Cap}: Foo: Cap % @float Foo & no caption: Foo % @float ,lbl & Caption{Cap}: 1.1: Cap % @float ,lbl & no caption: 1.1 % @float & @caption{Cap}: Cap % @float & no caption: % \def\Efloat{% \let\floatident = \empty % % In all cases, if we have a float type, it comes first. \ifx\floattype\empty \else \def\floatident{\floattype}\fi % % If we have an xref label, the number comes next. \ifx\floatlabel\empty \else \ifx\floattype\empty \else % if also had float type, need tie first. \appendtomacro\floatident{\tie}% \fi % the number. \appendtomacro\floatident{\chaplevelprefix\the\floatno}% \fi % % Start the printed caption with what we've constructed in % \floatident, but keep it separate; we need \floatident again. \let\captionline = \floatident % \ifx\thiscaption\empty \else \ifx\floatident\empty \else \appendtomacro\captionline{: }% had ident, so need a colon between \fi % % caption text. \appendtomacro\captionline{\scanexp\thiscaption}% \fi % % If we have anything to print, print it, with space before. % Eventually this needs to become an \insert. \ifx\captionline\empty \else \vskip.5\parskip \captionline % % Space below caption. \vskip\parskip \fi % % If have an xref label, write the list of floats info. Do this % after the caption, to avoid chance of it being a breakpoint. \ifx\floatlabel\empty \else % Write the text that goes in the lof to the aux file as % \floatlabel-lof. Besides \floatident, we include the short % caption if specified, else the full caption if specified, else nothing. {% \atdummies % % since we read the caption text in the macro world, where ^^M % is turned into a normal character, we have to scan it back, so % we don't write the literal three characters "^^M" into the aux file. \scanexp{% \xdef\noexpand\gtemp{% \ifx\thisshortcaption\empty \thiscaption \else \thisshortcaption \fi }% }% \immediate\write\auxfile{@xrdef{\floatlabel-lof}{\floatident \ifx\gtemp\empty \else : \gtemp \fi}}% }% \fi \egroup % end of \vtop % % place the captured inserts % % BEWARE: when the floats start floating, we have to issue warning % whenever an insert appears inside a float which could possibly % float. --kasal, 26may04 % \checkinserts } % Append the tokens #2 to the definition of macro #1, not expanding either. % \def\appendtomacro#1#2{% \expandafter\def\expandafter#1\expandafter{#1#2}% } % @caption, @shortcaption % \def\caption{\docaption\thiscaption} \def\shortcaption{\docaption\thisshortcaption} \def\docaption{\checkenv\float \bgroup\scanargctxt\defcaption} \def\defcaption#1#2{\egroup \def#1{#2}} % The parameter is the control sequence identifying the counter we are % going to use. Create it if it doesn't exist and assign it to \floatno. \def\getfloatno#1{% \ifx#1\relax % Haven't seen this figure type before. \csname newcount\endcsname #1% % % Remember to reset this floatno at the next chap. \expandafter\gdef\expandafter\resetallfloatnos \expandafter{\resetallfloatnos #1=0 }% \fi \let\floatno#1% } % \setref calls this to get the XREFLABEL-snt value. We want an @xref % to the FLOATLABEL to expand to "Figure 3.1". We call \setref when we % first read the @float command. % \def\Yfloat{\floattype@tie \chaplevelprefix\the\floatno}% % Magic string used for the XREFLABEL-title value, so \xrefX can % distinguish floats from other xref types. \def\floatmagic{!!float!!} % #1 is the control sequence we are passed; we expand into a conditional % which is true if #1 represents a float ref. That is, the magic % \lastsection value which we \setref above. % \def\iffloat#1{\expandafter\doiffloat#1==\finish} % % #1 is (maybe) the \floatmagic string. If so, #2 will be the % (safe) float type for this float. We set \iffloattype to #2. % \def\doiffloat#1=#2=#3\finish{% \def\temp{#1}% \def\iffloattype{#2}% \ifx\temp\floatmagic } % @listoffloats FLOATTYPE - print a list of floats like a table of contents. % \parseargdef\listoffloats{% \def\floattype{#1}% floattype {% % the floattype might have accents or other special characters, % but we need to use it in a control sequence name. \indexnofonts \turnoffactive \xdef\safefloattype{\floattype}% }% % % \xrdef saves the floats as a \do-list in \floatlistSAFEFLOATTYPE. \expandafter\ifx\csname floatlist\safefloattype\endcsname \relax \ifhavexrefs % if the user said @listoffloats foo but never @float foo. \message{\linenumber No `\safefloattype' floats to list.}% \fi \else \begingroup \leftskip=\tocindent % indent these entries like a toc \let\do=\listoffloatsdo \csname floatlist\safefloattype\endcsname \endgroup \fi } % This is called on each entry in a list of floats. We're passed the % xref label, in the form LABEL-title, which is how we save it in the % aux file. We strip off the -title and look up \XRLABEL-lof, which % has the text we're supposed to typeset here. % % Figures without xref labels will not be included in the list (since % they won't appear in the aux file). % \def\listoffloatsdo#1{\listoffloatsdoentry#1\finish} \def\listoffloatsdoentry#1-title\finish{{% % Can't fully expand XR#1-lof because it can contain anything. Just % pass the control sequence. On the other hand, XR#1-pg is just the % page number, and we want to fully expand that so we can get a link % in pdf output. \toksA = \expandafter{\csname XR#1-lof\endcsname}% % % use the same \entry macro we use to generate the TOC and index. \edef\writeentry{\noexpand\entry{\the\toksA}{\csname XR#1-pg\endcsname}}% \writeentry }} \message{localization,} % @documentlanguage is usually given very early, just after % @setfilename. If done too late, it may not override everything % properly. Single argument is the language (de) or locale (de_DE) % abbreviation. It would be nice if we could set up a hyphenation file. % { \catcode`\_ = \active \globaldefs=1 \parseargdef\documentlanguage{\begingroup \let_=\normalunderscore % normal _ character for filenames \tex % read txi-??.tex file in plain TeX. % Read the file by the name they passed if it exists. \openin 1 txi-#1.tex \ifeof 1 \documentlanguagetrywithoutunderscore{#1_\finish}% \else \input txi-#1.tex \fi \closein 1 \endgroup \endgroup} } % % If they passed de_DE, and txi-de_DE.tex doesn't exist, % try txi-de.tex. % \def\documentlanguagetrywithoutunderscore#1_#2\finish{% \openin 1 txi-#1.tex \ifeof 1 \errhelp = \nolanghelp \errmessage{Cannot read language file txi-#1.tex}% \else \input txi-#1.tex \fi \closein 1 } % \newhelp\nolanghelp{The given language definition file cannot be found or is empty. Maybe you need to install it? In the current directory should work if nowhere else does.} % Set the catcode of characters 128 through 255 to the specified number. % \def\setnonasciicharscatcode#1{% \count255=128 \loop\ifnum\count255<256 \global\catcode\count255=#1\relax \advance\count255 by 1 \repeat } \def\setnonasciicharscatcodenonglobal#1{% \count255=128 \loop\ifnum\count255<256 \catcode\count255=#1\relax \advance\count255 by 1 \repeat } % @documentencoding sets the definition of non-ASCII characters % according to the specified encoding. % \parseargdef\documentencoding{% % Encoding being declared for the document. \def\declaredencoding{\csname #1.enc\endcsname}% % % Supported encodings: names converted to tokens in order to be able % to compare them with \ifx. \def\ascii{\csname US-ASCII.enc\endcsname}% \def\latnine{\csname ISO-8859-15.enc\endcsname}% \def\latone{\csname ISO-8859-1.enc\endcsname}% \def\lattwo{\csname ISO-8859-2.enc\endcsname}% \def\utfeight{\csname UTF-8.enc\endcsname}% % \ifx \declaredencoding \ascii \asciichardefs % \else \ifx \declaredencoding \lattwo \setnonasciicharscatcode\active \lattwochardefs % \else \ifx \declaredencoding \latone \setnonasciicharscatcode\active \latonechardefs % \else \ifx \declaredencoding \latnine \setnonasciicharscatcode\active \latninechardefs % \else \ifx \declaredencoding \utfeight \setnonasciicharscatcode\active \utfeightchardefs % \else \message{Unknown document encoding #1, ignoring.}% % \fi % utfeight \fi % latnine \fi % latone \fi % lattwo \fi % ascii } % A message to be logged when using a character that isn't available % the default font encoding (OT1). % \def\missingcharmsg#1{\message{Character missing in OT1 encoding: #1.}} % Take account of \c (plain) vs. \, (Texinfo) difference. \def\cedilla#1{\ifx\c\ptexc\c{#1}\else\,{#1}\fi} % First, make active non-ASCII characters in order for them to be % correctly categorized when TeX reads the replacement text of % macros containing the character definitions. \setnonasciicharscatcode\active % % Latin1 (ISO-8859-1) character definitions. \def\latonechardefs{% \gdef^^a0{~} \gdef^^a1{\exclamdown} \gdef^^a2{\missingcharmsg{CENT SIGN}} \gdef^^a3{{\pounds}} \gdef^^a4{\missingcharmsg{CURRENCY SIGN}} \gdef^^a5{\missingcharmsg{YEN SIGN}} \gdef^^a6{\missingcharmsg{BROKEN BAR}} \gdef^^a7{\S} \gdef^^a8{\"{}} \gdef^^a9{\copyright} \gdef^^aa{\ordf} \gdef^^ab{\missingcharmsg{LEFT-POINTING DOUBLE ANGLE QUOTATION MARK}} \gdef^^ac{$\lnot$} \gdef^^ad{\-} \gdef^^ae{\registeredsymbol} \gdef^^af{\={}} % \gdef^^b0{\textdegree} \gdef^^b1{$\pm$} \gdef^^b2{$^2$} \gdef^^b3{$^3$} \gdef^^b4{\'{}} \gdef^^b5{$\mu$} \gdef^^b6{\P} % \gdef^^b7{$^.$} \gdef^^b8{\cedilla\ } \gdef^^b9{$^1$} \gdef^^ba{\ordm} % \gdef^^bb{\missingcharmsg{RIGHT-POINTING DOUBLE ANGLE QUOTATION MARK}} \gdef^^bc{$1\over4$} \gdef^^bd{$1\over2$} \gdef^^be{$3\over4$} \gdef^^bf{\questiondown} % \gdef^^c0{\`A} \gdef^^c1{\'A} \gdef^^c2{\^A} \gdef^^c3{\~A} \gdef^^c4{\"A} \gdef^^c5{\ringaccent A} \gdef^^c6{\AE} \gdef^^c7{\cedilla C} \gdef^^c8{\`E} \gdef^^c9{\'E} \gdef^^ca{\^E} \gdef^^cb{\"E} \gdef^^cc{\`I} \gdef^^cd{\'I} \gdef^^ce{\^I} \gdef^^cf{\"I} % \gdef^^d0{\missingcharmsg{LATIN CAPITAL LETTER ETH}} \gdef^^d1{\~N} \gdef^^d2{\`O} \gdef^^d3{\'O} \gdef^^d4{\^O} \gdef^^d5{\~O} \gdef^^d6{\"O} \gdef^^d7{$\times$} \gdef^^d8{\O} \gdef^^d9{\`U} \gdef^^da{\'U} \gdef^^db{\^U} \gdef^^dc{\"U} \gdef^^dd{\'Y} \gdef^^de{\missingcharmsg{LATIN CAPITAL LETTER THORN}} \gdef^^df{\ss} % \gdef^^e0{\`a} \gdef^^e1{\'a} \gdef^^e2{\^a} \gdef^^e3{\~a} \gdef^^e4{\"a} \gdef^^e5{\ringaccent a} \gdef^^e6{\ae} \gdef^^e7{\cedilla c} \gdef^^e8{\`e} \gdef^^e9{\'e} \gdef^^ea{\^e} \gdef^^eb{\"e} \gdef^^ec{\`{\dotless i}} \gdef^^ed{\'{\dotless i}} \gdef^^ee{\^{\dotless i}} \gdef^^ef{\"{\dotless i}} % \gdef^^f0{\missingcharmsg{LATIN SMALL LETTER ETH}} \gdef^^f1{\~n} \gdef^^f2{\`o} \gdef^^f3{\'o} \gdef^^f4{\^o} \gdef^^f5{\~o} \gdef^^f6{\"o} \gdef^^f7{$\div$} \gdef^^f8{\o} \gdef^^f9{\`u} \gdef^^fa{\'u} \gdef^^fb{\^u} \gdef^^fc{\"u} \gdef^^fd{\'y} \gdef^^fe{\missingcharmsg{LATIN SMALL LETTER THORN}} \gdef^^ff{\"y} } % Latin9 (ISO-8859-15) encoding character definitions. \def\latninechardefs{% % Encoding is almost identical to Latin1. \latonechardefs % \gdef^^a4{\euro} \gdef^^a6{\v S} \gdef^^a8{\v s} \gdef^^b4{\v Z} \gdef^^b8{\v z} \gdef^^bc{\OE} \gdef^^bd{\oe} \gdef^^be{\"Y} } % Latin2 (ISO-8859-2) character definitions. \def\lattwochardefs{% \gdef^^a0{~} \gdef^^a1{\missingcharmsg{LATIN CAPITAL LETTER A WITH OGONEK}} \gdef^^a2{\u{}} \gdef^^a3{\L} \gdef^^a4{\missingcharmsg{CURRENCY SIGN}} \gdef^^a5{\v L} \gdef^^a6{\'S} \gdef^^a7{\S} \gdef^^a8{\"{}} \gdef^^a9{\v S} \gdef^^aa{\cedilla S} \gdef^^ab{\v T} \gdef^^ac{\'Z} \gdef^^ad{\-} \gdef^^ae{\v Z} \gdef^^af{\dotaccent Z} % \gdef^^b0{\textdegree} \gdef^^b1{\missingcharmsg{LATIN SMALL LETTER A WITH OGONEK}} \gdef^^b2{\missingcharmsg{OGONEK}} \gdef^^b3{\l} \gdef^^b4{\'{}} \gdef^^b5{\v l} \gdef^^b6{\'s} \gdef^^b7{\v{}} \gdef^^b8{\cedilla\ } \gdef^^b9{\v s} \gdef^^ba{\cedilla s} \gdef^^bb{\v t} \gdef^^bc{\'z} \gdef^^bd{\H{}} \gdef^^be{\v z} \gdef^^bf{\dotaccent z} % \gdef^^c0{\'R} \gdef^^c1{\'A} \gdef^^c2{\^A} \gdef^^c3{\u A} \gdef^^c4{\"A} \gdef^^c5{\'L} \gdef^^c6{\'C} \gdef^^c7{\cedilla C} \gdef^^c8{\v C} \gdef^^c9{\'E} \gdef^^ca{\missingcharmsg{LATIN CAPITAL LETTER E WITH OGONEK}} \gdef^^cb{\"E} \gdef^^cc{\v E} \gdef^^cd{\'I} \gdef^^ce{\^I} \gdef^^cf{\v D} % \gdef^^d0{\missingcharmsg{LATIN CAPITAL LETTER D WITH STROKE}} \gdef^^d1{\'N} \gdef^^d2{\v N} \gdef^^d3{\'O} \gdef^^d4{\^O} \gdef^^d5{\H O} \gdef^^d6{\"O} \gdef^^d7{$\times$} \gdef^^d8{\v R} \gdef^^d9{\ringaccent U} \gdef^^da{\'U} \gdef^^db{\H U} \gdef^^dc{\"U} \gdef^^dd{\'Y} \gdef^^de{\cedilla T} \gdef^^df{\ss} % \gdef^^e0{\'r} \gdef^^e1{\'a} \gdef^^e2{\^a} \gdef^^e3{\u a} \gdef^^e4{\"a} \gdef^^e5{\'l} \gdef^^e6{\'c} \gdef^^e7{\cedilla c} \gdef^^e8{\v c} \gdef^^e9{\'e} \gdef^^ea{\missingcharmsg{LATIN SMALL LETTER E WITH OGONEK}} \gdef^^eb{\"e} \gdef^^ec{\v e} \gdef^^ed{\'\i} \gdef^^ee{\^\i} \gdef^^ef{\v d} % \gdef^^f0{\missingcharmsg{LATIN SMALL LETTER D WITH STROKE}} \gdef^^f1{\'n} \gdef^^f2{\v n} \gdef^^f3{\'o} \gdef^^f4{\^o} \gdef^^f5{\H o} \gdef^^f6{\"o} \gdef^^f7{$\div$} \gdef^^f8{\v r} \gdef^^f9{\ringaccent u} \gdef^^fa{\'u} \gdef^^fb{\H u} \gdef^^fc{\"u} \gdef^^fd{\'y} \gdef^^fe{\cedilla t} \gdef^^ff{\dotaccent{}} } % UTF-8 character definitions. % % This code to support UTF-8 is based on LaTeX's utf8.def, with some % changes for Texinfo conventions. It is included here under the GPL by % permission from Frank Mittelbach and the LaTeX team. % \newcount\countUTFx \newcount\countUTFy \newcount\countUTFz \gdef\UTFviiiTwoOctets#1#2{\expandafter \UTFviiiDefined\csname u8:#1\string #2\endcsname} % \gdef\UTFviiiThreeOctets#1#2#3{\expandafter \UTFviiiDefined\csname u8:#1\string #2\string #3\endcsname} % \gdef\UTFviiiFourOctets#1#2#3#4{\expandafter \UTFviiiDefined\csname u8:#1\string #2\string #3\string #4\endcsname} \gdef\UTFviiiDefined#1{% \ifx #1\relax \message{\linenumber Unicode char \string #1 not defined for Texinfo}% \else \expandafter #1% \fi } \begingroup \catcode`\~13 \catcode`\"12 \def\UTFviiiLoop{% \global\catcode\countUTFx\active \uccode`\~\countUTFx \uppercase\expandafter{\UTFviiiTmp}% \advance\countUTFx by 1 \ifnum\countUTFx < \countUTFy \expandafter\UTFviiiLoop \fi} \countUTFx = "C2 \countUTFy = "E0 \def\UTFviiiTmp{% \xdef~{\noexpand\UTFviiiTwoOctets\string~}} \UTFviiiLoop \countUTFx = "E0 \countUTFy = "F0 \def\UTFviiiTmp{% \xdef~{\noexpand\UTFviiiThreeOctets\string~}} \UTFviiiLoop \countUTFx = "F0 \countUTFy = "F4 \def\UTFviiiTmp{% \xdef~{\noexpand\UTFviiiFourOctets\string~}} \UTFviiiLoop \endgroup \begingroup \catcode`\"=12 \catcode`\<=12 \catcode`\.=12 \catcode`\,=12 \catcode`\;=12 \catcode`\!=12 \catcode`\~=13 \gdef\DeclareUnicodeCharacter#1#2{% \countUTFz = "#1\relax \wlog{\space\space defining Unicode char U+#1 (decimal \the\countUTFz)}% \begingroup \parseXMLCharref \def\UTFviiiTwoOctets##1##2{% \csname u8:##1\string ##2\endcsname}% \def\UTFviiiThreeOctets##1##2##3{% \csname u8:##1\string ##2\string ##3\endcsname}% \def\UTFviiiFourOctets##1##2##3##4{% \csname u8:##1\string ##2\string ##3\string ##4\endcsname}% \expandafter\expandafter\expandafter\expandafter \expandafter\expandafter\expandafter \gdef\UTFviiiTmp{#2}% \endgroup} \gdef\parseXMLCharref{% \ifnum\countUTFz < "A0\relax \errhelp = \EMsimple \errmessage{Cannot define Unicode char value < 00A0}% \else\ifnum\countUTFz < "800\relax \parseUTFviiiA,% \parseUTFviiiB C\UTFviiiTwoOctets.,% \else\ifnum\countUTFz < "10000\relax \parseUTFviiiA;% \parseUTFviiiA,% \parseUTFviiiB E\UTFviiiThreeOctets.{,;}% \else \parseUTFviiiA;% \parseUTFviiiA,% \parseUTFviiiA!% \parseUTFviiiB F\UTFviiiFourOctets.{!,;}% \fi\fi\fi } \gdef\parseUTFviiiA#1{% \countUTFx = \countUTFz \divide\countUTFz by 64 \countUTFy = \countUTFz \multiply\countUTFz by 64 \advance\countUTFx by -\countUTFz \advance\countUTFx by 128 \uccode `#1\countUTFx \countUTFz = \countUTFy} \gdef\parseUTFviiiB#1#2#3#4{% \advance\countUTFz by "#10\relax \uccode `#3\countUTFz \uppercase{\gdef\UTFviiiTmp{#2#3#4}}} \endgroup \def\utfeightchardefs{% \DeclareUnicodeCharacter{00A0}{\tie} \DeclareUnicodeCharacter{00A1}{\exclamdown} \DeclareUnicodeCharacter{00A3}{\pounds} \DeclareUnicodeCharacter{00A8}{\"{ }} \DeclareUnicodeCharacter{00A9}{\copyright} \DeclareUnicodeCharacter{00AA}{\ordf} \DeclareUnicodeCharacter{00AB}{\guillemetleft} \DeclareUnicodeCharacter{00AD}{\-} \DeclareUnicodeCharacter{00AE}{\registeredsymbol} \DeclareUnicodeCharacter{00AF}{\={ }} \DeclareUnicodeCharacter{00B0}{\ringaccent{ }} \DeclareUnicodeCharacter{00B4}{\'{ }} \DeclareUnicodeCharacter{00B8}{\cedilla{ }} \DeclareUnicodeCharacter{00BA}{\ordm} \DeclareUnicodeCharacter{00BB}{\guillemetright} \DeclareUnicodeCharacter{00BF}{\questiondown} \DeclareUnicodeCharacter{00C0}{\`A} \DeclareUnicodeCharacter{00C1}{\'A} \DeclareUnicodeCharacter{00C2}{\^A} \DeclareUnicodeCharacter{00C3}{\~A} \DeclareUnicodeCharacter{00C4}{\"A} \DeclareUnicodeCharacter{00C5}{\AA} \DeclareUnicodeCharacter{00C6}{\AE} \DeclareUnicodeCharacter{00C7}{\cedilla{C}} \DeclareUnicodeCharacter{00C8}{\`E} \DeclareUnicodeCharacter{00C9}{\'E} \DeclareUnicodeCharacter{00CA}{\^E} \DeclareUnicodeCharacter{00CB}{\"E} \DeclareUnicodeCharacter{00CC}{\`I} \DeclareUnicodeCharacter{00CD}{\'I} \DeclareUnicodeCharacter{00CE}{\^I} \DeclareUnicodeCharacter{00CF}{\"I} \DeclareUnicodeCharacter{00D1}{\~N} \DeclareUnicodeCharacter{00D2}{\`O} \DeclareUnicodeCharacter{00D3}{\'O} \DeclareUnicodeCharacter{00D4}{\^O} \DeclareUnicodeCharacter{00D5}{\~O} \DeclareUnicodeCharacter{00D6}{\"O} \DeclareUnicodeCharacter{00D8}{\O} \DeclareUnicodeCharacter{00D9}{\`U} \DeclareUnicodeCharacter{00DA}{\'U} \DeclareUnicodeCharacter{00DB}{\^U} \DeclareUnicodeCharacter{00DC}{\"U} \DeclareUnicodeCharacter{00DD}{\'Y} \DeclareUnicodeCharacter{00DF}{\ss} \DeclareUnicodeCharacter{00E0}{\`a} \DeclareUnicodeCharacter{00E1}{\'a} \DeclareUnicodeCharacter{00E2}{\^a} \DeclareUnicodeCharacter{00E3}{\~a} \DeclareUnicodeCharacter{00E4}{\"a} \DeclareUnicodeCharacter{00E5}{\aa} \DeclareUnicodeCharacter{00E6}{\ae} \DeclareUnicodeCharacter{00E7}{\cedilla{c}} \DeclareUnicodeCharacter{00E8}{\`e} \DeclareUnicodeCharacter{00E9}{\'e} \DeclareUnicodeCharacter{00EA}{\^e} \DeclareUnicodeCharacter{00EB}{\"e} \DeclareUnicodeCharacter{00EC}{\`{\dotless{i}}} \DeclareUnicodeCharacter{00ED}{\'{\dotless{i}}} \DeclareUnicodeCharacter{00EE}{\^{\dotless{i}}} \DeclareUnicodeCharacter{00EF}{\"{\dotless{i}}} \DeclareUnicodeCharacter{00F1}{\~n} \DeclareUnicodeCharacter{00F2}{\`o} \DeclareUnicodeCharacter{00F3}{\'o} \DeclareUnicodeCharacter{00F4}{\^o} \DeclareUnicodeCharacter{00F5}{\~o} \DeclareUnicodeCharacter{00F6}{\"o} \DeclareUnicodeCharacter{00F8}{\o} \DeclareUnicodeCharacter{00F9}{\`u} \DeclareUnicodeCharacter{00FA}{\'u} \DeclareUnicodeCharacter{00FB}{\^u} \DeclareUnicodeCharacter{00FC}{\"u} \DeclareUnicodeCharacter{00FD}{\'y} \DeclareUnicodeCharacter{00FF}{\"y} \DeclareUnicodeCharacter{0100}{\=A} \DeclareUnicodeCharacter{0101}{\=a} \DeclareUnicodeCharacter{0102}{\u{A}} \DeclareUnicodeCharacter{0103}{\u{a}} \DeclareUnicodeCharacter{0106}{\'C} \DeclareUnicodeCharacter{0107}{\'c} \DeclareUnicodeCharacter{0108}{\^C} \DeclareUnicodeCharacter{0109}{\^c} \DeclareUnicodeCharacter{010A}{\dotaccent{C}} \DeclareUnicodeCharacter{010B}{\dotaccent{c}} \DeclareUnicodeCharacter{010C}{\v{C}} \DeclareUnicodeCharacter{010D}{\v{c}} \DeclareUnicodeCharacter{010E}{\v{D}} \DeclareUnicodeCharacter{0112}{\=E} \DeclareUnicodeCharacter{0113}{\=e} \DeclareUnicodeCharacter{0114}{\u{E}} \DeclareUnicodeCharacter{0115}{\u{e}} \DeclareUnicodeCharacter{0116}{\dotaccent{E}} \DeclareUnicodeCharacter{0117}{\dotaccent{e}} \DeclareUnicodeCharacter{011A}{\v{E}} \DeclareUnicodeCharacter{011B}{\v{e}} \DeclareUnicodeCharacter{011C}{\^G} \DeclareUnicodeCharacter{011D}{\^g} \DeclareUnicodeCharacter{011E}{\u{G}} \DeclareUnicodeCharacter{011F}{\u{g}} \DeclareUnicodeCharacter{0120}{\dotaccent{G}} \DeclareUnicodeCharacter{0121}{\dotaccent{g}} \DeclareUnicodeCharacter{0124}{\^H} \DeclareUnicodeCharacter{0125}{\^h} \DeclareUnicodeCharacter{0128}{\~I} \DeclareUnicodeCharacter{0129}{\~{\dotless{i}}} \DeclareUnicodeCharacter{012A}{\=I} \DeclareUnicodeCharacter{012B}{\={\dotless{i}}} \DeclareUnicodeCharacter{012C}{\u{I}} \DeclareUnicodeCharacter{012D}{\u{\dotless{i}}} \DeclareUnicodeCharacter{0130}{\dotaccent{I}} \DeclareUnicodeCharacter{0131}{\dotless{i}} \DeclareUnicodeCharacter{0132}{IJ} \DeclareUnicodeCharacter{0133}{ij} \DeclareUnicodeCharacter{0134}{\^J} \DeclareUnicodeCharacter{0135}{\^{\dotless{j}}} \DeclareUnicodeCharacter{0139}{\'L} \DeclareUnicodeCharacter{013A}{\'l} \DeclareUnicodeCharacter{0141}{\L} \DeclareUnicodeCharacter{0142}{\l} \DeclareUnicodeCharacter{0143}{\'N} \DeclareUnicodeCharacter{0144}{\'n} \DeclareUnicodeCharacter{0147}{\v{N}} \DeclareUnicodeCharacter{0148}{\v{n}} \DeclareUnicodeCharacter{014C}{\=O} \DeclareUnicodeCharacter{014D}{\=o} \DeclareUnicodeCharacter{014E}{\u{O}} \DeclareUnicodeCharacter{014F}{\u{o}} \DeclareUnicodeCharacter{0150}{\H{O}} \DeclareUnicodeCharacter{0151}{\H{o}} \DeclareUnicodeCharacter{0152}{\OE} \DeclareUnicodeCharacter{0153}{\oe} \DeclareUnicodeCharacter{0154}{\'R} \DeclareUnicodeCharacter{0155}{\'r} \DeclareUnicodeCharacter{0158}{\v{R}} \DeclareUnicodeCharacter{0159}{\v{r}} \DeclareUnicodeCharacter{015A}{\'S} \DeclareUnicodeCharacter{015B}{\'s} \DeclareUnicodeCharacter{015C}{\^S} \DeclareUnicodeCharacter{015D}{\^s} \DeclareUnicodeCharacter{015E}{\cedilla{S}} \DeclareUnicodeCharacter{015F}{\cedilla{s}} \DeclareUnicodeCharacter{0160}{\v{S}} \DeclareUnicodeCharacter{0161}{\v{s}} \DeclareUnicodeCharacter{0162}{\cedilla{t}} \DeclareUnicodeCharacter{0163}{\cedilla{T}} \DeclareUnicodeCharacter{0164}{\v{T}} \DeclareUnicodeCharacter{0168}{\~U} \DeclareUnicodeCharacter{0169}{\~u} \DeclareUnicodeCharacter{016A}{\=U} \DeclareUnicodeCharacter{016B}{\=u} \DeclareUnicodeCharacter{016C}{\u{U}} \DeclareUnicodeCharacter{016D}{\u{u}} \DeclareUnicodeCharacter{016E}{\ringaccent{U}} \DeclareUnicodeCharacter{016F}{\ringaccent{u}} \DeclareUnicodeCharacter{0170}{\H{U}} \DeclareUnicodeCharacter{0171}{\H{u}} \DeclareUnicodeCharacter{0174}{\^W} \DeclareUnicodeCharacter{0175}{\^w} \DeclareUnicodeCharacter{0176}{\^Y} \DeclareUnicodeCharacter{0177}{\^y} \DeclareUnicodeCharacter{0178}{\"Y} \DeclareUnicodeCharacter{0179}{\'Z} \DeclareUnicodeCharacter{017A}{\'z} \DeclareUnicodeCharacter{017B}{\dotaccent{Z}} \DeclareUnicodeCharacter{017C}{\dotaccent{z}} \DeclareUnicodeCharacter{017D}{\v{Z}} \DeclareUnicodeCharacter{017E}{\v{z}} \DeclareUnicodeCharacter{01C4}{D\v{Z}} \DeclareUnicodeCharacter{01C5}{D\v{z}} \DeclareUnicodeCharacter{01C6}{d\v{z}} \DeclareUnicodeCharacter{01C7}{LJ} \DeclareUnicodeCharacter{01C8}{Lj} \DeclareUnicodeCharacter{01C9}{lj} \DeclareUnicodeCharacter{01CA}{NJ} \DeclareUnicodeCharacter{01CB}{Nj} \DeclareUnicodeCharacter{01CC}{nj} \DeclareUnicodeCharacter{01CD}{\v{A}} \DeclareUnicodeCharacter{01CE}{\v{a}} \DeclareUnicodeCharacter{01CF}{\v{I}} \DeclareUnicodeCharacter{01D0}{\v{\dotless{i}}} \DeclareUnicodeCharacter{01D1}{\v{O}} \DeclareUnicodeCharacter{01D2}{\v{o}} \DeclareUnicodeCharacter{01D3}{\v{U}} \DeclareUnicodeCharacter{01D4}{\v{u}} \DeclareUnicodeCharacter{01E2}{\={\AE}} \DeclareUnicodeCharacter{01E3}{\={\ae}} \DeclareUnicodeCharacter{01E6}{\v{G}} \DeclareUnicodeCharacter{01E7}{\v{g}} \DeclareUnicodeCharacter{01E8}{\v{K}} \DeclareUnicodeCharacter{01E9}{\v{k}} \DeclareUnicodeCharacter{01F0}{\v{\dotless{j}}} \DeclareUnicodeCharacter{01F1}{DZ} \DeclareUnicodeCharacter{01F2}{Dz} \DeclareUnicodeCharacter{01F3}{dz} \DeclareUnicodeCharacter{01F4}{\'G} \DeclareUnicodeCharacter{01F5}{\'g} \DeclareUnicodeCharacter{01F8}{\`N} \DeclareUnicodeCharacter{01F9}{\`n} \DeclareUnicodeCharacter{01FC}{\'{\AE}} \DeclareUnicodeCharacter{01FD}{\'{\ae}} \DeclareUnicodeCharacter{01FE}{\'{\O}} \DeclareUnicodeCharacter{01FF}{\'{\o}} \DeclareUnicodeCharacter{021E}{\v{H}} \DeclareUnicodeCharacter{021F}{\v{h}} \DeclareUnicodeCharacter{0226}{\dotaccent{A}} \DeclareUnicodeCharacter{0227}{\dotaccent{a}} \DeclareUnicodeCharacter{0228}{\cedilla{E}} \DeclareUnicodeCharacter{0229}{\cedilla{e}} \DeclareUnicodeCharacter{022E}{\dotaccent{O}} \DeclareUnicodeCharacter{022F}{\dotaccent{o}} \DeclareUnicodeCharacter{0232}{\=Y} \DeclareUnicodeCharacter{0233}{\=y} \DeclareUnicodeCharacter{0237}{\dotless{j}} \DeclareUnicodeCharacter{1E02}{\dotaccent{B}} \DeclareUnicodeCharacter{1E03}{\dotaccent{b}} \DeclareUnicodeCharacter{1E04}{\udotaccent{B}} \DeclareUnicodeCharacter{1E05}{\udotaccent{b}} \DeclareUnicodeCharacter{1E06}{\ubaraccent{B}} \DeclareUnicodeCharacter{1E07}{\ubaraccent{b}} \DeclareUnicodeCharacter{1E0A}{\dotaccent{D}} \DeclareUnicodeCharacter{1E0B}{\dotaccent{d}} \DeclareUnicodeCharacter{1E0C}{\udotaccent{D}} \DeclareUnicodeCharacter{1E0D}{\udotaccent{d}} \DeclareUnicodeCharacter{1E0E}{\ubaraccent{D}} \DeclareUnicodeCharacter{1E0F}{\ubaraccent{d}} \DeclareUnicodeCharacter{1E1E}{\dotaccent{F}} \DeclareUnicodeCharacter{1E1F}{\dotaccent{f}} \DeclareUnicodeCharacter{1E20}{\=G} \DeclareUnicodeCharacter{1E21}{\=g} \DeclareUnicodeCharacter{1E22}{\dotaccent{H}} \DeclareUnicodeCharacter{1E23}{\dotaccent{h}} \DeclareUnicodeCharacter{1E24}{\udotaccent{H}} \DeclareUnicodeCharacter{1E25}{\udotaccent{h}} \DeclareUnicodeCharacter{1E26}{\"H} \DeclareUnicodeCharacter{1E27}{\"h} \DeclareUnicodeCharacter{1E30}{\'K} \DeclareUnicodeCharacter{1E31}{\'k} \DeclareUnicodeCharacter{1E32}{\udotaccent{K}} \DeclareUnicodeCharacter{1E33}{\udotaccent{k}} \DeclareUnicodeCharacter{1E34}{\ubaraccent{K}} \DeclareUnicodeCharacter{1E35}{\ubaraccent{k}} \DeclareUnicodeCharacter{1E36}{\udotaccent{L}} \DeclareUnicodeCharacter{1E37}{\udotaccent{l}} \DeclareUnicodeCharacter{1E3A}{\ubaraccent{L}} \DeclareUnicodeCharacter{1E3B}{\ubaraccent{l}} \DeclareUnicodeCharacter{1E3E}{\'M} \DeclareUnicodeCharacter{1E3F}{\'m} \DeclareUnicodeCharacter{1E40}{\dotaccent{M}} \DeclareUnicodeCharacter{1E41}{\dotaccent{m}} \DeclareUnicodeCharacter{1E42}{\udotaccent{M}} \DeclareUnicodeCharacter{1E43}{\udotaccent{m}} \DeclareUnicodeCharacter{1E44}{\dotaccent{N}} \DeclareUnicodeCharacter{1E45}{\dotaccent{n}} \DeclareUnicodeCharacter{1E46}{\udotaccent{N}} \DeclareUnicodeCharacter{1E47}{\udotaccent{n}} \DeclareUnicodeCharacter{1E48}{\ubaraccent{N}} \DeclareUnicodeCharacter{1E49}{\ubaraccent{n}} \DeclareUnicodeCharacter{1E54}{\'P} \DeclareUnicodeCharacter{1E55}{\'p} \DeclareUnicodeCharacter{1E56}{\dotaccent{P}} \DeclareUnicodeCharacter{1E57}{\dotaccent{p}} \DeclareUnicodeCharacter{1E58}{\dotaccent{R}} \DeclareUnicodeCharacter{1E59}{\dotaccent{r}} \DeclareUnicodeCharacter{1E5A}{\udotaccent{R}} \DeclareUnicodeCharacter{1E5B}{\udotaccent{r}} \DeclareUnicodeCharacter{1E5E}{\ubaraccent{R}} \DeclareUnicodeCharacter{1E5F}{\ubaraccent{r}} \DeclareUnicodeCharacter{1E60}{\dotaccent{S}} \DeclareUnicodeCharacter{1E61}{\dotaccent{s}} \DeclareUnicodeCharacter{1E62}{\udotaccent{S}} \DeclareUnicodeCharacter{1E63}{\udotaccent{s}} \DeclareUnicodeCharacter{1E6A}{\dotaccent{T}} \DeclareUnicodeCharacter{1E6B}{\dotaccent{t}} \DeclareUnicodeCharacter{1E6C}{\udotaccent{T}} \DeclareUnicodeCharacter{1E6D}{\udotaccent{t}} \DeclareUnicodeCharacter{1E6E}{\ubaraccent{T}} \DeclareUnicodeCharacter{1E6F}{\ubaraccent{t}} \DeclareUnicodeCharacter{1E7C}{\~V} \DeclareUnicodeCharacter{1E7D}{\~v} \DeclareUnicodeCharacter{1E7E}{\udotaccent{V}} \DeclareUnicodeCharacter{1E7F}{\udotaccent{v}} \DeclareUnicodeCharacter{1E80}{\`W} \DeclareUnicodeCharacter{1E81}{\`w} \DeclareUnicodeCharacter{1E82}{\'W} \DeclareUnicodeCharacter{1E83}{\'w} \DeclareUnicodeCharacter{1E84}{\"W} \DeclareUnicodeCharacter{1E85}{\"w} \DeclareUnicodeCharacter{1E86}{\dotaccent{W}} \DeclareUnicodeCharacter{1E87}{\dotaccent{w}} \DeclareUnicodeCharacter{1E88}{\udotaccent{W}} \DeclareUnicodeCharacter{1E89}{\udotaccent{w}} \DeclareUnicodeCharacter{1E8A}{\dotaccent{X}} \DeclareUnicodeCharacter{1E8B}{\dotaccent{x}} \DeclareUnicodeCharacter{1E8C}{\"X} \DeclareUnicodeCharacter{1E8D}{\"x} \DeclareUnicodeCharacter{1E8E}{\dotaccent{Y}} \DeclareUnicodeCharacter{1E8F}{\dotaccent{y}} \DeclareUnicodeCharacter{1E90}{\^Z} \DeclareUnicodeCharacter{1E91}{\^z} \DeclareUnicodeCharacter{1E92}{\udotaccent{Z}} \DeclareUnicodeCharacter{1E93}{\udotaccent{z}} \DeclareUnicodeCharacter{1E94}{\ubaraccent{Z}} \DeclareUnicodeCharacter{1E95}{\ubaraccent{z}} \DeclareUnicodeCharacter{1E96}{\ubaraccent{h}} \DeclareUnicodeCharacter{1E97}{\"t} \DeclareUnicodeCharacter{1E98}{\ringaccent{w}} \DeclareUnicodeCharacter{1E99}{\ringaccent{y}} \DeclareUnicodeCharacter{1EA0}{\udotaccent{A}} \DeclareUnicodeCharacter{1EA1}{\udotaccent{a}} \DeclareUnicodeCharacter{1EB8}{\udotaccent{E}} \DeclareUnicodeCharacter{1EB9}{\udotaccent{e}} \DeclareUnicodeCharacter{1EBC}{\~E} \DeclareUnicodeCharacter{1EBD}{\~e} \DeclareUnicodeCharacter{1ECA}{\udotaccent{I}} \DeclareUnicodeCharacter{1ECB}{\udotaccent{i}} \DeclareUnicodeCharacter{1ECC}{\udotaccent{O}} \DeclareUnicodeCharacter{1ECD}{\udotaccent{o}} \DeclareUnicodeCharacter{1EE4}{\udotaccent{U}} \DeclareUnicodeCharacter{1EE5}{\udotaccent{u}} \DeclareUnicodeCharacter{1EF2}{\`Y} \DeclareUnicodeCharacter{1EF3}{\`y} \DeclareUnicodeCharacter{1EF4}{\udotaccent{Y}} \DeclareUnicodeCharacter{1EF8}{\~Y} \DeclareUnicodeCharacter{1EF9}{\~y} \DeclareUnicodeCharacter{2013}{--} \DeclareUnicodeCharacter{2014}{---} \DeclareUnicodeCharacter{2018}{\quoteleft} \DeclareUnicodeCharacter{2019}{\quoteright} \DeclareUnicodeCharacter{201A}{\quotesinglbase} \DeclareUnicodeCharacter{201C}{\quotedblleft} \DeclareUnicodeCharacter{201D}{\quotedblright} \DeclareUnicodeCharacter{201E}{\quotedblbase} \DeclareUnicodeCharacter{2022}{\bullet} \DeclareUnicodeCharacter{2026}{\dots} \DeclareUnicodeCharacter{2039}{\guilsinglleft} \DeclareUnicodeCharacter{203A}{\guilsinglright} \DeclareUnicodeCharacter{20AC}{\euro} \DeclareUnicodeCharacter{2192}{\expansion} \DeclareUnicodeCharacter{21D2}{\result} \DeclareUnicodeCharacter{2212}{\minus} \DeclareUnicodeCharacter{2217}{\point} \DeclareUnicodeCharacter{2261}{\equiv} }% end of \utfeightchardefs % US-ASCII character definitions. \def\asciichardefs{% nothing need be done \relax } % Make non-ASCII characters printable again for compatibility with % existing Texinfo documents that may use them, even without declaring a % document encoding. % \setnonasciicharscatcode \other \message{formatting,} \newdimen\defaultparindent \defaultparindent = 15pt \chapheadingskip = 15pt plus 4pt minus 2pt \secheadingskip = 12pt plus 3pt minus 2pt \subsecheadingskip = 9pt plus 2pt minus 2pt % Prevent underfull vbox error messages. \vbadness = 10000 % Don't be so finicky about underfull hboxes, either. \hbadness = 2000 % Following George Bush, get rid of widows and orphans. \widowpenalty=10000 \clubpenalty=10000 % Use TeX 3.0's \emergencystretch to help line breaking, but if we're % using an old version of TeX, don't do anything. We want the amount of % stretch added to depend on the line length, hence the dependence on % \hsize. We call this whenever the paper size is set. % \def\setemergencystretch{% \ifx\emergencystretch\thisisundefined % Allow us to assign to \emergencystretch anyway. \def\emergencystretch{\dimen0}% \else \emergencystretch = .15\hsize \fi } % Parameters in order: 1) textheight; 2) textwidth; % 3) voffset; 4) hoffset; 5) binding offset; 6) topskip; % 7) physical page height; 8) physical page width. % % We also call \setleading{\textleading}, so the caller should define % \textleading. The caller should also set \parskip. % \def\internalpagesizes#1#2#3#4#5#6#7#8{% \voffset = #3\relax \topskip = #6\relax \splittopskip = \topskip % \vsize = #1\relax \advance\vsize by \topskip \outervsize = \vsize \advance\outervsize by 2\topandbottommargin \pageheight = \vsize % \hsize = #2\relax \outerhsize = \hsize \advance\outerhsize by 0.5in \pagewidth = \hsize % \normaloffset = #4\relax \bindingoffset = #5\relax % \ifpdf \pdfpageheight #7\relax \pdfpagewidth #8\relax % if we don't reset these, they will remain at "1 true in" of % whatever layout pdftex was dumped with. \pdfhorigin = 1 true in \pdfvorigin = 1 true in \fi % \setleading{\textleading} % \parindent = \defaultparindent \setemergencystretch } % @letterpaper (the default). \def\letterpaper{{\globaldefs = 1 \parskip = 3pt plus 2pt minus 1pt \textleading = 13.2pt % % If page is nothing but text, make it come out even. \internalpagesizes{607.2pt}{6in}% that's 46 lines {\voffset}{.25in}% {\bindingoffset}{36pt}% {11in}{8.5in}% }} % Use @smallbook to reset parameters for 7x9.25 trim size. \def\smallbook{{\globaldefs = 1 \parskip = 2pt plus 1pt \textleading = 12pt % \internalpagesizes{7.5in}{5in}% {-.2in}{0in}% {\bindingoffset}{16pt}% {9.25in}{7in}% % \lispnarrowing = 0.3in \tolerance = 700 \hfuzz = 1pt \contentsrightmargin = 0pt \defbodyindent = .5cm }} % Use @smallerbook to reset parameters for 6x9 trim size. % (Just testing, parameters still in flux.) \def\smallerbook{{\globaldefs = 1 \parskip = 1.5pt plus 1pt \textleading = 12pt % \internalpagesizes{7.4in}{4.8in}% {-.2in}{-.4in}% {0pt}{14pt}% {9in}{6in}% % \lispnarrowing = 0.25in \tolerance = 700 \hfuzz = 1pt \contentsrightmargin = 0pt \defbodyindent = .4cm }} % Use @afourpaper to print on European A4 paper. \def\afourpaper{{\globaldefs = 1 \parskip = 3pt plus 2pt minus 1pt \textleading = 13.2pt % % Double-side printing via postscript on Laserjet 4050 % prints double-sided nicely when \bindingoffset=10mm and \hoffset=-6mm. % To change the settings for a different printer or situation, adjust % \normaloffset until the front-side and back-side texts align. Then % do the same for \bindingoffset. You can set these for testing in % your texinfo source file like this: % @tex % \global\normaloffset = -6mm % \global\bindingoffset = 10mm % @end tex \internalpagesizes{673.2pt}{160mm}% that's 51 lines {\voffset}{\hoffset}% {\bindingoffset}{44pt}% {297mm}{210mm}% % \tolerance = 700 \hfuzz = 1pt \contentsrightmargin = 0pt \defbodyindent = 5mm }} % Use @afivepaper to print on European A5 paper. % From romildo@urano.iceb.ufop.br, 2 July 2000. % He also recommends making @example and @lisp be small. \def\afivepaper{{\globaldefs = 1 \parskip = 2pt plus 1pt minus 0.1pt \textleading = 12.5pt % \internalpagesizes{160mm}{120mm}% {\voffset}{\hoffset}% {\bindingoffset}{8pt}% {210mm}{148mm}% % \lispnarrowing = 0.2in \tolerance = 800 \hfuzz = 1.2pt \contentsrightmargin = 0pt \defbodyindent = 2mm \tableindent = 12mm }} % A specific text layout, 24x15cm overall, intended for A4 paper. \def\afourlatex{{\globaldefs = 1 \afourpaper \internalpagesizes{237mm}{150mm}% {\voffset}{4.6mm}% {\bindingoffset}{7mm}% {297mm}{210mm}% % % Must explicitly reset to 0 because we call \afourpaper. \globaldefs = 0 }} % Use @afourwide to print on A4 paper in landscape format. \def\afourwide{{\globaldefs = 1 \afourpaper \internalpagesizes{241mm}{165mm}% {\voffset}{-2.95mm}% {\bindingoffset}{7mm}% {297mm}{210mm}% \globaldefs = 0 }} % @pagesizes TEXTHEIGHT[,TEXTWIDTH] % Perhaps we should allow setting the margins, \topskip, \parskip, % and/or leading, also. Or perhaps we should compute them somehow. % \parseargdef\pagesizes{\pagesizesyyy #1,,\finish} \def\pagesizesyyy#1,#2,#3\finish{{% \setbox0 = \hbox{\ignorespaces #2}\ifdim\wd0 > 0pt \hsize=#2\relax \fi \globaldefs = 1 % \parskip = 3pt plus 2pt minus 1pt \setleading{\textleading}% % \dimen0 = #1\relax \advance\dimen0 by \voffset % \dimen2 = \hsize \advance\dimen2 by \normaloffset % \internalpagesizes{#1}{\hsize}% {\voffset}{\normaloffset}% {\bindingoffset}{44pt}% {\dimen0}{\dimen2}% }} % Set default to letter. % \letterpaper \message{and turning on texinfo input format.} % Define macros to output various characters with catcode for normal text. \catcode`\"=\other \catcode`\~=\other \catcode`\^=\other \catcode`\_=\other \catcode`\|=\other \catcode`\<=\other \catcode`\>=\other \catcode`\+=\other \catcode`\$=\other \def\normaldoublequote{"} \def\normaltilde{~} \def\normalcaret{^} \def\normalunderscore{_} \def\normalverticalbar{|} \def\normalless{<} \def\normalgreater{>} \def\normalplus{+} \def\normaldollar{$}%$ font-lock fix % This macro is used to make a character print one way in \tt % (where it can probably be output as-is), and another way in other fonts, % where something hairier probably needs to be done. % % #1 is what to print if we are indeed using \tt; #2 is what to print % otherwise. Since all the Computer Modern typewriter fonts have zero % interword stretch (and shrink), and it is reasonable to expect all % typewriter fonts to have this, we can check that font parameter. % \def\ifusingtt#1#2{\ifdim \fontdimen3\font=0pt #1\else #2\fi} % Same as above, but check for italic font. Actually this also catches % non-italic slanted fonts since it is impossible to distinguish them from % italic fonts. But since this is only used by $ and it uses \sl anyway % this is not a problem. \def\ifusingit#1#2{\ifdim \fontdimen1\font>0pt #1\else #2\fi} % Turn off all special characters except @ % (and those which the user can use as if they were ordinary). % Most of these we simply print from the \tt font, but for some, we can % use math or other variants that look better in normal text. \catcode`\"=\active \def\activedoublequote{{\tt\char34}} \let"=\activedoublequote \catcode`\~=\active \def~{{\tt\char126}} \chardef\hat=`\^ \catcode`\^=\active \def^{{\tt \hat}} \catcode`\_=\active \def_{\ifusingtt\normalunderscore\_} \let\realunder=_ % Subroutine for the previous macro. \def\_{\leavevmode \kern.07em \vbox{\hrule width.3em height.1ex}\kern .07em } \catcode`\|=\active \def|{{\tt\char124}} \chardef \less=`\< \catcode`\<=\active \def<{{\tt \less}} \chardef \gtr=`\> \catcode`\>=\active \def>{{\tt \gtr}} \catcode`\+=\active \def+{{\tt \char 43}} \catcode`\$=\active \def${\ifusingit{{\sl\$}}\normaldollar}%$ font-lock fix % If a .fmt file is being used, characters that might appear in a file % name cannot be active until we have parsed the command line. % So turn them off again, and have \everyjob (or @setfilename) turn them on. % \otherifyactive is called near the end of this file. \def\otherifyactive{\catcode`+=\other \catcode`\_=\other} % Used sometimes to turn off (effectively) the active characters even after % parsing them. \def\turnoffactive{% \normalturnoffactive \otherbackslash } \catcode`\@=0 % \backslashcurfont outputs one backslash character in current font, % as in \char`\\. \global\chardef\backslashcurfont=`\\ \global\let\rawbackslashxx=\backslashcurfont % let existing .??s files work % \realbackslash is an actual character `\' with catcode other, and % \doublebackslash is two of them (for the pdf outlines). {\catcode`\\=\other @gdef@realbackslash{\} @gdef@doublebackslash{\\}} % In texinfo, backslash is an active character; it prints the backslash % in fixed width font. \catcode`\\=\active @def@normalbackslash{{@tt@backslashcurfont}} % On startup, @fixbackslash assigns: % @let \ = @normalbackslash % \rawbackslash defines an active \ to do \backslashcurfont. % \otherbackslash defines an active \ to be a literal `\' character with % catcode other. @gdef@rawbackslash{@let\=@backslashcurfont} @gdef@otherbackslash{@let\=@realbackslash} % Same as @turnoffactive except outputs \ as {\tt\char`\\} instead of % the literal character `\'. % @def@normalturnoffactive{% @let\=@normalbackslash @let"=@normaldoublequote @let~=@normaltilde @let^=@normalcaret @let_=@normalunderscore @let|=@normalverticalbar @let<=@normalless @let>=@normalgreater @let+=@normalplus @let$=@normaldollar %$ font-lock fix @unsepspaces } % Make _ and + \other characters, temporarily. % This is canceled by @fixbackslash. @otherifyactive % If a .fmt file is being used, we don't want the `\input texinfo' to show up. % That is what \eatinput is for; after that, the `\' should revert to printing % a backslash. % @gdef@eatinput input texinfo{@fixbackslash} @global@let\ = @eatinput % On the other hand, perhaps the file did not have a `\input texinfo'. Then % the first `\' in the file would cause an error. This macro tries to fix % that, assuming it is called before the first `\' could plausibly occur. % Also turn back on active characters that might appear in the input % file name, in case not using a pre-dumped format. % @gdef@fixbackslash{% @ifx\@eatinput @let\ = @normalbackslash @fi @catcode`+=@active @catcode`@_=@active } % Say @foo, not \foo, in error messages. @escapechar = `@@ % These look ok in all fonts, so just make them not special. @catcode`@& = @other @catcode`@# = @other @catcode`@% = @other @c Local variables: @c eval: (add-hook 'write-file-hooks 'time-stamp) @c page-delimiter: "^\\\\message" @c time-stamp-start: "def\\\\texinfoversion{" @c time-stamp-format: "%:y-%02m-%02d.%02H" @c time-stamp-end: "}" @c End: @c vim:sw=2: @ignore arch-tag: e1b36e32-c96e-4135-a41a-0b2efa2ea115 @end ignore gmp-doc-5.1.2/doc/mdate-sh0000755000175000000620000001212512146435154014256 0ustar stevestaff#!/bin/sh # Get modification time of a file or directory and pretty-print it. scriptversion=2003-11-09.00 # Copyright (C) 1995, 1996, 1997, 2003 Free Software Foundation, Inc. # written by Ulrich Drepper , June 1995 # # This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2, or (at your option) # any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program; if not, write to the Free Software Foundation, # Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. # As a special exception to the GNU General Public License, if you # distribute this file as part of a program that contains a # configuration script generated by Autoconf, you may include it under # the same distribution terms that you use for the rest of that program. # This file is maintained in Automake, please report # bugs to or send patches to # . case $1 in '') echo "$0: No file. Try \`$0 --help' for more information." 1>&2 exit 1; ;; -h | --h*) cat <<\EOF Usage: mdate-sh [--help] [--version] FILE Pretty-print the modification time of FILE. Report bugs to . EOF exit 0 ;; -v | --v*) echo "mdate-sh $scriptversion" exit 0 ;; esac # Prevent date giving response in another language. LANG=C export LANG LC_ALL=C export LC_ALL LC_TIME=C export LC_TIME # GNU ls changes its time format in response to the TIME_STYLE variable, but # we cannot unset it since the V7 shell did not have an "unset" command. # The documentation says that the default is "posix-long-iso". # test "${TIME_STYLE+set}" = set && TIME_STYLE=posix-long-iso save_arg1="$1" # Find out how to get the extended ls output of a file or directory. if ls -L /dev/null 1>/dev/null 2>&1; then ls_command='ls -L -l -d' else ls_command='ls -l -d' fi # A `ls -l' line looks as follows on OS/2. # drwxrwx--- 0 Aug 11 2001 foo # This differs from Unix, which adds ownership information. # drwxrwx--- 2 root root 4096 Aug 11 2001 foo # # To find the date, we split the line on spaces and iterate on words # until we find a month. This cannot work with files whose owner is a # user named `Jan', or `Feb', etc. However, it's unlikely that `/' # will be owned by a user whose name is a month. So we first look at # the extended ls output of the root directory to decide how many # words should be skipped to get the date. # On HPUX /bin/sh, "set" interprets "-rw-r--r--" as options, so the "x" below. set - x`$ls_command /` # Find which argument is the month. month= command= until test $month do shift # Add another shift to the command. command="$command shift;" case $1 in Jan) month=January; nummonth=1;; Feb) month=February; nummonth=2;; Mar) month=March; nummonth=3;; Apr) month=April; nummonth=4;; May) month=May; nummonth=5;; Jun) month=June; nummonth=6;; Jul) month=July; nummonth=7;; Aug) month=August; nummonth=8;; Sep) month=September; nummonth=9;; Oct) month=October; nummonth=10;; Nov) month=November; nummonth=11;; Dec) month=December; nummonth=12;; esac done # Get the extended ls output of the file or directory. set - x`eval "$ls_command \"\$save_arg1\""` # Remove all preceding arguments eval $command # Get the month. Next argument is day, followed by the year or time. case $1 in Jan) month=January; nummonth=1;; Feb) month=February; nummonth=2;; Mar) month=March; nummonth=3;; Apr) month=April; nummonth=4;; May) month=May; nummonth=5;; Jun) month=June; nummonth=6;; Jul) month=July; nummonth=7;; Aug) month=August; nummonth=8;; Sep) month=September; nummonth=9;; Oct) month=October; nummonth=10;; Nov) month=November; nummonth=11;; Dec) month=December; nummonth=12;; esac day=$2 # Here we have to deal with the problem that the ls output gives either # the time of day or the year. case $3 in *:*) set `date`; eval year=\$$# case $2 in Jan) nummonthtod=1;; Feb) nummonthtod=2;; Mar) nummonthtod=3;; Apr) nummonthtod=4;; May) nummonthtod=5;; Jun) nummonthtod=6;; Jul) nummonthtod=7;; Aug) nummonthtod=8;; Sep) nummonthtod=9;; Oct) nummonthtod=10;; Nov) nummonthtod=11;; Dec) nummonthtod=12;; esac # For the first six month of the year the time notation can also # be used for files modified in the last year. if (expr $nummonth \> $nummonthtod) > /dev/null; then year=`expr $year - 1` fi;; *) year=$3;; esac # The result. echo $day $month $year # Local Variables: # mode: shell-script # sh-indentation: 2 # eval: (add-hook 'write-file-hooks 'time-stamp) # time-stamp-start: "scriptversion=" # time-stamp-format: "%:y-%02m-%02d.%02H" # time-stamp-end: "$" # End: gmp-doc-5.1.2/doc/configuration0000644000175000000620000003142012146435154015417 0ustar stevestaff/* doc/configuration (in Emacs -*-outline-*- format). */ Copyright 2000, 2001, 2002, 2003, 2004 Free Software Foundation, Inc. This file is part of the GNU MP Library. The GNU MP Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. The GNU MP Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. * Adding a new file ** Adding a top-level file i) Add it to libgmp_la_SOURCES in Makefile.am. ii) If libmp.la needs it (usually doesn't), then add it to libmp_la_SOURCES too. ** Adding a subdirectory file For instance for mpz, i) Add file.c to libmpz_la_SOURCES in mpz/Makefile.am. ii) Add mpz/file$U.lo to MPZ_OBJECTS in the top-level Makefile.am iii) If for some reason libmp.la needs it (usually doesn't) then add mpz/file$U.lo to libmp_la_DEPENDENCIES in the top-level Makefile.am too. The same applies to mpf, mpq, scanf and printf. ** Adding an mpn file The way we build libmpn (in the `mpn' subdirectory) is quite special. Currently only mpn/mp_bases.c is truely generic and included in every configuration. All other files are linked at build time into the mpn build directory from one of the CPU specific sub-directories, or from the mpn/generic directory. There are four types of mpn source files. .asm Assembly code preprocessed with m4 .S Assembly code preprocessed with cpp .s Assembly code not preprocessed at all .c C code There are two types of .asm files. i) ``Normal'' files containing one function, though possibly with more than one entry point. ii) Multi-function files that generate one of a set of functions according to build options. To add a new implementation of an existing function, i) Put it in the appropriate CPU-specific mpn subdirectory, it'll be detected and used. ii) Any entrypoints tested by HAVE_NATIVE_func in other code must have PROLOGUE(func) for configure to grep. This is normal for .asm or .S files, but for .c files a dummy comment like the following will be needed. /* PROLOGUE(func) */ To add a new implementation using a multi-function file, in addition do the following, i) Use a MULFUNC_PROLOGUE(func1 func2 ...) in the .asm, declaring all the functions implemented, including carry-in variants. If there's a separate PROLOGUE(func) for each possible function (but this is usually not the case), then MULFUNC_PROLOGUE isn't necessary. To add a new style of multi-function file, in addition do the following, i) Add to the GMP_MULFUNC_CHOICES "case" statement in configure.in which lists each multi-function filename and what function files it can provide. To add a completely new mpn function file, do the following, i) Ensure the filename is a valid C identifier, due to the -DOPERATION_$* used to support multi-function files. This means "-" can't be used (but "_" can). ii) Add it to configure.in under one of the following a) `gmp_mpn_functions' if it exists for every target. This means there must be a C version in mpn/generic. (Eg. mul_1) b) `gmp_mpn_functions_optional' if it's a standard function, but doesn't need to exist for every target. Code wanting to use this will test HAVE_NATIVE_func to see if it's available. (Eg. copyi) c) `extra_functions' for some targets, if it's a special function that only ever needs to exist for certain targets. Code wanting to use it can test either HAVE_NATIVE_func or HAVE_HOST_CPU_foo, as desired. iii) If HAVE_NATIVE_func is going to be used, then add a #undef to the AH_VERBATIM([HAVE_NATIVE] block in configure.in. iv) If the function can be provided by a multi-function file, then add to the "case" statement in configure.in which lists each multi-function filename and what function files it can provide. ** Adding a test program i) Tests to be run early in the testing can be added to the main "tests" sub-directory. ii) Tests for mpn, mpz, mpq and mpf can be added under the corresponding tests subdirectory. iii) Generic tests for late in the testing can be added to "tests/misc". printf and scanf tests currently live there too. iv) Random number function tests can be added to "tests/rand". That directory has some development-time programs too. v) C++ test programs can be added to "tests/cxx". A line like the following must be added for each, since by default automake looks for a .c file. t_foo_SOURCES = t-foo.cc In all cases the name of the program should be added to check_PROGRAMS in the Makefile.am. TESTS is equal to check_PROGRAMS, so all those programs get run. "tests/devel" has a number of programs which are only for development purposes and are not for use in "make check". These should be listed in EXTRA_PROGRAMS to get Makefile rules created, but they're never built or run unless an explicit "make someprog" is used. * Adding a new CPU In general it's policy to use proper names for each CPU type supported. If two CPUs are quite similar and perhaps don't have any actual differences in GMP then they're still given separate names, for example alphaev67 and alphaev68. Canonical names: i) Decide the canonical CPU names GMP will accept. ii) Add these to the config.sub wrapper if configfsf.sub doesn't already accept them. iii) Document the names in gmp.texi. Aliases (optional): i) Any aliases can be added to the config.sub wrapper, unless configfsf.sub already does the right thing with them. ii) Leave configure.in and everywhere else using only the canonical names. Aliases shouldn't appear anywhere except config.sub. iii) Document in gmp.texi, if desired. Usually this isn't a good idea, better encourage users to know just the canonical names. Configure: i) Add patterns to configure.in for the new CPU names. Include the following (see configure.in for the variables to set up), a) ABI choices (if any). b) Compiler choices. c) mpn path for CPU specific code. d) Good default CFLAGS for each likely compiler. d) Any special tests necessary on the compiler or assembler capabilities. ii) M4 macros to be shared by asm files in a CPU family are by convention in a foo-defs.m4 like mpn/x86/x86-defs.m4. They're likely to use settings from config.m4 generated by configure. Fat binaries: i) In configure.in, add CPU specific directory(s) to fat_path. ii) In mpn//fat.c, identify the CPU at runtime and use suitable CPUVEC_SETUP_subdir macros to select the function pointers for it. iii) For the x86s, add to the "$tmp_prefix" setups in configure.in which abbreviates subdirectory names to fit an 8.3 filesystem. (No need to restrict to 8.3, just ensure uniqueness when truncated.) * The configure system ** Installing tools The current versions of automake, autoconf and libtool in use can be checked in the ChangeLog. Look for "Update to ...". Patches may have been applied, look for "Regenerate ...". The GMP build system is in places somewhat dependent on the internals of the build tools. Obviously that's avoided as much as possible, but where it can't it creates a problem when upgrading or attempting to use different tools versions. ** Updating gmp The following files need to be updated when going to a new version of the build tools. Unfortunately the tools generally don't identify when an out-of-date version is present. aclocal.m4 is updated by running "aclocal". (Only needed for a new automake or libtool.) INSTALL.autoconf can be copied from INSTALL in autoconf. ltmain.sh comes from libtool. Remove it and run "libtoolize --copy", or just copy the file by hand. texinfo.tex can be updated from ftp.gnu.org. Check it still works with "make gmp.dvi", "make gmp.ps" and "make gmp.pdf". configfsf.guess and configfsf.sub can be updated from ftp.gnu.org (or from the "config" cvs module at subversions.gnu.org). The gmp config.guess and config.sub wrappers are supposed to make such an update fairly painless. depcomp from automake is not needed because configure.in specifies automake with "no-dependencies". ** How it works During development: Input files Tool Output files --------------------------------------------------------- aclocal $prefix/share/aclocal*/*.m4 ----------------> aclocal.m4 configure.in \ autoconf aclocal.m4 / -----------------------------> configure */Makefile.am \ automake configure.in | ----------------------------> Makefile.in aclocal.m4 / configure.in \ autoheader aclocal.m4 / -----------------------------> config.in At build time: Input files Tool Output files -------------------------------------------- */Makefile.in \ configure / */Makefile config.in | -------------> | config.h gmp-h.in | | config.m4 mp-h.in / | gmp.h | mp.h \ fat.h (fat binary build only) When configured with --enable-maintainer-mode the Makefiles include rules to re-run the necessary tools if the input files are changed. This can end up running a lot more things than are really necessary. If a build tree is in too much of a mess for those rules to work properly then a bootstrap can be done from the source directory with aclocal autoconf automake autoheader The autom4te.cache directory is created by autoconf to save some work in subsequent automake or autoheader runs. It's recreated automatically if removed, it doesn't get distributed. ** C++ configuration It's intended that the contents of libgmp.la won't vary according to whether --enable-cxx is selected. This means that if C++ shared libraries don't work properly then a shared+static with --disable-cxx can be done for the C parts, then a static-only with --enable-cxx to get libgmpxx. libgmpxx.la uses some internals from libgmp.la, in order to share code between C and C++. It's intended that libgmpxx can only be expected to work with libgmp from the same version of GMP. If some of the shared internals change their interface, then it's proposed to rename them, for instance __gmp_doprint2 or the like, so as to provoke link errors rather than mysterious failures from a mismatch. * Development setups ** General --disable-shared will make builds go much faster, though of course shared or shared+static should be tested too. --prefix to a dummy directory followed by "make install" will show what's installed. "make check" acts on the libgmp just built, and will ignore any other /usr/lib/libgmp, or at least it should do. Libtool does various hairy things to ensure it hits the just-built library. ** Long long limb testing On systems where gcc supports long long, but a limb is normally just a long, the following can be used to force long long for testing purposes. It will probably run quite slowly. ./configure --host=none ABI=longlong ** Function argument conversions When using gcc, configuring with something like ./configure CFLAGS="-g -Wall -Wconversion -Wno-sign-compare" can show where function parameters are being converted due to having function prototypes available, which won't happen in a K&R compiler. Doing this in combination with the long long limb setups above is good. Conversions between int and long aren't warned about by gcc when they're the same size, which is unfortunate because casts should be used in such cases, for the benefit of K&R compilers with int!=long and where the difference matters in function calls. * Other Notes ** Compatibility compat.c is the home of functions retained for binary compatibility, but now done by other means (like a macro). struct __mpz_struct etc - this must be retained for C++ compatibility. C++ applications defining functions taking mpz_t etc parameters will get this in the mangled name because C++ "sees though" the typedef mpz_t to the underlying struct. __gmpn - note that glibc defines some __mpn symbols, old versions of some mpn routines, which it uses for floating point printfs. Local variables: mode: outline fill-column: 70 End: /* eof doc/configuration */ gmp-doc-5.1.2/doc/fdl-1.3.texi0000644000175000000620000005601512146435154014573 0ustar stevestaff@c The GNU Free Documentation License. @center Version 1.3, 3 November 2008 @c This file is intended to be included within another document, @c hence no sectioning command or @node. @display Copyright @copyright{} 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. @uref{http://fsf.org/} Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. @end display @enumerate 0 @item PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document @dfn{free} in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. 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The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing. @end enumerate @page @heading ADDENDUM: How to use this License for your documents To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: @smallexample @group Copyright (C) @var{year} @var{your name}. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''. @end group @end smallexample If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the ``with@dots{}Texts.'' line with this: @smallexample @group with the Invariant Sections being @var{list their titles}, with the Front-Cover Texts being @var{list}, and with the Back-Cover Texts being @var{list}. @end group @end smallexample If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. @c Local Variables: @c ispell-local-pdict: "ispell-dict" @c End: gmp-doc-5.1.2/doc/isa_abi_headache0000644000175000000620000001113212146435154015737 0ustar stevestaffCopyright 2000 Free Software Foundation, Inc. This file is part of the GNU MP Library. The GNU MP Library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. The GNU MP Library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. Terms Used In This Document: ISA = Instruction Set Architecture. The instructions the current processor provides. ABI = Application Binary Interface. Specifies calling convention, type sizes, etc. AR64 = Arithmetic operations are 64-bit using 64-bit instructions (E.g., addition, subtraction, load, store, of 64-bit integer types are done with single instructions, not 32 bits at a time.) Environment = The operating system and compiler. GMP is a very complex package to build since its speed is very sensitive to the ISA and ABI. For example, if the ISA provides 64-bit instructions, it is crucial that GMP is configured to use them. Most environments that run on a 64-bit ISA provide more than one ABI. Typically one of the supported ABI's is a backward compatible 32-bit ABI, and one ABI provides 64-bit addressing and `long' (sometimes known as LP64). But a few environments (IRIX, HP-UX) provide intermediate ABI's using 32-bit addressing but allow efficient 64-bit operations through a `long long' type. For the latter to be useful to GMP, the ABI must allow operations using the native 64-bit instructions provided by the ISA, and allow passing of 64-bit quantities atomically. The ABI is typically chosen by means of command line options to the compiler tools (gcc, cc, c89, nm, ar, ld, as). Different environments use different defaults, but as of this writing (May 2000) the dominating default is to the plain 32-bit ABI in its most arcane form. The GMP 3.0.x approach was to compile using the ABI that gives the best performance. That places the burden on users to pass special options to the compiler when they compile their GMP applications. That approach has its advantages and disadvantages. The main advantage is that users don't unknowingly get bad GMP performance. The main disadvantage is that users' compiles (actually links) will fail unless they pass special compiler options. ** SPARC System vendors often confuse ABI, ISA, and implementation. The worst case is Solaris, were the unbundled compiler confuses ISA and ABI, and the options have very confusing names. option interpretation ====== ============== cc -xarch=v8plus ISA=sparcv9, ABI=V8plus (PTR=32, see below) gcc -mv8plus ISA=sparcv9, ABI=V8plus (see below) cc -xarch=v9 ISA=sparcv9, ABI=V9 (implying AR=64, PTR=64) It's hard to believe, but the option v8plus really means ISA=V9! Solaris releases prior to version 7 running on a V9 CPU fails to save/restore the upper 32 bits of the `i' and `l' registers. The `v8plus' option generates code that use as many V9 features as possible under such circumstances. ** MIPS The IRIX 6 compilers gets things right. They have a clear understanding of the differences between ABI and ISA. The option names are descriptive. option interpretation ====== ============== cc -n32 ABI=n32 (implying AR=64, PTR=32) gcc -mabi=n32 ABI=n32 (implying AR=64, PTR=32) cc -64 ABI=64 (implying AR=64, PTR=64) gcc -mabi=64 ABI=64 (implying AR=64, PTR=64) cc -mips3 ISA=mips3 gcc -mips3 ISA=mips3 cc -mips4 ISA=mips4 gcc -mips4 ISA=mips4 ** HP-PA HP-UX is somewhat weird, but not as broken as Solaris. option interpretation ====== ============== cc +DA2.0 ABI=32bit (implying AR=64, PTR=32) cc +DD64 ABI=64bit (implying AR=64, PTR=64) Code performing 64-bit arithmetic in the HP-UX 32-bit is not compatible with the 64-bit ABI; the former has a calling convention that passes/returns 64-bit integer quantities as two 32-bit chunks. ** PowerPC While the PowerPC ABI's are capable of supporting 64-bit registers/operations, the compilers under AIX are similar to Solaris' cc in that they don't currently provide any 32-bit addressing with 64-bit arithmetic. option interpretation ====== ============== cc -q64 ABI=64bit (implying AR=64, PTR=64) gcc -maix64 -mpowerpc64 ABI=64bit (implying AR=64, PTR=64) gmp-doc-5.1.2/doc/gmp.info-10000644000175000000620000111200112146435202014411 0ustar stevestaffThis is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from ../../gmp/doc/gmp.texi. This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover Texts being "You have freedom to copy and modify this GNU Manual, like GNU software". A copy of the license is included in *note GNU Free Documentation License::. INFO-DIR-SECTION GNU libraries START-INFO-DIR-ENTRY * gmp: (gmp). GNU Multiple Precision Arithmetic Library. END-INFO-DIR-ENTRY  File: gmp.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir) GNU MP ****** This manual describes how to install and use the GNU multiple precision arithmetic library, version 5.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover Texts being "You have freedom to copy and modify this GNU Manual, like GNU software". A copy of the license is included in *note GNU Free Documentation License::. * Menu: * Copying:: GMP Copying Conditions (LGPL). * Introduction to GMP:: Brief introduction to GNU MP. * Installing GMP:: How to configure and compile the GMP library. * GMP Basics:: What every GMP user should know. * Reporting Bugs:: How to usefully report bugs. * Integer Functions:: Functions for arithmetic on signed integers. * Rational Number Functions:: Functions for arithmetic on rational numbers. * Floating-point Functions:: Functions for arithmetic on floats. * Low-level Functions:: Fast functions for natural numbers. * Random Number Functions:: Functions for generating random numbers. * Formatted Output:: `printf' style output. * Formatted Input:: `scanf' style input. * C++ Class Interface:: Class wrappers around GMP types. * Custom Allocation:: How to customize the internal allocation. * Language Bindings:: Using GMP from other languages. * Algorithms:: What happens behind the scenes. * Internals:: How values are represented behind the scenes. * Contributors:: Who brings you this library? * References:: Some useful papers and books to read. * GNU Free Documentation License:: * Concept Index:: * Function Index::  File: gmp.info, Node: Copying, Next: Introduction to GMP, Prev: Top, Up: Top GNU MP Copying Conditions ************************* This library is "free"; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you. Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things. To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MP library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights. Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MP library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. The precise conditions of the license for the GNU MP library are found in the Lesser General Public License version 3 that accompanies the source code, see `COPYING.LIB'. Certain demonstration programs are provided under the terms of the plain General Public License version 3, see `COPYING'.  File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top 1 Introduction to GNU MP ************************ GNU MP is a portable library written in C for arbitrary precision arithmetic on integers, rational numbers, and floating-point numbers. It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types. Many applications use just a few hundred bits of precision; but some applications may need thousands or even millions of bits. GMP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum. The speed of GMP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance). There is assembly code for these CPUs: ARM, DEC Alpha 21064, 21164, and 21264, AMD 29000, AMD K6, K6-2, Athlon, and Athlon64, Hitachi SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and Pyramid AP/XP. For up-to-date information on GMP, please see the GMP web pages at `http://gmplib.org/' The latest version of the library is available at `ftp://ftp.gnu.org/gnu/gmp/' Many sites around the world mirror `ftp.gnu.org', please use a mirror near you, see `http://www.gnu.org/order/ftp.html' for a full list. There are three public mailing lists of interest. One for release announcements, one for general questions and discussions about usage of the GMP library and one for bug reports. For more information, see `http://gmplib.org/mailman/listinfo/'. The proper place for bug reports is . See *note Reporting Bugs:: for information about reporting bugs. 1.1 How to use this Manual ========================== Everyone should read *note GMP Basics::. If you need to install the library yourself, then read *note Installing GMP::. If you have a system with multiple ABIs, then read *note ABI and ISA::, for the compiler options that must be used on applications. The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.  File: gmp.info, Node: Installing GMP, Next: GMP Basics, Prev: Introduction to GMP, Up: Top 2 Installing GMP **************** GMP has an autoconf/automake/libtool based configuration system. On a Unix-like system a basic build can be done with ./configure make Some self-tests can be run with make check And you can install (under `/usr/local' by default) with make install If you experience problems, please report them to . See *note Reporting Bugs::, for information on what to include in useful bug reports. * Menu: * Build Options:: * ABI and ISA:: * Notes for Package Builds:: * Notes for Particular Systems:: * Known Build Problems:: * Performance optimization::  File: gmp.info, Node: Build Options, Next: ABI and ISA, Prev: Installing GMP, Up: Installing GMP 2.1 Build Options ================= All the usual autoconf configure options are available, run `./configure --help' for a summary. The file `INSTALL.autoconf' has some generic installation information too. Tools `configure' requires various Unix-like tools. See *note Notes for Particular Systems::, for some options on non-Unix systems. It might be possible to build without the help of `configure', certainly all the code is there, but unfortunately you'll be on your own. Build Directory To compile in a separate build directory, `cd' to that directory, and prefix the configure command with the path to the GMP source directory. For example cd /my/build/dir /my/sources/gmp-5.1.2/configure Not all `make' programs have the necessary features (`VPATH') to support this. In particular, SunOS and Slowaris `make' have bugs that make them unable to build in a separate directory. Use GNU `make' instead. `--prefix' and `--exec-prefix' The `--prefix' option can be used in the normal way to direct GMP to install under a particular tree. The default is `/usr/local'. `--exec-prefix' can be used to direct architecture-dependent files like `libgmp.a' to a different location. This can be used to share architecture-independent parts like the documentation, but separate the dependent parts. Note however that `gmp.h' and `mp.h' are architecture-dependent since they encode certain aspects of `libgmp', so it will be necessary to ensure both `$prefix/include' and `$exec_prefix/include' are available to the compiler. `--disable-shared', `--disable-static' By default both shared and static libraries are built (where possible), but one or other can be disabled. Shared libraries result in smaller executables and permit code sharing between separate running processes, but on some CPUs are slightly slower, having a small cost on each function call. Native Compilation, `--build=CPU-VENDOR-OS' For normal native compilation, the system can be specified with `--build'. By default `./configure' uses the output from running `./config.guess'. On some systems `./config.guess' can determine the exact CPU type, on others it will be necessary to give it explicitly. For example, ./configure --build=ultrasparc-sun-solaris2.7 In all cases the `OS' part is important, since it controls how libtool generates shared libraries. Running `./config.guess' is the simplest way to see what it should be, if you don't know already. Cross Compilation, `--host=CPU-VENDOR-OS' When cross-compiling, the system used for compiling is given by `--build' and the system where the library will run is given by `--host'. For example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries, ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu Compiler tools are sought first with the host system type as a prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then plain `ranlib'. This makes it possible for a set of cross-compiling tools to co-exist with native tools. The prefix is the argument to `--host', and this can be an alias, such as `m68k-linux'. But note that tools don't have to be setup this way, it's enough to just have a `PATH' with a suitable cross-compiling `cc' etc. Compiling for a different CPU in the same family as the build system is a form of cross-compilation, though very possibly this would merely be special options on a native compiler. In any case `./configure' avoids depending on being able to run code on the build system, which is important when creating binaries for a newer CPU since they very possibly won't run on the build system. In all cases the compiler must be able to produce an executable (of whatever format) from a standard C `main'. Although only object files will go to make up `libgmp', `./configure' uses linking tests for various purposes, such as determining what functions are available on the host system. Currently a warning is given unless an explicit `--build' is used when cross-compiling, because it may not be possible to correctly guess the build system type if the `PATH' has only a cross-compiling `cc'. Note that the `--target' option is not appropriate for GMP. It's for use when building compiler tools, with `--host' being where they will run, and `--target' what they'll produce code for. Ordinary programs or libraries like GMP are only interested in the `--host' part, being where they'll run. (Some past versions of GMP used `--target' incorrectly.) CPU types In general, if you want a library that runs as fast as possible, you should configure GMP for the exact CPU type your system uses. However, this may mean the binaries won't run on older members of the family, and might run slower on other members, older or newer. The best idea is always to build GMP for the exact machine type you intend to run it on. The following CPUs have specific support. See `configure.in' for details of what code and compiler options they select. * Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57, alphaev6, alphaev67, alphaev68 alphaev7 * Cray: c90, j90, t90, sv1 * HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w, hppa64 * IA-64: ia64, itanium, itanium2 * MIPS: mips, mips3, mips64 * Motorola: m68k, m68000, m68010, m68020, m68030, m68040, m68060, m68302, m68360, m88k, m88110 * POWER: power, power1, power2, power2sc * PowerPC: powerpc, powerpc64, powerpc401, powerpc403, powerpc405, powerpc505, powerpc601, powerpc602, powerpc603, powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630, powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801, powerpc821, powerpc823, powerpc860, powerpc970 * SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9, ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64 * x86 family: i386, i486, i586, pentium, pentiummmx, pentiumpro, pentium2, pentium3, pentium4, k6, k62, k63, athlon, amd64, viac3, viac32 * Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax, z8k CPUs not listed will use generic C code. Generic C Build If some of the assembly code causes problems, or if otherwise desired, the generic C code can be selected with the configure `--disable-assembly'. Note that this will run quite slowly, but it should be portable and should at least make it possible to get something running if all else fails. Fat binary, `--enable-fat' Using `--enable-fat' selects a "fat binary" build on x86, where optimized low level subroutines are chosen at runtime according to the CPU detected. This means more code, but gives good performance on all x86 chips. (This option might become available for more architectures in the future.) `ABI' On some systems GMP supports multiple ABIs (application binary interfaces), meaning data type sizes and calling conventions. By default GMP chooses the best ABI available, but a particular ABI can be selected. For example ./configure --host=mips64-sgi-irix6 ABI=n32 See *note ABI and ISA::, for the available choices on relevant CPUs, and what applications need to do. `CC', `CFLAGS' By default the C compiler used is chosen from among some likely candidates, with `gcc' normally preferred if it's present. The usual `CC=whatever' can be passed to `./configure' to choose something different. For various systems, default compiler flags are set based on the CPU and compiler. The usual `CFLAGS="-whatever"' can be passed to `./configure' to use something different or to set good flags for systems GMP doesn't otherwise know. The `CC' and `CFLAGS' used are printed during `./configure', and can be found in each generated `Makefile'. This is the easiest way to check the defaults when considering changing or adding something. Note that when `CC' and `CFLAGS' are specified on a system supporting multiple ABIs it's important to give an explicit `ABI=whatever', since GMP can't determine the ABI just from the flags and won't be able to select the correct assembly code. If just `CC' is selected then normal default `CFLAGS' for that compiler will be used (if GMP recognises it). For example `CC=gcc' can be used to force the use of GCC, with default flags (and default ABI). `CPPFLAGS' Any flags like `-D' defines or `-I' includes required by the preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'. Compiling is done with both `CPPFLAGS' and `CFLAGS', but preprocessing uses just `CPPFLAGS'. This distinction is because most preprocessors won't accept all the flags the compiler does. Preprocessing is done separately in some configure tests. `CC_FOR_BUILD' Some build-time programs are compiled and run to generate host-specific data tables. `CC_FOR_BUILD' is the compiler used for this. It doesn't need to be in any particular ABI or mode, it merely needs to generate executables that can run. The default is to try the selected `CC' and some likely candidates such as `cc' and `gcc', looking for something that works. No flags are used with `CC_FOR_BUILD' because a simple invocation like `cc foo.c' should be enough. If some particular options are required they can be included as for instance `CC_FOR_BUILD="cc -whatever"'. C++ Support, `--enable-cxx' C++ support in GMP can be enabled with `--enable-cxx', in which case a C++ compiler will be required. As a convenience `--enable-cxx=detect' can be used to enable C++ support only if a compiler can be found. The C++ support consists of a library `libgmpxx.la' and header file `gmpxx.h' (*note Headers and Libraries::). A separate `libgmpxx.la' has been adopted rather than having C++ objects within `libgmp.la' in order to ensure dynamic linked C programs aren't bloated by a dependency on the C++ standard library, and to avoid any chance that the C++ compiler could be required when linking plain C programs. `libgmpxx.la' will use certain internals from `libgmp.la' and can only be expected to work with `libgmp.la' from the same GMP version. Future changes to the relevant internals will be accompanied by renaming, so a mismatch will cause unresolved symbols rather than perhaps mysterious misbehaviour. In general `libgmpxx.la' will be usable only with the C++ compiler that built it, since name mangling and runtime support are usually incompatible between different compilers. `CXX', `CXXFLAGS' When C++ support is enabled, the C++ compiler and its flags can be set with variables `CXX' and `CXXFLAGS' in the usual way. The default for `CXX' is the first compiler that works from a list of likely candidates, with `g++' normally preferred when available. The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without `-g', then for `g++' either `-g -O2' or `-O2', or for other compilers `-g' or nothing. Trying `CFLAGS' this way is convenient when using `gcc' and `g++' together, since the flags for `gcc' will usually suit `g++'. It's important that the C and C++ compilers match, meaning their startup and runtime support routines are compatible and that they generate code in the same ABI (if there's a choice of ABIs on the system). `./configure' isn't currently able to check these things very well itself, so for that reason `--disable-cxx' is the default, to avoid a build failure due to a compiler mismatch. Perhaps this will change in the future. Incidentally, it's normally not good enough to set `CXX' to the same as `CC'. Although `gcc' for instance recognises `foo.cc' as C++ code, only `g++' will invoke the linker the right way when building an executable or shared library from C++ object files. Temporary Memory, `--enable-alloca=' GMP allocates temporary workspace using one of the following three methods, which can be selected with for instance `--enable-alloca=malloc-reentrant'. * `alloca' - C library or compiler builtin. * `malloc-reentrant' - the heap, in a re-entrant fashion. * `malloc-notreentrant' - the heap, with global variables. For convenience, the following choices are also available. `--disable-alloca' is the same as `no'. * `yes' - a synonym for `alloca'. * `no' - a synonym for `malloc-reentrant'. * `reentrant' - `alloca' if available, otherwise `malloc-reentrant'. This is the default. * `notreentrant' - `alloca' if available, otherwise `malloc-notreentrant'. `alloca' is reentrant and fast, and is recommended. It actually allocates just small blocks on the stack; larger ones use malloc-reentrant. `malloc-reentrant' is, as the name suggests, reentrant and thread safe, but `malloc-notreentrant' is faster and should be used if reentrancy is not required. The two malloc methods in fact use the memory allocation functions selected by `mp_set_memory_functions', these being `malloc' and friends by default. *Note Custom Allocation::. An additional choice `--enable-alloca=debug' is available, to help when debugging memory related problems (*note Debugging::). FFT Multiplication, `--disable-fft' By default multiplications are done using Karatsuba, 3-way Toom, higher degree Toom, and Fermat FFT. The FFT is only used on large to very large operands and can be disabled to save code size if desired. Assertion Checking, `--enable-assert' This option enables some consistency checking within the library. This can be of use while debugging, *note Debugging::. Execution Profiling, `--enable-profiling=prof/gprof/instrument' Enable profiling support, in one of various styles, *note Profiling::. `MPN_PATH' Various assembly versions of each mpn subroutines are provided. For a given CPU, a search is made though a path to choose a version of each. For example `sparcv8' has MPN_PATH="sparc32/v8 sparc32 generic" which means look first for v8 code, then plain sparc32 (which is v7), and finally fall back on generic C. Knowledgeable users with special requirements can specify a different path. Normally this is completely unnecessary. Documentation The source for the document you're now reading is `doc/gmp.texi', in Texinfo format, see *note Texinfo: (texinfo)Top. Info format `doc/gmp.info' is included in the distribution. The usual automake targets are available to make PostScript, DVI, PDF and HTML (these will require various TeX and Texinfo tools). DocBook and XML can be generated by the Texinfo `makeinfo' program too, see *note Options for `makeinfo': (texinfo)makeinfo options. Some supplementary notes can also be found in the `doc' subdirectory.  File: gmp.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing GMP 2.2 ABI and ISA =============== ABI (Application Binary Interface) refers to the calling conventions between functions, meaning what registers are used and what sizes the various C data types are. ISA (Instruction Set Architecture) refers to the instructions and registers a CPU has available. Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the latter for compatibility with older CPUs in the family. GMP supports some CPUs like this in both ABIs. In fact within GMP `ABI' means a combination of chip ABI, plus how GMP chooses to use it. For example in some 32-bit ABIs, GMP may support a limb as either a 32-bit `long' or a 64-bit `long long'. By default GMP chooses the best ABI available for a given system, and this generally gives significantly greater speed. But an ABI can be chosen explicitly to make GMP compatible with other libraries, or particular application requirements. For example, ./configure ABI=32 In all cases it's vital that all object code used in a given program is compiled for the same ABI. Usually a limb is implemented as a `long'. When a `long long' limb is used this is encoded in the generated `gmp.h'. This is convenient for applications, but it does mean that `gmp.h' will vary, and can't be just copied around. `gmp.h' remains compiler independent though, since all compilers for a particular ABI will be expected to use the same limb type. Currently no attempt is made to follow whatever conventions a system has for installing library or header files built for a particular ABI. This will probably only matter when installing multiple builds of GMP, and it might be as simple as configuring with a special `libdir', or it might require more than that. Note that builds for different ABIs need to done separately, with a fresh `./configure' and `make' each. AMD64 (`x86_64') On AMD64 systems supporting both 32-bit and 64-bit modes for applications, the following ABI choices are available. `ABI=64' The 64-bit ABI uses 64-bit limbs and pointers and makes full use of the chip architecture. This is the default. Applications will usually not need special compiler flags, but for reference the option is gcc -m64 `ABI=32' The 32-bit ABI is the usual i386 conventions. This will be slower, and is not recommended except for inter-operating with other code not yet 64-bit capable. Applications must be compiled with gcc -m32 (In GCC 2.95 and earlier there's no `-m32' option, it's the only mode.) HPPA 2.0 (`hppa2.0*', `hppa64') `ABI=2.0w' The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or up. Applications must be compiled with gcc [built for 2.0w] cc +DD64 `ABI=2.0n' The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal calling conventions, but with 64-bit instructions permitted within functions. GMP uses a 64-bit `long long' for a limb. This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or higher. Applications must be compiled with gcc [built for 2.0n] cc +DA2.0 +e Note that current versions of GCC (eg. 3.2) don't generate 64-bit instructions for `long long' operations and so may be slower than for 2.0w. (The GMP assembly code is the same though.) `ABI=1.0' HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI. No special compiler options are needed for applications. All three ABIs are available for CPU types `hppa2.0w', `hppa2.0' and `hppa64', but for CPU type `hppa2.0n' only 2.0n or 1.0 are considered. Note that GCC on HP-UX has no options to choose between 2.0n and 2.0w modes, unlike HP `cc'. Instead it must be built for one or the other ABI. GMP will detect how it was built, and skip to the corresponding `ABI'. IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*') HP-UX supports two ABIs for IA-64. GMP performance is the same in both. `ABI=32' In the 32-bit ABI, pointers, `int's and `long's are 32 bits and GMP uses a 64 bit `long long' for a limb. Applications can be compiled without any special flags since this ABI is the default in both HP C and GCC, but for reference the flags are gcc -milp32 cc +DD32 `ABI=64' In the 64-bit ABI, `long's and pointers are 64 bits and GMP uses a `long' for a limb. Applications must be compiled with gcc -mlp64 cc +DD64 On other IA-64 systems, GNU/Linux for instance, `ABI=64' is the only choice. MIPS under IRIX 6 (`mips*-*-irix[6789]') IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs o32, n32, and 64. n32 or 64 are recommended, and GMP performance will be the same in each. The default is n32. `ABI=o32' The o32 ABI is 32-bit pointers and integers, and no 64-bit operations. GMP will be slower than in n32 or 64, this option only exists to support old compilers, eg. GCC 2.7.2. Applications can be compiled with no special flags on an old compiler, or on a newer compiler with gcc -mabi=32 cc -32 `ABI=n32' The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a `long long'. Applications must be compiled with gcc -mabi=n32 cc -n32 `ABI=64' The 64-bit ABI is 64-bit pointers and integers. Applications must be compiled with gcc -mabi=64 cc -64 Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code. PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970', `power4', `power5') `ABI=mode64' The AIX 64 ABI uses 64-bit limbs and pointers and is the default on PowerPC 64 `*-*-aix*' systems. Applications must be compiled with gcc -maix64 xlc -q64 On 64-bit GNU/Linux, BSD, and Mac OS X/Darwin systems, the applications must be compiled with gcc -m64 `ABI=mode32' The `mode32' ABI uses a 64-bit `long long' limb but with the chip still in 32-bit mode and using 32-bit calling conventions. This is the default for systems where the true 64-bit ABI is unavailable. No special compiler options are typically needed for applications. This ABI is not available under AIX. `ABI=32' This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No special compiler options are needed for applications. GMP's speed is greatest for the `mode64' ABI, the `mode32' ABI is 2nd best. In `ABI=32' only the 32-bit ISA is used and this doesn't make full use of a 64-bit chip. Sparc V9 (`sparc64', `sparcv9', `ultrasparc*') `ABI=64' The 64-bit V9 ABI is available on the various BSD sparc64 ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7 and up (when the kernel is in 64-bit mode). GCC 3.2 or higher, or Sun `cc' is required. On GNU/Linux, depending on the default `gcc' mode, applications must be compiled with gcc -m64 On Solaris applications must be compiled with gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9 cc -xarch=v9 On the BSD sparc64 systems no special options are required, since 64-bits is the only ABI available. `ABI=32' For the basic 32-bit ABI, GMP still uses as much of the V9 ISA as it can. In the Sun documentation this combination is known as "v8plus". On GNU/Linux, depending on the default `gcc' mode, applications may need to be compiled with gcc -m32 On Solaris, no special compiler options are required for applications, though using something like the following is recommended. (`gcc' 2.8 and earlier only support `-mv8' though.) gcc -mv8plus cc -xarch=v8plus GMP speed is greatest in `ABI=64', so it's the default where available. The speed is partly because there are extra registers available and partly because 64-bits is considered the more important case and has therefore had better code written for it. Don't be confused by the names of the `-m' and `-x' compiler options, they're called `arch' but effectively control both ABI and ISA. On Solaris 2.6 and earlier, only `ABI=32' is available since the kernel doesn't save all registers. On Solaris 2.7 with the kernel in 32-bit mode, a normal native build will reject `ABI=64' because the resulting executables won't run. `ABI=64' can still be built if desired by making it look like a cross-compile, for example ./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64  File: gmp.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing GMP 2.3 Notes for Package Builds ============================ GMP should present no great difficulties for packaging in a binary distribution. Libtool is used to build the library and `-version-info' is set appropriately, having started from `3:0:0' in GMP 3.0 (*note Library interface versions: (libtool)Versioning.). The GMP 4 series will be upwardly binary compatible in each release and will be upwardly binary compatible with all of the GMP 3 series. Additional function interfaces may be added in each release, so on systems where libtool versioning is not fully checked by the loader an auxiliary mechanism may be needed to express that a dynamic linked application depends on a new enough GMP. An auxiliary mechanism may also be needed to express that `libgmpxx.la' (from `--enable-cxx', *note Build Options::) requires `libgmp.la' from the same GMP version, since this is not done by the libtool versioning, nor otherwise. A mismatch will result in unresolved symbols from the linker, or perhaps the loader. When building a package for a CPU family, care should be taken to use `--host' (or `--build') to choose the least common denominator among the CPUs which might use the package. For example this might mean plain `sparc' (meaning V7) for SPARCs. For x86s, `--enable-fat' sets things up for a fat binary build, making a runtime selection of optimized low level routines. This is a good choice for packaging to run on a range of x86 chips. Users who care about speed will want GMP built for their exact CPU type, to make best use of the available optimizations. Providing a way to suitably rebuild a package may be useful. This could be as simple as making it possible for a user to omit `--build' (and `--host') so `./config.guess' will detect the CPU. But a way to manually specify a `--build' will be wanted for systems where `./config.guess' is inexact. On systems with multiple ABIs, a packaged build will need to decide which among the choices is to be provided, see *note ABI and ISA::. A given run of `./configure' etc will only build one ABI. If a second ABI is also required then a second run of `./configure' etc must be made, starting from a clean directory tree (`make distclean'). As noted under "ABI and ISA", currently no attempt is made to follow system conventions for install locations that vary with ABI, such as `/usr/lib/sparcv9' for `ABI=64' as opposed to `/usr/lib' for `ABI=32'. A package build can override `libdir' and other standard variables as necessary. Note that `gmp.h' is a generated file, and will be architecture and ABI dependent. When attempting to install two ABIs simultaneously it will be important that an application compile gets the correct `gmp.h' for its desired ABI. If compiler include paths don't vary with ABI options then it might be necessary to create a `/usr/include/gmp.h' which tests preprocessor symbols and chooses the correct actual `gmp.h'.  File: gmp.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing GMP 2.4 Notes for Particular Systems ================================ AIX 3 and 4 On systems `*-*-aix[34]*' shared libraries are disabled by default, since some versions of the native `ar' fail on the convenience libraries used. A shared build can be attempted with ./configure --enable-shared --disable-static Note that the `--disable-static' is necessary because in a shared build libtool makes `libgmp.a' a symlink to `libgmp.so', apparently for the benefit of old versions of `ld' which only recognise `.a', but unfortunately this is done even if a fully functional `ld' is available. ARM On systems `arm*-*-*', versions of GCC up to and including 2.95.3 have a bug in unsigned division, giving wrong results for some operands. GMP `./configure' will demand GCC 2.95.4 or later. Compaq C++ Compaq C++ on OSF 5.1 has two flavours of `iostream', a standard one and an old pre-standard one (see `man iostream_intro'). GMP can only use the standard one, which unfortunately is not the default but must be selected by defining `__USE_STD_IOSTREAM'. Configure with for instance ./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM Floating Point Mode On some systems, the hardware floating point has a control mode which can set all operations to be done in a particular precision, for instance single, double or extended on x86 systems (x87 floating point). The GMP functions involving a `double' cannot be expected to operate to their full precision when the hardware is in single precision mode. Of course this affects all code, including application code, not just GMP. MS-DOS and MS Windows On an MS-DOS system DJGPP can be used to build GMP, and on an MS Windows system Cygwin, DJGPP and MINGW can be used. All three are excellent ports of GCC and the various GNU tools. `http://www.cygwin.com/' `http://www.delorie.com/djgpp/' `http://www.mingw.org/' Microsoft also publishes an Interix "Services for Unix" which can be used to build GMP on Windows (with a normal `./configure'), but it's not free software. MS Windows DLLs On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default GMP builds only a static library, but a DLL can be built instead using ./configure --disable-static --enable-shared Static and DLL libraries can't both be built, since certain export directives in `gmp.h' must be different. A MINGW DLL build of GMP can be used with Microsoft C. Libtool doesn't install a `.lib' format import library, but it can be created with MS `lib' as follows, and copied to the install directory. Similarly for `libmp' and `libgmpxx'. cd .libs lib /def:libgmp-3.dll.def /out:libgmp-3.lib MINGW uses the C runtime library `msvcrt.dll' for I/O, so applications wanting to use the GMP I/O routines must be compiled with `cl /MD' to do the same. If one of the other C runtime library choices provided by MS C is desired then the suggestion is to use the GMP string functions and confine I/O to the application. Motorola 68k CPU Types `m68k' is taken to mean 68000. `m68020' or higher will give a performance boost on applicable CPUs. `m68360' can be used for CPU32 series chips. `m68302' can be used for "Dragonball" series chips, though this is merely a synonym for `m68000'. OpenBSD 2.6 `m4' in this release of OpenBSD has a bug in `eval' that makes it unsuitable for `.asm' file processing. `./configure' will detect the problem and either abort or choose another m4 in the `PATH'. The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4. Power CPU Types In GMP, CPU types `power*' and `powerpc*' will each use instructions not available on the other, so it's important to choose the right one for the CPU that will be used. Currently GMP has no assembly code support for using just the common instruction subset. To get executables that run on both, the current suggestion is to use the generic C code (`--disable-assembly'), possibly with appropriate compiler options (like `-mcpu=common' for `gcc'). CPU `rs6000' (which is not a CPU but a family of workstations) is accepted by `config.sub', but is currently equivalent to `--disable-assembly'. Sparc CPU Types `sparcv8' or `supersparc' on relevant systems will give a significant performance increase over the V7 code selected by plain `sparc'. Sparc App Regs The GMP assembly code for both 32-bit and 64-bit Sparc clobbers the "application registers" `g2', `g3' and `g4', the same way that the GCC default `-mapp-regs' does (*note SPARC Options: (gcc)SPARC Options.). This makes that code unsuitable for use with the special V9 `-mcmodel=embmedany' (which uses `g4' as a data segment pointer), and for applications wanting to use those registers for special purposes. In these cases the only suggestion currently is to build GMP with `--disable-assembly' to avoid the assembly code. SunOS 4 `/usr/bin/m4' lacks various features needed to process `.asm' files, and instead `./configure' will automatically use `/usr/5bin/m4', which we believe is always available (if not then use GNU m4). x86 CPU Types `i586', `pentium' or `pentiummmx' code is good for its intended P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II, P-III). `i386' is a better choice when making binaries that must run on both. x86 MMX and SSE2 Code If the CPU selected has MMX code but the assembler doesn't support it, a warning is given and non-MMX code is used instead. This will be an inferior build, since the MMX code that's present is there because it's faster than the corresponding plain integer code. The same applies to SSE2. Old versions of `gas' don't support MMX instructions, in particular version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent OpenBSD 3.1 doesn't. Solaris 2.6 and 2.7 `as' generate incorrect object code for register to register `movq' instructions, and so can't be used for MMX code. Install a recent `gas' if MMX code is wanted on these systems.  File: gmp.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing GMP 2.5 Known Build Problems ======================== You might find more up-to-date information at `http://gmplib.org/'. Compiler link options The version of libtool currently in use rather aggressively strips compiler options when linking a shared library. This will hopefully be relaxed in the future, but for now if this is a problem the suggestion is to create a little script to hide them, and for instance configure with ./configure CC=gcc-with-my-options DJGPP (`*-*-msdosdjgpp*') The DJGPP port of `bash' 2.03 is unable to run the `configure' script, it exits silently, having died writing a preamble to `config.log'. Use `bash' 2.04 or higher. `make all' was found to run out of memory during the final `libgmp.la' link on one system tested, despite having 64Mb available. Running `make libgmp.la' directly helped, perhaps recursing into the various subdirectories uses up memory. GNU binutils `strip' prior to 2.12 `strip' from GNU binutils 2.11 and earlier should not be used on the static libraries `libgmp.a' and `libmp.a' since it will discard all but the last of multiple archive members with the same name, like the three versions of `init.o' in `libgmp.a'. Binutils 2.12 or higher can be used successfully. The shared libraries `libgmp.so' and `libmp.so' are not affected by this and any version of `strip' can be used on them. `make' syntax error On certain versions of SCO OpenServer 5 and IRIX 6.5 the native `make' is unable to handle the long dependencies list for `libgmp.la'. The symptom is a "syntax error" on the following line of the top-level `Makefile'. libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES) Either use GNU Make, or as a workaround remove `$(libgmp_la_DEPENDENCIES)' from that line (which will make the initial build work, but if any recompiling is done `libgmp.la' might not be rebuilt). MacOS X (`*-*-darwin*') Libtool currently only knows how to create shared libraries on MacOS X using the native `cc' (which is a modified GCC), not a plain GCC. A static-only build should work though (`--disable-shared'). NeXT prior to 3.3 The system compiler on old versions of NeXT was a massacred and old GCC, even if it called itself `cc'. This compiler cannot be used to build GMP, you need to get a real GCC, and install that. (NeXT may have fixed this in release 3.3 of their system.) POWER and PowerPC Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP on POWER or PowerPC. If you want to use GCC for these machines, get GCC 2.7.2.1 (or later). Sequent Symmetry Use the GNU assembler instead of the system assembler, since the latter has serious bugs. Solaris 2.6 The system `sed' prints an error "Output line too long" when libtool builds `libgmp.la'. This doesn't seem to cause any obvious ill effects, but GNU `sed' is recommended, to avoid any doubt. Sparc Solaris 2.7 with gcc 2.95.2 in `ABI=32' A shared library build of GMP seems to fail in this combination, it builds but then fails the tests, apparently due to some incorrect data relocations within `gmp_randinit_lc_2exp_size'. The exact cause is unknown, `--disable-shared' is recommended.  File: gmp.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing GMP 2.6 Performance optimization ============================ For optimal performance, build GMP for the exact CPU type of the target computer, see *note Build Options::. Unlike what is the case for most other programs, the compiler typically doesn't matter much, since GMP uses assembly language for the most critical operation. In particular for long-running GMP applications, and applications demanding extremely large numbers, building and running the `tuneup' program in the `tune' subdirectory, can be important. For example, cd tune make tuneup ./tuneup will generate better contents for the `gmp-mparam.h' parameter file. To use the results, put the output in the file indicated in the `Parameters for ...' header. Then recompile from scratch. The `tuneup' program takes one useful parameter, `-f NNN', which instructs the program how long to check FFT multiply parameters. If you're going to use GMP for extremely large numbers, you may want to run `tuneup' with a large NNN value.  File: gmp.info, Node: GMP Basics, Next: Reporting Bugs, Prev: Installing GMP, Up: Top 3 GMP Basics ************ *Using functions, macros, data types, etc. not documented in this manual is strongly discouraged. If you do so your application is guaranteed to be incompatible with future versions of GMP.* * Menu: * Headers and Libraries:: * Nomenclature and Types:: * Function Classes:: * Variable Conventions:: * Parameter Conventions:: * Memory Management:: * Reentrancy:: * Useful Macros and Constants:: * Compatibility with older versions:: * Demonstration Programs:: * Efficiency:: * Debugging:: * Profiling:: * Autoconf:: * Emacs::  File: gmp.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: GMP Basics, Up: GMP Basics 3.1 Headers and Libraries ========================= All declarations needed to use GMP are collected in the include file `gmp.h'. It is designed to work with both C and C++ compilers. #include Note however that prototypes for GMP functions with `FILE *' parameters are only provided if `' is included too. #include #include Likewise `' (or `') is required for prototypes with `va_list' parameters, such as `gmp_vprintf'. And `' for prototypes with `struct obstack' parameters, such as `gmp_obstack_printf', when available. All programs using GMP must link against the `libgmp' library. On a typical Unix-like system this can be done with `-lgmp', for example gcc myprogram.c -lgmp GMP C++ functions are in a separate `libgmpxx' library. This is built and installed if C++ support has been enabled (*note Build Options::). For example, g++ mycxxprog.cc -lgmpxx -lgmp GMP is built using Libtool and an application can use that to link if desired, *note GNU Libtool: (libtool)Top. If GMP has been installed to a non-standard location then it may be necessary to use `-I' and `-L' compiler options to point to the right directories, and some sort of run-time path for a shared library.  File: gmp.info, Node: Nomenclature and Types, Next: Function Classes, Prev: Headers and Libraries, Up: GMP Basics 3.2 Nomenclature and Types ========================== In this manual, "integer" usually means a multiple precision integer, as defined by the GMP library. The C data type for such integers is `mpz_t'. Here are some examples of how to declare such integers: mpz_t sum; struct foo { mpz_t x, y; }; mpz_t vec[20]; "Rational number" means a multiple precision fraction. The C data type for these fractions is `mpq_t'. For example: mpq_t quotient; "Floating point number" or "Float" for short, is an arbitrary precision mantissa with a limited precision exponent. The C data type for such objects is `mpf_t'. For example: mpf_t fp; The floating point functions accept and return exponents in the C type `mp_exp_t'. Currently this is usually a `long', but on some systems it's an `int' for efficiency. A "limb" means the part of a multi-precision number that fits in a single machine word. (We chose this word because a limb of the human body is analogous to a digit, only larger, and containing several digits.) Normally a limb is 32 or 64 bits. The C data type for a limb is `mp_limb_t'. Counts of limbs of a multi-precision number represented in the C type `mp_size_t'. Currently this is normally a `long', but on some systems it's an `int' for efficiency, and on some systems it will be `long long' in the future. Counts of bits of a multi-precision number are represented in the C type `mp_bitcnt_t'. Currently this is always an `unsigned long', but on some systems it will be an `unsigned long long' in the future. "Random state" means an algorithm selection and current state data. The C data type for such objects is `gmp_randstate_t'. For example: gmp_randstate_t rstate; Also, in general `mp_bitcnt_t' is used for bit counts and ranges, and `size_t' is used for byte or character counts.  File: gmp.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: GMP Basics 3.3 Function Classes ==================== There are six classes of functions in the GMP library: 1. Functions for signed integer arithmetic, with names beginning with `mpz_'. The associated type is `mpz_t'. There are about 150 functions in this class. (*note Integer Functions::) 2. Functions for rational number arithmetic, with names beginning with `mpq_'. The associated type is `mpq_t'. There are about 40 functions in this class, but the integer functions can be used for arithmetic on the numerator and denominator separately. (*note Rational Number Functions::) 3. Functions for floating-point arithmetic, with names beginning with `mpf_'. The associated type is `mpf_t'. There are about 60 functions is this class. (*note Floating-point Functions::) 4. Fast low-level functions that operate on natural numbers. These are used by the functions in the preceding groups, and you can also call them directly from very time-critical user programs. These functions' names begin with `mpn_'. The associated type is array of `mp_limb_t'. There are about 30 (hard-to-use) functions in this class. (*note Low-level Functions::) 5. Miscellaneous functions. Functions for setting up custom allocation and functions for generating random numbers. (*note Custom Allocation::, and *note Random Number Functions::)  File: gmp.info, Node: Variable Conventions, Next: Parameter Conventions, Prev: Function Classes, Up: GMP Basics 3.4 Variable Conventions ======================== GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last. GMP lets you use the same variable for both input and output in one call. For example, the main function for integer multiplication, `mpz_mul', can be used to square `x' and put the result back in `x' with mpz_mul (x, x, x); Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details. A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid excessive initializing and clearing. In general, initialize near the start of a function and clear near the end. For example, void foo (void) { mpz_t n; int i; mpz_init (n); for (i = 1; i < 100; i++) { mpz_mul (n, ...); mpz_fdiv_q (n, ...); ... } mpz_clear (n); }  File: gmp.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: GMP Basics 3.5 Parameter Conventions ========================= When a GMP variable is used as a function parameter, it's effectively a call-by-reference, meaning if the function stores a value there it will change the original in the caller. Parameters which are input-only can be designated `const' to provoke a compiler error or warning on attempting to modify them. When a function is going to return a GMP result, it should designate a parameter that it sets, like the library functions do. More than one value can be returned by having more than one output parameter, again like the library functions. A `return' of an `mpz_t' etc doesn't return the object, only a pointer, and this is almost certainly not what's wanted. Here's an example accepting an `mpz_t' parameter, doing a calculation, and storing the result to the indicated parameter. void foo (mpz_t result, const mpz_t param, unsigned long n) { unsigned long i; mpz_mul_ui (result, param, n); for (i = 1; i < n; i++) mpz_add_ui (result, result, i*7); } int main (void) { mpz_t r, n; mpz_init (r); mpz_init_set_str (n, "123456", 0); foo (r, n, 20L); gmp_printf ("%Zd\n", r); return 0; } `foo' works even if the mainline passes the same variable for `param' and `result', just like the library functions. But sometimes it's tricky to make that work, and an application might not want to bother supporting that sort of thing. For interest, the GMP types `mpz_t' etc are implemented as one-element arrays of certain structures. This is why declaring a variable creates an object with the fields GMP needs, but then using it as a parameter passes a pointer to the object. Note that the actual fields in each `mpz_t' etc are for internal use only and should not be accessed directly by code that expects to be compatible with future GMP releases.  File: gmp.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: GMP Basics 3.6 Memory Management ===================== The GMP types like `mpz_t' are small, containing only a couple of sizes, and pointers to allocated data. Once a variable is initialized, GMP takes care of all space allocation. Additional space is allocated whenever a variable doesn't have enough. `mpz_t' and `mpq_t' variables never reduce their allocated space. Normally this is the best policy, since it avoids frequent reallocation. Applications that need to return memory to the heap at some particular point can use `mpz_realloc2', or clear variables no longer needed. `mpf_t' variables, in the current implementation, use a fixed amount of space, determined by the chosen precision and allocated at initialization, so their size doesn't change. All memory is allocated using `malloc' and friends by default, but this can be changed, see *note Custom Allocation::. Temporary memory on the stack is also used (via `alloca'), but this can be changed at build-time if desired, see *note Build Options::.  File: gmp.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: GMP Basics 3.7 Reentrancy ============== GMP is reentrant and thread-safe, with some exceptions: * If configured with `--enable-alloca=malloc-notreentrant' (or with `--enable-alloca=notreentrant' when `alloca' is not available), then naturally GMP is not reentrant. * `mpf_set_default_prec' and `mpf_init' use a global variable for the selected precision. `mpf_init2' can be used instead, and in the C++ interface an explicit precision to the `mpf_class' constructor. * `mpz_random' and the other old random number functions use a global random state and are hence not reentrant. The newer random number functions that accept a `gmp_randstate_t' parameter can be used instead. * `gmp_randinit' (obsolete) returns an error indication through a global variable, which is not thread safe. Applications are advised to use `gmp_randinit_default' or `gmp_randinit_lc_2exp' instead. * `mp_set_memory_functions' uses global variables to store the selected memory allocation functions. * If the memory allocation functions set by a call to `mp_set_memory_functions' (or `malloc' and friends by default) are not reentrant, then GMP will not be reentrant either. * If the standard I/O functions such as `fwrite' are not reentrant then the GMP I/O functions using them will not be reentrant either. * It's safe for two threads to read from the same GMP variable simultaneously, but it's not safe for one to read while the another might be writing, nor for two threads to write simultaneously. It's not safe for two threads to generate a random number from the same `gmp_randstate_t' simultaneously, since this involves an update of that variable.  File: gmp.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: GMP Basics 3.8 Useful Macros and Constants =============================== -- Global Constant: const int mp_bits_per_limb The number of bits per limb. -- Macro: __GNU_MP_VERSION -- Macro: __GNU_MP_VERSION_MINOR -- Macro: __GNU_MP_VERSION_PATCHLEVEL The major and minor GMP version, and patch level, respectively, as integers. For GMP i.j, these numbers will be i, j, and 0, respectively. For GMP i.j.k, these numbers will be i, j, and k, respectively. -- Global Constant: const char * const gmp_version The GMP version number, as a null-terminated string, in the form "i.j.k". This release is "5.1.2". Note that the format "i.j" was used, before version 4.3.0, when k was zero. -- Macro: __GMP_CC -- Macro: __GMP_CFLAGS The compiler and compiler flags, respectively, used when compiling GMP, as strings.  File: gmp.info, Node: Compatibility with older versions, Next: Demonstration Programs, Prev: Useful Macros and Constants, Up: GMP Basics 3.9 Compatibility with older versions ===================================== This version of GMP is upwardly binary compatible with all 5.x, 4.x, and 3.x versions, and upwardly compatible at the source level with all 2.x versions, with the following exceptions. * `mpn_gcd' had its source arguments swapped as of GMP 3.0, for consistency with other `mpn' functions. * `mpf_get_prec' counted precision slightly differently in GMP 3.0 and 3.0.1, but in 3.1 reverted to the 2.x style. * `mpn_bdivmod', documented as preliminary in GMP 4, has been removed. There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 5. Please see the GMP 2 manual for details.  File: gmp.info, Node: Demonstration Programs, Next: Efficiency, Prev: Compatibility with older versions, Up: GMP Basics 3.10 Demonstration programs =========================== The `demos' subdirectory has some sample programs using GMP. These aren't built or installed, but there's a `Makefile' with rules for them. For instance, make pexpr ./pexpr 68^975+10 The following programs are provided * `pexpr' is an expression evaluator, the program used on the GMP web page. * The `calc' subdirectory has a similar but simpler evaluator using `lex' and `yacc'. * The `expr' subdirectory is yet another expression evaluator, a library designed for ease of use within a C program. See `demos/expr/README' for more information. * `factorize' is a Pollard-Rho factorization program. * `isprime' is a command-line interface to the `mpz_probab_prime_p' function. * `primes' counts or lists primes in an interval, using a sieve. * `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic class numbers. * The `perl' subdirectory is a comprehensive perl interface to GMP. See `demos/perl/INSTALL' for more information. Documentation is in POD format in `demos/perl/GMP.pm'. As an aside, consideration has been given at various times to some sort of expression evaluation within the main GMP library. Going beyond something minimal quickly leads to matters like user-defined functions, looping, fixnums for control variables, etc, which are considered outside the scope of GMP (much closer to language interpreters or compilers, *Note Language Bindings::.) Something simple for program input convenience may yet be a possibility, a combination of the `expr' demo and the `pexpr' tree back-end perhaps. But for now the above evaluators are offered as illustrations.  File: gmp.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: GMP Basics 3.11 Efficiency =============== Small Operands On small operands, the time for function call overheads and memory allocation can be significant in comparison to actual calculation. This is unavoidable in a general purpose variable precision library, although GMP attempts to be as efficient as it can on both large and small operands. Static Linking On some CPUs, in particular the x86s, the static `libgmp.a' should be used for maximum speed, since the PIC code in the shared `libgmp.so' will have a small overhead on each function call and global data address. For many programs this will be insignificant, but for long calculations there's a gain to be had. Initializing and Clearing Avoid excessive initializing and clearing of variables, since this can be quite time consuming, especially in comparison to otherwise fast operations like addition. A language interpreter might want to keep a free list or stack of initialized variables ready for use. It should be possible to integrate something like that with a garbage collector too. Reallocations An `mpz_t' or `mpq_t' variable used to hold successively increasing values will have its memory repeatedly `realloc'ed, which could be quite slow or could fragment memory, depending on the C library. If an application can estimate the final size then `mpz_init2' or `mpz_realloc2' can be called to allocate the necessary space from the beginning (*note Initializing Integers::). It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2' is too small, since all functions will do a further reallocation if necessary. Badly overestimating memory required will waste space though. `2exp' Functions It's up to an application to call functions like `mpz_mul_2exp' when appropriate. General purpose functions like `mpz_mul' make no attempt to identify powers of two or other special forms, because such inputs will usually be very rare and testing every time would be wasteful. `ui' and `si' Functions The `ui' functions and the small number of `si' functions exist for convenience and should be used where applicable. But if for example an `mpz_t' contains a value that fits in an `unsigned long' there's no need extract it and call a `ui' function, just use the regular `mpz' function. In-Place Operations `mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and `mpf_neg' are fast when used for in-place operations like `mpz_abs(x,x)', since in the current implementation only a single field of `x' needs changing. On suitable compilers (GCC for instance) this is inlined too. `mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit from an in-place operation like `mpz_add_ui(x,x,y)', since usually only one or two limbs of `x' will need to be changed. The same applies to the full precision `mpz_add' etc if `y' is small. If `y' is big then cache locality may be helped, but that's all. `mpz_mul' is currently the opposite, a separate destination is slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is only one limb, make a temporary copy of `x' before forming the result. Normally that copying will only be a tiny fraction of the time for the multiply, so this is not a particularly important consideration. `mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no attempt to recognise a copy of something to itself, so a call like `mpz_set(x,x)' will be wasteful. Naturally that would never be written deliberately, but if it might arise from two pointers to the same object then a test to avoid it might be desirable. if (x != y) mpz_set (x, y); Note that it's never worth introducing extra `mpz_set' calls just to get in-place operations. If a result should go to a particular variable then just direct it there and let GMP take care of data movement. Divisibility Testing (Small Integers) `mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best functions for testing whether an `mpz_t' is divisible by an individual small integer. They use an algorithm which is faster than `mpz_tdiv_ui', but which gives no useful information about the actual remainder, only whether it's zero (or a particular value). However when testing divisibility by several small integers, it's best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo 23*29*31 = 20677 and then test that. The division functions like `mpz_tdiv_q_ui' which give a quotient as well as a remainder are generally a little slower than the remainder-only functions like `mpz_tdiv_ui'. If the quotient is only rarely wanted then it's probably best to just take a remainder and then go back and calculate the quotient if and when it's wanted (`mpz_divexact_ui' can be used if the remainder is zero). Rational Arithmetic The `mpq' functions operate on `mpq_t' values with no common factors in the numerator and denominator. Common factors are checked-for and cast out as necessary. In general, cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations. However, applications that know something about the factorization of the values they're working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it's enough to check for factors of it in the denominator instead of doing a full GCD. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end. The `mpq_numref' and `mpq_denref' macros give access to the numerator and denominator to do things outside the scope of the supplied `mpq' functions. *Note Applying Integer Functions::. The canonical form for rationals allows mixed-type `mpq_t' and integer additions or subtractions to be done directly with multiples of the denominator. This will be somewhat faster than `mpq_add'. For example, /* mpq increment */ mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q)); /* mpq += unsigned long */ mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL); /* mpq -= mpz */ mpz_submul (mpq_numref(q), mpq_denref(q), z); Number Sequences Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are designed for calculating isolated values. If a range of values is wanted it's probably best to call to get a starting point and iterate from there. Text Input/Output Hexadecimal or octal are suggested for input or output in text form. Power-of-2 bases like these can be converted much more efficiently than other bases, like decimal. For big numbers there's usually nothing of particular interest to be seen in the digits, so the base doesn't matter much. Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers.  File: gmp.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: GMP Basics 3.12 Debugging ============== Stack Overflow Depending on the system, a segmentation violation or bus error might be the only indication of stack overflow. See `--enable-alloca' choices in *note Build Options::, for how to address this. In new enough versions of GCC, `-fstack-check' may be able to ensure an overflow is recognised by the system before too much damage is done, or `-fstack-limit-symbol' or `-fstack-limit-register' may be able to add checking if the system itself doesn't do any (*note Options for Code Generation: (gcc)Code Gen Options.). These options must be added to the `CFLAGS' used in the GMP build (*note Build Options::), adding them just to an application will have no effect. Note also they're a slowdown, adding overhead to each function call and each stack allocation. Heap Problems The most likely cause of application problems with GMP is heap corruption. Failing to `init' GMP variables will have unpredictable effects, and corruption arising elsewhere in a program may well affect GMP. Initializing GMP variables more than once or failing to clear them will cause memory leaks. In all such cases a `malloc' debugger is recommended. On a GNU or BSD system the standard C library `malloc' has some diagnostic facilities, see *note Allocation Debugging: (libc)Allocation Debugging, or `man 3 malloc'. Other possibilities, in no particular order, include `http://www.inf.ethz.ch/personal/biere/projects/ccmalloc/' `http://dmalloc.com/' `http://www.perens.com/FreeSoftware/' (electric fence) `http://packages.debian.org/stable/devel/fda' `http://www.gnupdate.org/components/leakbug/' `http://people.redhat.com/~otaylor/memprof/' `http://www.cbmamiga.demon.co.uk/mpatrol/' The GMP default allocation routines in `memory.c' also have a simple sentinel scheme which can be enabled with `#define DEBUG' in that file. This is mainly designed for detecting buffer overruns during GMP development, but might find other uses. Stack Backtraces On some systems the compiler options GMP uses by default can interfere with debugging. In particular on x86 and 68k systems `-fomit-frame-pointer' is used and this generally inhibits stack backtracing. Recompiling without such options may help while debugging, though the usual caveats about it potentially moving a memory problem or hiding a compiler bug will apply. GDB, the GNU Debugger A sample `.gdbinit' is included in the distribution, showing how to call some undocumented dump functions to print GMP variables from within GDB. Note that these functions shouldn't be used in final application code since they're undocumented and may be subject to incompatible changes in future versions of GMP. Source File Paths GMP has multiple source files with the same name, in different directories. For example `mpz', `mpq' and `mpf' each have an `init.c'. If the debugger can't already determine the right one it may help to build with absolute paths on each C file. One way to do that is to use a separate object directory with an absolute path to the source directory. cd /my/build/dir /my/source/dir/gmp-5.1.2/configure This works via `VPATH', and might require GNU `make'. Alternately it might be possible to change the `.c.lo' rules appropriately. Assertion Checking The build option `--enable-assert' is available to add some consistency checks to the library (see *note Build Options::). These are likely to be of limited value to most applications. Assertion failures are just as likely to indicate memory corruption as a library or compiler bug. Applications using the low-level `mpn' functions, however, will benefit from `--enable-assert' since it adds checks on the parameters of most such functions, many of which have subtle restrictions on their usage. Note however that only the generic C code has checks, not the assembly code, so `--disable-assembly' should be used for maximum checking. Temporary Memory Checking The build option `--enable-alloca=debug' arranges that each block of temporary memory in GMP is allocated with a separate call to `malloc' (or the allocation function set with `mp_set_memory_functions'). This can help a malloc debugger detect accesses outside the intended bounds, or detect memory not released. In a normal build, on the other hand, temporary memory is allocated in blocks which GMP divides up for its own use, or may be allocated with a compiler builtin `alloca' which will go nowhere near any malloc debugger hooks. Maximum Debuggability To summarize the above, a GMP build for maximum debuggability would be ./configure --disable-shared --enable-assert \ --enable-alloca=debug --disable-assembly CFLAGS=-g For C++, add `--enable-cxx CXXFLAGS=-g'. Checker The GCC checker (`http://savannah.nongnu.org/projects/checker/') can be used with GMP. It contains a stub library which means GMP applications compiled with checker can use a normal GMP build. A build of GMP with checking within GMP itself can be made. This will run very very slowly. On GNU/Linux for example, ./configure --disable-assembly CC=checkergcc `--disable-assembly' must be used, since the GMP assembly code doesn't support the checking scheme. The GMP C++ features cannot be used, since current versions of checker (0.9.9.1) don't yet support the standard C++ library. Valgrind The valgrind program (`http://valgrind.org/') is a memory checker for x86s. It translates and emulates machine instructions to do strong checks for uninitialized data (at the level of individual bits), memory accesses through bad pointers, and memory leaks. Recent versions of Valgrind are getting support for MMX and SSE/SSE2 instructions, for past versions GMP will need to be configured not to use those, i.e. for an x86 without them (for instance plain `i486'). GMP's assembly code sometimes promotes a read of the limbs to some larger size, for efficiency. GMP will do this even at the start and end of a multilimb operand, using naturaly aligned operations on the larger type. This may lead to benign reads outside of allocated areas, triggering complants from Valgrind. Other Problems Any suspected bug in GMP itself should be isolated to make sure it's not an application problem, see *note Reporting Bugs::.  File: gmp.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: GMP Basics 3.13 Profiling ============== Running a program under a profiler is a good way to find where it's spending most time and where improvements can be best sought. The profiling choices for a GMP build are as follows. `--disable-profiling' The default is to add nothing special for profiling. It should be possible to just compile the mainline of a program with `-p' and use `prof' to get a profile consisting of timer-based sampling of the program counter. Most of the GMP assembly code has the necessary symbol information. This approach has the advantage of minimizing interference with normal program operation, but on most systems the resolution of the sampling is quite low (10 milliseconds for instance), requiring long runs to get accurate information. `--enable-profiling=prof' Build with support for the system `prof', which means `-p' added to the `CFLAGS'. This provides call counting in addition to program counter sampling, which allows the most frequently called routines to be identified, and an average time spent in each routine to be determined. The x86 assembly code has support for this option, but on other processors the assembly routines will be as if compiled without `-p' and therefore won't appear in the call counts. On some systems, such as GNU/Linux, `-p' in fact means `-pg' and in this case `--enable-profiling=gprof' described below should be used instead. `--enable-profiling=gprof' Build with support for `gprof', which means `-pg' added to the `CFLAGS'. This provides call graph construction in addition to call counting and program counter sampling, which makes it possible to count calls coming from different locations. For example the number of calls to `mpn_mul' from `mpz_mul' versus the number from `mpf_mul'. The program counter sampling is still flat though, so only a total time in `mpn_mul' would be accumulated, not a separate amount for each call site. The x86 assembly code has support for this option, but on other processors the assembly routines will be as if compiled without `-pg' and therefore not be included in the call counts. On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are incompatible, so the latter is omitted from the default flags in that case, which might result in poorer code generation. Incidentally, it should be possible to use the `gprof' program with a plain `--enable-profiling=prof' build. But in that case only the `gprof -p' flat profile and call counts can be expected to be valid, not the `gprof -q' call graph. `--enable-profiling=instrument' Build with the GCC option `-finstrument-functions' added to the `CFLAGS' (*note Options for Code Generation: (gcc)Code Gen Options.). This inserts special instrumenting calls at the start and end of each function, allowing exact timing and full call graph construction. This instrumenting is not normally a standard system feature and will require support from an external library, such as `http://sourceforge.net/projects/fnccheck/' This should be included in `LIBS' during the GMP configure so that test programs will link. For example, ./configure --enable-profiling=instrument LIBS=-lfc On a GNU system the C library provides dummy instrumenting functions, so programs compiled with this option will link. In this case it's only necessary to ensure the correct library is added when linking an application. The x86 assembly code supports this option, but on other processors the assembly routines will be as if compiled without `-finstrument-functions' meaning time spent in them will effectively be attributed to their caller.  File: gmp.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: GMP Basics 3.14 Autoconf ============= Autoconf based applications can easily check whether GMP is installed. The only thing to be noted is that GMP library symbols from version 3 onwards have prefixes like `__gmpz'. The following therefore would be a simple test, AC_CHECK_LIB(gmp, __gmpz_init) This just uses the default `AC_CHECK_LIB' actions for found or not found, but an application that must have GMP would want to generate an error if not found. For example, AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR([GNU MP not found, see http://gmplib.org/])]) If functions added in some particular version of GMP are required, then one of those can be used when checking. For example `mpz_mul_si' was added in GMP 3.1, AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR( [GNU MP not found, or not 3.1 or up, see http://gmplib.org/])]) An alternative would be to test the version number in `gmp.h' using say `AC_EGREP_CPP'. That would make it possible to test the exact version, if some particular sub-minor release is known to be necessary. In general it's recommended that applications should simply demand a new enough GMP rather than trying to provide supplements for features not available in past versions. Occasionally an application will need or want to know the size of a type at configuration or preprocessing time, not just with `sizeof' in the code. This can be done in the normal way with `mp_limb_t' etc, but GMP 4.0 or up is best for this, since prior versions needed certain `-D' defines on systems using a `long long' limb. The following would suit Autoconf 2.50 or up, AC_CHECK_SIZEOF(mp_limb_t, , [#include ])  File: gmp.info, Node: Emacs, Prev: Autoconf, Up: GMP Basics 3.15 Emacs ========== (`info-lookup-symbol') is a good way to find documentation on C functions while editing (*note Info Documentation Lookup: (emacs)Info Lookup.). The GMP manual can be included in such lookups by putting the following in your `.emacs', (eval-after-load "info-look" '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist)))) (setcar (nthcdr 3 mode-value) (cons '("(gmp)Function Index" nil "^ -.* " "\\>") (nth 3 mode-value)))))  File: gmp.info, Node: Reporting Bugs, Next: Integer Functions, Prev: GMP Basics, Up: Top 4 Reporting Bugs **************** If you think you have found a bug in the GMP library, please investigate it and report it. We have made this library available to you, and it is not too much to ask you to report the bugs you find. Before you report a bug, check it's not already addressed in *note Known Build Problems::, or perhaps *note Notes for Particular Systems::. You may also want to check `http://gmplib.org/' for patches for this release. Please include the following in any report, * The GMP version number, and if pre-packaged or patched then say so. * A test program that makes it possible for us to reproduce the bug. Include instructions on how to run the program. * A description of what is wrong. If the results are incorrect, in what way. If you get a crash, say so. * If you get a crash, include a stack backtrace from the debugger if it's informative (`where' in `gdb', or `$C' in `adb'). * Please do not send core dumps, executables or `strace's. * The configuration options you used when building GMP, if any. * The name of the compiler and its version. For `gcc', get the version with `gcc -v', otherwise perhaps `what `which cc`', or similar. * The output from running `uname -a'. * The output from running `./config.guess', and from running `./configfsf.guess' (might be the same). * If the bug is related to `configure', then the compressed contents of `config.log'. * If the bug is related to an `asm' file not assembling, then the contents of `config.m4' and the offending line or lines from the temporary `mpn/tmp-.s'. Please make an effort to produce a self-contained report, with something definite that can be tested or debugged. Vague queries or piecemeal messages are difficult to act on and don't help the development effort. It is not uncommon that an observed problem is actually due to a bug in the compiler; the GMP code tends to explore interesting corners in compilers. If your bug report is good, we will do our best to help you get a corrected version of the library; if the bug report is poor, we won't do anything about it (except maybe ask you to send a better report). Send your report to: . If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.  File: gmp.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top 5 Integer Functions ******************* This chapter describes the GMP functions for performing integer arithmetic. These functions start with the prefix `mpz_'. GMP integers are stored in objects of type `mpz_t'. * Menu: * Initializing Integers:: * Assigning Integers:: * Simultaneous Integer Init & Assign:: * Converting Integers:: * Integer Arithmetic:: * Integer Division:: * Integer Exponentiation:: * Integer Roots:: * Number Theoretic Functions:: * Integer Comparisons:: * Integer Logic and Bit Fiddling:: * I/O of Integers:: * Integer Random Numbers:: * Integer Import and Export:: * Miscellaneous Integer Functions:: * Integer Special Functions::  File: gmp.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions 5.1 Initialization Functions ============================ The functions for integer arithmetic assume that all integer objects are initialized. You do that by calling the function `mpz_init'. For example, { mpz_t integ; mpz_init (integ); ... mpz_add (integ, ...); ... mpz_sub (integ, ...); /* Unless the program is about to exit, do ... */ mpz_clear (integ); } As you can see, you can store new values any number of times, once an object is initialized. -- Function: void mpz_init (mpz_t X) Initialize X, and set its value to 0. -- Function: void mpz_inits (mpz_t X, ...) Initialize a NULL-terminated list of `mpz_t' variables, and set their values to 0. -- Function: void mpz_init2 (mpz_t X, mp_bitcnt_t N) Initialize X, with space for N-bit numbers, and set its value to 0. Calling this function instead of `mpz_init' or `mpz_inits' is never necessary; reallocation is handled automatically by GMP when needed. While N defines the initial space, X will grow automatically in the normal way, if necessary, for subsequent values stored. `mpz_init2' makes it possible to avoid such reallocations if a maximum size is known in advance. In preparation for an operation, GMP often allocates one limb more than ultimately needed. To make sure GMP will not perform reallocation for X, you need to add the number of bits in `mp_limb_t' to N. -- Function: void mpz_clear (mpz_t X) Free the space occupied by X. Call this function for all `mpz_t' variables when you are done with them. -- Function: void mpz_clears (mpz_t X, ...) Free the space occupied by a NULL-terminated list of `mpz_t' variables. -- Function: void mpz_realloc2 (mpz_t X, mp_bitcnt_t N) Change the space allocated for X to N bits. The value in X is preserved if it fits, or is set to 0 if not. Calling this function is never necessary; reallocation is handled automatically by GMP when needed. But this function can be used to increase the space for a variable in order to avoid repeated automatic reallocations, or to decrease it to give memory back to the heap.  File: gmp.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions 5.2 Assignment Functions ======================== These functions assign new values to already initialized integers (*note Initializing Integers::). -- Function: void mpz_set (mpz_t ROP, mpz_t OP) -- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP) -- Function: void mpz_set_si (mpz_t ROP, signed long int OP) -- Function: void mpz_set_d (mpz_t ROP, double OP) -- Function: void mpz_set_q (mpz_t ROP, mpq_t OP) -- Function: void mpz_set_f (mpz_t ROP, mpf_t OP) Set the value of ROP from OP. `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an integer. -- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE) Set the value of ROP from STR, a null-terminated C string in base BASE. White space is allowed in the string, and is simply ignored. The BASE may vary from 2 to 62, or if BASE is 0, then the leading characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B' for binary, `0' for octal, or decimal otherwise. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value. For bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. This function returns 0 if the entire string is a valid number in base BASE. Otherwise it returns -1. -- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2) Swap the values ROP1 and ROP2 efficiently.  File: gmp.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions 5.3 Combined Initialization and Assignment Functions ==================================================== For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions' names have the form `mpz_init_set...' Here is an example of using one: { mpz_t pie; mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10); ... mpz_sub (pie, ...); ... mpz_clear (pie); } Once the integer has been initialized by any of the `mpz_init_set...' functions, it can be used as the source or destination operand for the ordinary integer functions. Don't use an initialize-and-set function on a variable already initialized! -- Function: void mpz_init_set (mpz_t ROP, mpz_t OP) -- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP) -- Function: void mpz_init_set_si (mpz_t ROP, signed long int OP) -- Function: void mpz_init_set_d (mpz_t ROP, double OP) Initialize ROP with limb space and set the initial numeric value from OP. -- Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE) Initialize ROP and set its value like `mpz_set_str' (see its documentation above for details). If the string is a correct base BASE number, the function returns 0; if an error occurs it returns -1. ROP is initialized even if an error occurs. (I.e., you have to call `mpz_clear' for it.)  File: gmp.info, Node: Converting Integers, Next: Integer Arithmetic, Prev: Simultaneous Integer Init & Assign, Up: Integer Functions 5.4 Conversion Functions ======================== This section describes functions for converting GMP integers to standard C types. Functions for converting _to_ GMP integers are described in *note Assigning Integers:: and *note I/O of Integers::. -- Function: unsigned long int mpz_get_ui (mpz_t OP) Return the value of OP as an `unsigned long'. If OP is too big to fit an `unsigned long' then just the least significant bits that do fit are returned. The sign of OP is ignored, only the absolute value is used. -- Function: signed long int mpz_get_si (mpz_t OP) If OP fits into a `signed long int' return the value of OP. Otherwise return the least significant part of OP, with the same sign as OP. If OP is too big to fit in a `signed long int', the returned result is probably not very useful. To find out if the value will fit, use the function `mpz_fits_slong_p'. -- Function: double mpz_get_d (mpz_t OP) Convert OP to a `double', truncating if necessary (i.e. rounding towards zero). If the exponent from the conversion is too big, the result is system dependent. An infinity is returned where available. A hardware overflow trap may or may not occur. -- Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP) Convert OP to a `double', truncating if necessary (i.e. rounding towards zero), and returning the exponent separately. The return value is in the range 0.5<=abs(D)<1 and the exponent is stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP is zero, the return is 0.0 and 0 is stored to `*EXP'. This is similar to the standard C `frexp' function (*note Normalization Functions: (libc)Normalization Functions.). -- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP) Convert OP to a string of digits in base BASE. The base argument may vary from 2 to 62 or from -2 to -36. For BASE in the range 2..36, digits and lower-case letters are used; for -2..-36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of storage large enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'. The two extra bytes are for a possible minus sign, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given STR.  File: gmp.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions 5.5 Arithmetic Functions ======================== -- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2) -- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 + OP2. -- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2) -- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) -- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t OP2) Set ROP to OP1 - OP2. -- Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2) -- Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2) -- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 times OP2. -- Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2) -- Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to ROP + OP1 times OP2. -- Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2) -- Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to ROP - OP1 times OP2. -- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, mp_bitcnt_t OP2) Set ROP to OP1 times 2 raised to OP2. This operation can also be defined as a left shift by OP2 bits. -- Function: void mpz_neg (mpz_t ROP, mpz_t OP) Set ROP to -OP. -- Function: void mpz_abs (mpz_t ROP, mpz_t OP) Set ROP to the absolute value of OP.  File: gmp.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions 5.6 Division Functions ====================== Division is undefined if the divisor is zero. Passing a zero divisor to the division or modulo functions (including the modular powering functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional division by zero. This lets a program handle arithmetic exceptions in these functions the same way as for normal C `int' arithmetic. -- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D) -- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D) -- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) -- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_cdiv_ui (mpz_t N, unsigned long int D) -- Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B) -- Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B) -- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D) -- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D) -- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) -- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_fdiv_ui (mpz_t N, unsigned long int D) -- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B) -- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B) -- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D) -- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D) -- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) -- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) -- Function: unsigned long int mpz_tdiv_ui (mpz_t N, unsigned long int D) -- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B) -- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B) Divide N by D, forming a quotient Q and/or remainder R. For the `2exp' functions, D=2^B. The rounding is in three styles, each suiting different applications. * `cdiv' rounds Q up towards +infinity, and R will have the opposite sign to D. The `c' stands for "ceil". * `fdiv' rounds Q down towards -infinity, and R will have the same sign as D. The `f' stands for "floor". * `tdiv' rounds Q towards zero, and R will have the same sign as N. The `t' stands for "truncate". In all cases Q and R will satisfy N=Q*D+R, and R will satisfy 0<=abs(R) 0 and that MOD is odd. This function is designed to take the same time and have the same cache access patterns for any two same-size arguments, assuming that function arguments are placed at the same position and that the machine state is identical upon function entry. This function is intended for cryptographic purposes, where resilience to side-channel attacks is desired. -- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int EXP) -- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE, unsigned long int EXP) Set ROP to BASE raised to EXP. The case 0^0 yields 1.  File: gmp.info, Node: Integer Roots, Next: Number Theoretic Functions, Prev: Integer Exponentiation, Up: Integer Functions 5.8 Root Extraction Functions ============================= -- Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N) Set ROP to the truncated integer part of the Nth root of OP. Return non-zero if the computation was exact, i.e., if OP is ROP to the Nth power. -- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, mpz_t U, unsigned long int N) Set ROOT to the truncated integer part of the Nth root of U. Set REM to the remainder, U-ROOT**N. -- Function: void mpz_sqrt (mpz_t ROP, mpz_t OP) Set ROP to the truncated integer part of the square root of OP. -- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP) Set ROP1 to the truncated integer part of the square root of OP, like `mpz_sqrt'. Set ROP2 to the remainder OP-ROP1*ROP1, which will be zero if OP is a perfect square. If ROP1 and ROP2 are the same variable, the results are undefined. -- Function: int mpz_perfect_power_p (mpz_t OP) Return non-zero if OP is a perfect power, i.e., if there exist integers A and B, with B>1, such that OP equals A raised to the power B. Under this definition both 0 and 1 are considered to be perfect powers. Negative values of OP are accepted, but of course can only be odd perfect powers. -- Function: int mpz_perfect_square_p (mpz_t OP) Return non-zero if OP is a perfect square, i.e., if the square root of OP is an integer. Under this definition both 0 and 1 are considered to be perfect squares.  File: gmp.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions 5.9 Number Theoretic Functions ============================== -- Function: int mpz_probab_prime_p (mpz_t N, int REPS) Determine whether N is prime. Return 2 if N is definitely prime, return 1 if N is probably prime (without being certain), or return 0 if N is definitely composite. This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. The argument REPS controls how many such tests are done; a higher value will reduce the chances of a composite being returned as "probably prime". 25 is a reasonable number; a composite number will then be identified as a prime with a probability of less than 2^(-50). Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime. -- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP) Set ROP to the next prime greater than OP. This function uses a probabilistic algorithm to identify primes. For practical purposes it's adequate, the chance of a composite passing will be extremely small. -- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to the greatest common divisor of OP1 and OP2. The result is always positive even if one or both input operands are negative. Except if both inputs are zero; then this function defines gcd(0,0) = 0. -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Compute the greatest common divisor of OP1 and OP2. If ROP is not `NULL', store the result there. If the result is small enough to fit in an `unsigned long int', it is returned. If the result does not fit, 0 is returned, and the result is equal to the argument OP1. Note that the result will always fit if OP2 is non-zero. -- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t B) Set G to the greatest common divisor of A and B, and in addition set S and T to coefficients satisfying A*S + B*T = G. The value in G is always positive, even if one or both of A and B are negative (or zero if both inputs are zero). The values in S and T are chosen such that normally, abs(S) < abs(B) / (2 G) and abs(T) < abs(A) / (2 G), and these relations define S and T uniquely. There are a few exceptional cases: If abs(A) = abs(B), then S = 0, T = sgn(B). Otherwise, S = sgn(A) if B = 0 or abs(B) = 2 G, and T = sgn(B) if A = 0 or abs(A) = 2 G. In all cases, S = 0 if and only if G = abs(B), i.e., if B divides A or A = B = 0. If T is `NULL' then that value is not computed. -- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2) -- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2) Set ROP to the least common multiple of OP1 and OP2. ROP is always positive, irrespective of the signs of OP1 and OP2. ROP will be zero if either OP1 or OP2 is zero. -- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2) Compute the inverse of OP1 modulo OP2 and put the result in ROP. If the inverse exists, the return value is non-zero and ROP will satisfy 0 < ROP < abs(OP2). If an inverse doesn't exist the return value is zero and ROP is undefined. The behaviour of this function is undefined when OP2 is zero. -- Function: int mpz_jacobi (mpz_t A, mpz_t B) Calculate the Jacobi symbol (A/B). This is defined only for B odd. -- Function: int mpz_legendre (mpz_t A, mpz_t P) Calculate the Legendre symbol (A/P). This is defined only for P an odd positive prime, and for such P it's identical to the Jacobi symbol. -- Function: int mpz_kronecker (mpz_t A, mpz_t B) -- Function: int mpz_kronecker_si (mpz_t A, long B) -- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B) -- Function: int mpz_si_kronecker (long A, mpz_t B) -- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B) Calculate the Jacobi symbol (A/B) with the Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even. When B is odd the Jacobi symbol and Kronecker symbol are identical, so `mpz_kronecker_ui' etc can be used for mixed precision Jacobi symbols too. For more information see Henri Cohen section 1.4.2 (*note References::), or any number theory textbook. See also the example program `demos/qcn.c' which uses `mpz_kronecker_ui'. -- Function: mp_bitcnt_t mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F) Remove all occurrences of the factor F from OP and store the result in ROP. The return value is how many such occurrences were removed. -- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int N) -- Function: void mpz_2fac_ui (mpz_t ROP, unsigned long int N) -- Function: void mpz_mfac_uiui (mpz_t ROP, unsigned long int N, unsigned long int M) Set ROP to the factorial of N: `mpz_fac_ui' computes the plain factorial N!, `mpz_2fac_ui' computes the double-factorial N!!, and `mpz_mfac_uiui' the M-multi-factorial N!^(M). -- Function: void mpz_primorial_ui (mpz_t ROP, unsigned long int N) Set ROP to the primorial of N, i.e. the product of all positive prime numbers <=N. -- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K) -- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N, unsigned long int K) Compute the binomial coefficient N over K and store the result in ROP. Negative values of N are supported by `mpz_bin_ui', using the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1 section 1.2.6 part G. -- Function: void mpz_fib_ui (mpz_t FN, unsigned long int N) -- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long int N) `mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number. `mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1]. These functions are designed for calculating isolated Fibonacci numbers. When a sequence of values is wanted it's best to start with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or similar. -- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N) -- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned long int N) `mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number. `mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1]. These functions are designed for calculating isolated Lucas numbers. When a sequence of values is wanted it's best to start with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1] or similar. The Fibonacci numbers and Lucas numbers are related sequences, so it's never necessary to call both `mpz_fib2_ui' and `mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas can be found in *note Lucas Numbers Algorithm::, the reverse is straightforward too.  File: gmp.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions 5.10 Comparison Functions ========================= -- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2) -- Function: int mpz_cmp_d (mpz_t OP1, double OP2) -- Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2) -- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, or a negative value if OP1 < OP2. `mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their arguments more than once. `mpz_cmp_d' can be called with an infinity, but results are undefined for a NaN. -- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2) -- Function: int mpz_cmpabs_d (mpz_t OP1, double OP2) -- Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2) Compare the absolute values of OP1 and OP2. Return a positive value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a negative value if abs(OP1) < abs(OP2). `mpz_cmpabs_d' can be called with an infinity, but results are undefined for a NaN. -- Macro: int mpz_sgn (mpz_t OP) Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. This function is actually implemented as a macro. It evaluates its argument multiple times.  File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions 5.11 Logical and Bit Manipulation Functions =========================================== These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0. -- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 bitwise-and OP2. -- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 bitwise inclusive-or OP2. -- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 bitwise exclusive-or OP2. -- Function: void mpz_com (mpz_t ROP, mpz_t OP) Set ROP to the one's complement of OP. -- Function: mp_bitcnt_t mpz_popcount (mpz_t OP) If OP>=0, return the population count of OP, which is the number of 1 bits in the binary representation. If OP<0, the number of 1s is infinite, and the return value is the largest possible `mp_bitcnt_t'. -- Function: mp_bitcnt_t mpz_hamdist (mpz_t OP1, mpz_t OP2) If OP1 and OP2 are both >=0 or both <0, return the hamming distance between the two operands, which is the number of bit positions where OP1 and OP2 have different bit values. If one operand is >=0 and the other <0 then the number of bits different is infinite, and the return value is the largest possible `mp_bitcnt_t'. -- Function: mp_bitcnt_t mpz_scan0 (mpz_t OP, mp_bitcnt_t STARTING_BIT) -- Function: mp_bitcnt_t mpz_scan1 (mpz_t OP, mp_bitcnt_t STARTING_BIT) Scan OP, starting from bit STARTING_BIT, towards more significant bits, until the first 0 or 1 bit (respectively) is found. Return the index of the found bit. If the bit at STARTING_BIT is already what's sought, then STARTING_BIT is returned. If there's no bit found, then the largest possible `mp_bitcnt_t' is returned. This will happen in `mpz_scan0' past the end of a negative number, or `mpz_scan1' past the end of a nonnegative number. -- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX) Set bit BIT_INDEX in ROP. -- Function: void mpz_clrbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX) Clear bit BIT_INDEX in ROP. -- Function: void mpz_combit (mpz_t ROP, mp_bitcnt_t BIT_INDEX) Complement bit BIT_INDEX in ROP. -- Function: int mpz_tstbit (mpz_t OP, mp_bitcnt_t BIT_INDEX) Test bit BIT_INDEX in OP and return 0 or 1 accordingly.  File: gmp.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions 5.12 Input and Output Functions =============================== Functions that perform input from a stdio stream, and functions that output to a stdio stream, of `mpz' numbers. Passing a `NULL' pointer for a STREAM argument to any of these functions will make them read from `stdin' and write to `stdout', respectively. When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions. See also *note Formatted Output:: and *note Formatted Input::. -- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP) Output OP on stdio stream STREAM, as a string of digits in base BASE. The base argument may vary from 2 to 62 or from -2 to -36. For BASE in the range 2..36, digits and lower-case letters are used; for -2..-36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. Return the number of bytes written, or if an error occurred, return 0. -- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE) Input a possibly white-space preceded string in base BASE from stdio stream STREAM, and put the read integer in ROP. The BASE may vary from 2 to 62, or if BASE is 0, then the leading characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B' for binary, `0' for octal, or decimal otherwise. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value. For bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. Return the number of bytes read, or if an error occurred, return 0. -- Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP) Output OP on stdio stream STREAM, in raw binary format. The integer is written in a portable format, with 4 bytes of size information, and that many bytes of limbs. Both the size and the limbs are written in decreasing significance order (i.e., in big-endian). The output can be read with `mpz_inp_raw'. Return the number of bytes written, or if an error occurred, return 0. The output of this can not be read by `mpz_inp_raw' from GMP 1, because of changes necessary for compatibility between 32-bit and 64-bit machines. -- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM) Input from stdio stream STREAM in the format written by `mpz_out_raw', and put the result in ROP. Return the number of bytes read, or if an error occurred, return 0. This routine can read the output from `mpz_out_raw' also from GMP 1, in spite of changes necessary for compatibility between 32-bit and 64-bit machines.  File: gmp.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions 5.13 Random Number Functions ============================ The random number functions of GMP come in two groups; older function that rely on a global state, and newer functions that accept a state parameter that is read and modified. Please see the *note Random Number Functions:: for more information on how to use and not to use random number functions. -- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE, mp_bitcnt_t N) Generate a uniformly distributed random integer in the range 0 to 2^N-1, inclusive. The variable STATE must be initialized by calling one of the `gmp_randinit' functions (*note Random State Initialization::) before invoking this function. -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE, mpz_t N) Generate a uniform random integer in the range 0 to N-1, inclusive. The variable STATE must be initialized by calling one of the `gmp_randinit' functions (*note Random State Initialization::) before invoking this function. -- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE, mp_bitcnt_t N) Generate a random integer with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. The random number will be in the range 0 to 2^N-1, inclusive. The variable STATE must be initialized by calling one of the `gmp_randinit' functions (*note Random State Initialization::) before invoking this function. -- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE) Generate a random integer of at most MAX_SIZE limbs. The generated random number doesn't satisfy any particular requirements of randomness. Negative random numbers are generated when MAX_SIZE is negative. This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm' instead. -- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE) Generate a random integer of at most MAX_SIZE limbs, with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when MAX_SIZE is negative. This function is obsolete. Use `mpz_rrandomb' instead.  File: gmp.info, Node: Integer Import and Export, Next: Miscellaneous Integer Functions, Prev: Integer Random Numbers, Up: Integer Functions 5.14 Integer Import and Export ============================== `mpz_t' variables can be converted to and from arbitrary words of binary data with the following functions. -- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER, size_t SIZE, int ENDIAN, size_t NAILS, const void *OP) Set ROP from an array of word data at OP. The parameters specify the format of the data. COUNT many words are read, each SIZE bytes. ORDER can be 1 for most significant word first or -1 for least significant first. Within each word ENDIAN can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant NAILS bits of each word are skipped, this can be 0 to use the full words. There is no sign taken from the data, ROP will simply be a positive integer. An application can handle any sign itself, and apply it for instance with `mpz_neg'. There are no data alignment restrictions on OP, any address is allowed. Here's an example converting an array of `unsigned long' data, most significant element first, and host byte order within each value. unsigned long a[20]; /* Initialize Z and A */ mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a); This example assumes the full `sizeof' bytes are used for data in the given type, which is usually true, and certainly true for `unsigned long' everywhere we know of. However on Cray vector systems it may be noted that `short' and `int' are always stored in 8 bytes (and with `sizeof' indicating that) but use only 32 or 46 bits. The NAILS feature can account for this, by passing for instance `8*sizeof(int)-INT_BIT'. -- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER, size_t SIZE, int ENDIAN, size_t NAILS, mpz_t OP) Fill ROP with word data from OP. The parameters specify the format of the data produced. Each word will be SIZE bytes and ORDER can be 1 for most significant word first or -1 for least significant first. Within each word ENDIAN can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant NAILS bits of each word are unused and set to zero, this can be 0 to produce full words. The number of words produced is written to `*COUNTP', or COUNTP can be `NULL' to discard the count. ROP must have enough space for the data, or if ROP is `NULL' then a result array of the necessary size is allocated using the current GMP allocation function (*note Custom Allocation::). In either case the return value is the destination used, either ROP or the allocated block. If OP is non-zero then the most significant word produced will be non-zero. If OP is zero then the count returned will be zero and nothing written to ROP. If ROP is `NULL' in this case, no block is allocated, just `NULL' is returned. The sign of OP is ignored, just the absolute value is exported. An application can use `mpz_sgn' to get the sign and handle it as desired. (*note Integer Comparisons::) There are no data alignment restrictions on ROP, any address is allowed. When an application is allocating space itself the required size can be determined with a calculation like the following. Since `mpz_sizeinbase' always returns at least 1, `count' here will be at least one, which avoids any portability problems with `malloc(0)', though if `z' is zero no space at all is actually needed (or written). numb = 8*size - nail; count = (mpz_sizeinbase (z, 2) + numb-1) / numb; p = malloc (count * size);  File: gmp.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions 5.15 Miscellaneous Functions ============================ -- Function: int mpz_fits_ulong_p (mpz_t OP) -- Function: int mpz_fits_slong_p (mpz_t OP) -- Function: int mpz_fits_uint_p (mpz_t OP) -- Function: int mpz_fits_sint_p (mpz_t OP) -- Function: int mpz_fits_ushort_p (mpz_t OP) -- Function: int mpz_fits_sshort_p (mpz_t OP) Return non-zero iff the value of OP fits in an `unsigned long int', `signed long int', `unsigned int', `signed int', `unsigned short int', or `signed short int', respectively. Otherwise, return zero. -- Macro: int mpz_odd_p (mpz_t OP) -- Macro: int mpz_even_p (mpz_t OP) Determine whether OP is odd or even, respectively. Return non-zero if yes, zero if no. These macros evaluate their argument more than once. -- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE) Return the size of OP measured in number of digits in the given BASE. BASE can vary from 2 to 62. The sign of OP is ignored, just the absolute value is used. The result will be either exact or 1 too big. If BASE is a power of 2, the result is always exact. If OP is zero the return value is always 1. This function can be used to determine the space required when converting OP to a string. The right amount of allocation is normally two more than the value returned by `mpz_sizeinbase', one extra for a minus sign and one for the null-terminator. It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate the most significant 1 bit in OP, counting from 1. (Unlike the bitwise functions which start from 0, *Note Logical and Bit Manipulation Functions: Integer Logic and Bit Fiddling.)  File: gmp.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions 5.16 Special Functions ====================== The functions in this section are for various special purposes. Most applications will not need them. -- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, mp_size_t ARRAY_SIZE, mp_size_t FIXED_NUM_BITS) This is a special type of initialization. *Fixed* space of FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in INTEGER_ARRAY. There is no way to free the storage allocated by this function. Don't call `mpz_clear'! The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For example, mpz_t arr[20000]; mpz_array_init (arr[0], 20000, 512); This function is only intended for programs that create a large number of integers and need to reduce memory usage by avoiding the overheads of allocating and reallocating lots of small blocks. In normal programs this function is not recommended. The space allocated to each integer by this function will not be automatically increased, unlike the normal `mpz_init', so an application must ensure it is sufficient for any value stored. The following space requirements apply to various routines, * `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and `mpz_set_ui' need room for the value they store. * `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room for the larger of the two operands, plus an extra `mp_bits_per_limb'. * `mpz_mul', `mpz_mul_ui' and `mpz_mul_si' need room for the sum of the number of bits in their operands, but each rounded up to a multiple of `mp_bits_per_limb'. * `mpz_swap' can be used between two array variables, but not between an array and a normal variable. For other functions, or if in doubt, the suggestion is to calculate in a regular `mpz_init' variable and copy the result to an array variable with `mpz_set'. -- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC) Change the space for INTEGER to NEW_ALLOC limbs. The value in INTEGER is preserved if it fits, or is set to 0 if not. The return value is not useful to applications and should be ignored. `mpz_realloc2' is the preferred way to accomplish allocation changes like this. `mpz_realloc2' and `_mpz_realloc' are the same except that `_mpz_realloc' takes its size in limbs. -- Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N) Return limb number N from OP. The sign of OP is ignored, just the absolute value is used. The least significant limb is number 0. `mpz_size' can be used to find how many limbs make up OP. `mpz_getlimbn' returns zero if N is outside the range 0 to `mpz_size(OP)-1'. -- Function: size_t mpz_size (mpz_t OP) Return the size of OP measured in number of limbs. If OP is zero, the returned value will be zero.  File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top 6 Rational Number Functions *************************** This chapter describes the GMP functions for performing arithmetic on rational numbers. These functions start with the prefix `mpq_'. Rational numbers are stored in objects of type `mpq_t'. All rational arithmetic functions assume operands have a canonical form, and canonicalize their result. The canonical from means that the denominator and the numerator have no common factors, and that the denominator is positive. Zero has the unique representation 0/1. Pure assignment functions do not canonicalize the assigned variable. It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable. -- Function: void mpq_canonicalize (mpq_t OP) Remove any factors that are common to the numerator and denominator of OP, and make the denominator positive. * Menu: * Initializing Rationals:: * Rational Conversions:: * Rational Arithmetic:: * Comparing Rationals:: * Applying Integer Functions:: * I/O of Rationals::  File: gmp.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions 6.1 Initialization and Assignment Functions =========================================== -- Function: void mpq_init (mpq_t X) Initialize X and set it to 0/1. Each variable should normally only be initialized once, or at least cleared out (using the function `mpq_clear') between each initialization. -- Function: void mpq_inits (mpq_t X, ...) Initialize a NULL-terminated list of `mpq_t' variables, and set their values to 0/1. -- Function: void mpq_clear (mpq_t X) Free the space occupied by X. Make sure to call this function for all `mpq_t' variables when you are done with them. -- Function: void mpq_clears (mpq_t X, ...) Free the space occupied by a NULL-terminated list of `mpq_t' variables. -- Function: void mpq_set (mpq_t ROP, mpq_t OP) -- Function: void mpq_set_z (mpq_t ROP, mpz_t OP) Assign ROP from OP. -- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1, unsigned long int OP2) -- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned long int OP2) Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have common factors, ROP has to be passed to `mpq_canonicalize' before any operations are performed on ROP. -- Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE) Set ROP from a null-terminated string STR in the given BASE. The string can be an integer like "41" or a fraction like "41/152". The fraction must be in canonical form (*note Rational Number Functions::), or if not then `mpq_canonicalize' must be called. The numerator and optional denominator are parsed the same as in `mpz_set_str' (*note Assigning Integers::). White space is allowed in the string, and is simply ignored. The BASE can vary from 2 to 62, or if BASE is 0 then the leading characters are used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for octal, or decimal otherwise. Note that this is done separately for the numerator and denominator, so for instance `0xEF/100' is 239/100, whereas `0xEF/0x100' is 239/256. The return value is 0 if the entire string is a valid number, or -1 if not. -- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2) Swap the values ROP1 and ROP2 efficiently.  File: gmp.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions 6.2 Conversion Functions ======================== -- Function: double mpq_get_d (mpq_t OP) Convert OP to a `double', truncating if necessary (i.e. rounding towards zero). If the exponent from the conversion is too big or too small to fit a `double' then the result is system dependent. For too big an infinity is returned when available. For too small 0.0 is normally returned. Hardware overflow, underflow and denorm traps may or may not occur. -- Function: void mpq_set_d (mpq_t ROP, double OP) -- Function: void mpq_set_f (mpq_t ROP, mpf_t OP) Set ROP to the value of OP. There is no rounding, this conversion is exact. -- Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP) Convert OP to a string of digits in base BASE. The base may vary from 2 to 36. The string will be of the form `num/den', or if the denominator is 1 then just `num'. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of storage large enough for the result, that being mpz_sizeinbase (mpq_numref(OP), BASE) + mpz_sizeinbase (mpq_denref(OP), BASE) + 3 The three extra bytes are for a possible minus sign, possible slash, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given STR.  File: gmp.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions 6.3 Arithmetic Functions ======================== -- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2) Set SUM to ADDEND1 + ADDEND2. -- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t SUBTRAHEND) Set DIFFERENCE to MINUEND - SUBTRAHEND. -- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t MULTIPLICAND) Set PRODUCT to MULTIPLIER times MULTIPLICAND. -- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2) Set ROP to OP1 times 2 raised to OP2. -- Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t DIVISOR) Set QUOTIENT to DIVIDEND/DIVISOR. -- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2) Set ROP to OP1 divided by 2 raised to OP2. -- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND) Set NEGATED_OPERAND to -OPERAND. -- Function: void mpq_abs (mpq_t ROP, mpq_t OP) Set ROP to the absolute value of OP. -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER) Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero, this routine will divide by zero.  File: gmp.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions 6.4 Comparison Functions ======================== -- Function: int mpq_cmp (mpq_t OP1, mpq_t OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. To determine if two rationals are equal, `mpq_equal' is faster than `mpq_cmp'. -- Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned long int DEN2) -- Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int DEN2) Compare OP1 and NUM2/DEN2. Return a positive value if OP1 > NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 < NUM2/DEN2. NUM2 and DEN2 are allowed to have common factors. These functions are implemented as a macros and evaluate their arguments multiple times. -- Macro: int mpq_sgn (mpq_t OP) Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. This function is actually implemented as a macro. It evaluates its argument multiple times. -- Function: int mpq_equal (mpq_t OP1, mpq_t OP2) Return non-zero if OP1 and OP2 are equal, zero if they are non-equal. Although `mpq_cmp' can be used for the same purpose, this function is much faster.  File: gmp.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions 6.5 Applying Integer Functions to Rationals =========================================== The set of `mpq' functions is quite small. In particular, there are few functions for either input or output. The following functions give direct access to the numerator and denominator of an `mpq_t'. Note that if an assignment to the numerator and/or denominator could take an `mpq_t' out of the canonical form described at the start of this chapter (*note Rational Number Functions::) then `mpq_canonicalize' must be called before any other `mpq' functions are applied to that `mpq_t'. -- Macro: mpz_t mpq_numref (mpq_t OP) -- Macro: mpz_t mpq_denref (mpq_t OP) Return a reference to the numerator and denominator of OP, respectively. The `mpz' functions can be used on the result of these macros. -- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL) -- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL) -- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR) -- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR) Get or set the numerator or denominator of a rational. These functions are equivalent to calling `mpz_set' with an appropriate `mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or `mpq_denref' is recommended instead of these functions.  File: gmp.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions 6.6 Input and Output Functions ============================== Functions that perform input from a stdio stream, and functions that output to a stdio stream, of `mpq' numbers. Passing a `NULL' pointer for a STREAM argument to any of these functions will make them read from `stdin' and write to `stdout', respectively. When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions. See also *note Formatted Output:: and *note Formatted Input::. -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP) Output OP on stdio stream STREAM, as a string of digits in base BASE. The base may vary from 2 to 36. Output is in the form `num/den' or if the denominator is 1 then just `num'. Return the number of bytes written, or if an error occurred, return 0. -- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE) Read a string of digits from STREAM and convert them to a rational in ROP. Any initial white-space characters are read and discarded. Return the number of characters read (including white space), or 0 if a rational could not be read. The input can be a fraction like `17/63' or just an integer like `123'. Reading stops at the first character not in this form, and white space is not permitted within the string. If the input might not be in canonical form, then `mpq_canonicalize' must be called (*note Rational Number Functions::). The BASE can be between 2 and 36, or can be 0 in which case the leading characters of the string determine the base, `0x' or `0X' for hexadecimal, `0' for octal, or decimal otherwise. The leading characters are examined separately for the numerator and denominator of a fraction, so for instance `0x10/11' is 16/11, whereas `0x10/0x11' is 16/17.  File: gmp.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top 7 Floating-point Functions ************************** GMP floating point numbers are stored in objects of type `mpf_t' and functions operating on them have an `mpf_' prefix. The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time. The exponent of each float is a fixed precision, one machine word on most systems. In the current implementation the exponent is a count of limbs, so for example on a 32-bit system this means a range of roughly 2^-68719476768 to 2^68719476736, or on a 64-bit system this will be greater. Note however `mpf_get_str' can only return an exponent which fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents bigger than a `long'. Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high. All calculations are performed to the precision of the destination variable. Each function is defined to calculate with "infinite precision" followed by a truncation to the destination precision, but of course the work done is only what's needed to determine a result under that definition. The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn't be concerned by such details. The mantissa in stored in binary, as might be imagined from the fact precisions are expressed in bits. One consequence of this is that decimal fractions like 0.1 cannot be represented exactly. The same is true of plain IEEE `double' floats. This makes both highly unsuitable for calculations involving money or other values that should be exact decimal fractions. (Suitably scaled integers, or perhaps rationals, are better choices.) `mpf' functions and variables have no special notion of infinity or not-a-number, and applications must take care not to overflow the exponent or results will be unpredictable. This might change in a future release. Note that the `mpf' functions are _not_ intended as a smooth extension to IEEE P754 arithmetic. In particular results obtained on one computer often differ from the results on a computer with a different word size. * Menu: * Initializing Floats:: * Assigning Floats:: * Simultaneous Float Init & Assign:: * Converting Floats:: * Float Arithmetic:: * Float Comparison:: * I/O of Floats:: * Miscellaneous Float Functions::  File: gmp.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions 7.1 Initialization Functions ============================ -- Function: void mpf_set_default_prec (mp_bitcnt_t PREC) Set the default precision to be *at least* PREC bits. All subsequent calls to `mpf_init' will use this precision, but previously initialized variables are unaffected. -- Function: mp_bitcnt_t mpf_get_default_prec (void) Return the default precision actually used. An `mpf_t' object must be initialized before storing the first value in it. The functions `mpf_init' and `mpf_init2' are used for that purpose. -- Function: void mpf_init (mpf_t X) Initialize X to 0. Normally, a variable should be initialized once only or at least be cleared, using `mpf_clear', between initializations. The precision of X is undefined unless a default precision has already been established by a call to `mpf_set_default_prec'. -- Function: void mpf_init2 (mpf_t X, mp_bitcnt_t PREC) Initialize X to 0 and set its precision to be *at least* PREC bits. Normally, a variable should be initialized once only or at least be cleared, using `mpf_clear', between initializations. -- Function: void mpf_inits (mpf_t X, ...) Initialize a NULL-terminated list of `mpf_t' variables, and set their values to 0. The precision of the initialized variables is undefined unless a default precision has already been established by a call to `mpf_set_default_prec'. -- Function: void mpf_clear (mpf_t X) Free the space occupied by X. Make sure to call this function for all `mpf_t' variables when you are done with them. -- Function: void mpf_clears (mpf_t X, ...) Free the space occupied by a NULL-terminated list of `mpf_t' variables. Here is an example on how to initialize floating-point variables: { mpf_t x, y; mpf_init (x); /* use default precision */ mpf_init2 (y, 256); /* precision _at least_ 256 bits */ ... /* Unless the program is about to exit, do ... */ mpf_clear (x); mpf_clear (y); } The following three functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers. -- Function: mp_bitcnt_t mpf_get_prec (mpf_t OP) Return the current precision of OP, in bits. -- Function: void mpf_set_prec (mpf_t ROP, mp_bitcnt_t PREC) Set the precision of ROP to be *at least* PREC bits. The value in ROP will be truncated to the new precision. This function requires a call to `realloc', and so should not be used in a tight loop. -- Function: void mpf_set_prec_raw (mpf_t ROP, mp_bitcnt_t PREC) Set the precision of ROP to be *at least* PREC bits, without changing the memory allocated. PREC must be no more than the allocated precision for ROP, that being the precision when ROP was initialized, or in the most recent `mpf_set_prec'. The value in ROP is unchanged, and in particular if it had a higher precision than PREC it will retain that higher precision. New values written to ROP will use the new PREC. Before calling `mpf_clear' or the full `mpf_set_prec', another `mpf_set_prec_raw' call must be made to restore ROP to its original allocated precision. Failing to do so will have unpredictable results. `mpf_get_prec' can be used before `mpf_set_prec_raw' to get the original allocated precision. After `mpf_set_prec_raw' it reflects the PREC value set. `mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable at different precisions during a calculation, perhaps to gradually increase precision in an iteration, or just to use various different precisions for different purposes during a calculation.  File: gmp.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions 7.2 Assignment Functions ======================== These functions assign new values to already initialized floats (*note Initializing Floats::). -- Function: void mpf_set (mpf_t ROP, mpf_t OP) -- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP) -- Function: void mpf_set_si (mpf_t ROP, signed long int OP) -- Function: void mpf_set_d (mpf_t ROP, double OP) -- Function: void mpf_set_z (mpf_t ROP, mpz_t OP) -- Function: void mpf_set_q (mpf_t ROP, mpq_t OP) Set the value of ROP from OP. -- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE) Set the value of ROP from the string in STR. The string is of the form `M@N' or, if the base is 10 or less, alternatively `MeN'. `M' is the mantissa and `N' is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if BASE is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing `localeconv'. The argument BASE may be in the ranges 2 to 62, or -62 to -2. Negative values are used to specify that the exponent is in decimal. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value; for bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. Unlike the corresponding `mpz' function, the base will not be determined from the leading characters of the string if BASE is 0. This is so that numbers like `0.23' are not interpreted as octal. White space is allowed in the string, and is simply ignored. [This is not really true; white-space is ignored in the beginning of the string and within the mantissa, but not in other places, such as after a minus sign or in the exponent. We are considering changing the definition of this function, making it fail when there is any white-space in the input, since that makes a lot of sense. Please tell us your opinion about this change. Do you really want it to accept "3 14" as meaning 314 as it does now?] This function returns 0 if the entire string is a valid number in base BASE. Otherwise it returns -1. -- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2) Swap ROP1 and ROP2 efficiently. Both the values and the precisions of the two variables are swapped.  File: gmp.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions 7.3 Combined Initialization and Assignment Functions ==================================================== For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions' names have the form `mpf_init_set...' Once the float has been initialized by any of the `mpf_init_set...' functions, it can be used as the source or destination operand for the ordinary float functions. Don't use an initialize-and-set function on a variable already initialized! -- Function: void mpf_init_set (mpf_t ROP, mpf_t OP) -- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP) -- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP) -- Function: void mpf_init_set_d (mpf_t ROP, double OP) Initialize ROP and set its value from OP. The precision of ROP will be taken from the active default precision, as set by `mpf_set_default_prec'. -- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE) Initialize ROP and set its value from the string in STR. See `mpf_set_str' above for details on the assignment operation. Note that ROP is initialized even if an error occurs. (I.e., you have to call `mpf_clear' for it.) The precision of ROP will be taken from the active default precision, as set by `mpf_set_default_prec'.  File: gmp.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions 7.4 Conversion Functions ======================== -- Function: double mpf_get_d (mpf_t OP) Convert OP to a `double', truncating if necessary (i.e. rounding towards zero). If the exponent in OP is too big or too small to fit a `double' then the result is system dependent. For too big an infinity is returned when available. For too small 0.0 is normally returned. Hardware overflow, underflow and denorm traps may or may not occur. -- Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP) Convert OP to a `double', truncating if necessary (i.e. rounding towards zero), and with an exponent returned separately. The return value is in the range 0.5<=abs(D)<1 and the exponent is stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP is zero, the return is 0.0 and 0 is stored to `*EXP'. This is similar to the standard C `frexp' function (*note Normalization Functions: (libc)Normalization Functions.). -- Function: long mpf_get_si (mpf_t OP) -- Function: unsigned long mpf_get_ui (mpf_t OP) Convert OP to a `long' or `unsigned long', truncating any fraction part. If OP is too big for the return type, the result is undefined. See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note Miscellaneous Float Functions::). -- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE, size_t N_DIGITS, mpf_t OP) Convert OP to a string of digits in base BASE. The base argument may vary from 2 to 62 or from -2 to -36. Up to N_DIGITS digits will be generated. Trailing zeros are not returned. No more digits than can be accurately represented by OP are ever generated. If N_DIGITS is 0 then that accurate maximum number of digits are generated. For BASE in the range 2..36, digits and lower-case letters are used; for -2..-36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of N_DIGITS + 2 bytes, that being enough for the mantissa, a possible minus sign, and a null-terminator. When N_DIGITS is 0 to get all significant digits, an application won't be able to know the space required, and STR should be `NULL' in that case. The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. The applicable exponent is written through the EXPPTR pointer. For example, the number 3.1416 would be returned as string "31416" and exponent 1. When OP is zero, an empty string is produced and the exponent returned is 0. A pointer to the result string is returned, being either the allocated block or the given STR.  File: gmp.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions 7.5 Arithmetic Functions ======================== -- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2) -- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 + OP2. -- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2) -- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t OP2) -- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 - OP2. -- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2) -- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 times OP2. Division is undefined if the divisor is zero, and passing a zero divisor to the divide functions will make these functions intentionally divide by zero. This lets the user handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions. -- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2) -- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t OP2) -- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1/OP2. -- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP) -- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP) Set ROP to the square root of OP. -- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 raised to the power OP2. -- Function: void mpf_neg (mpf_t ROP, mpf_t OP) Set ROP to -OP. -- Function: void mpf_abs (mpf_t ROP, mpf_t OP) Set ROP to the absolute value of OP. -- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2) Set ROP to OP1 times 2 raised to OP2. -- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2) Set ROP to OP1 divided by 2 raised to OP2.  File: gmp.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions 7.6 Comparison Functions ======================== -- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2) -- Function: int mpf_cmp_d (mpf_t OP1, double OP2) -- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2) -- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. `mpf_cmp_d' can be called with an infinity, but results are undefined for a NaN. -- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, mp_bitcnt_t op3) Return non-zero if the first OP3 bits of OP1 and OP2 are equal, zero otherwise. I.e., test if OP1 and OP2 are approximately equal. Caution 1: All version of GMP up to version 4.2.4 compared just whole limbs, meaning sometimes more than OP3 bits, sometimes fewer. Caution 2: This function will consider XXX11...111 and XX100...000 different, even if ... is replaced by a semi-infinite number of bits. Such numbers are really just one ulp off, and should be considered equal. -- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2) Compute the relative difference between OP1 and OP2 and store the result in ROP. This is abs(OP1-OP2)/OP1. -- Macro: int mpf_sgn (mpf_t OP) Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0. This function is actually implemented as a macro. It evaluates its argument multiple times.  File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions 7.7 Input and Output Functions ============================== Functions that perform input from a stdio stream, and functions that output to a stdio stream, of `mpf' numbers. Passing a `NULL' pointer for a STREAM argument to any of these functions will make them read from `stdin' and write to `stdout', respectively. When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions. See also *note Formatted Output:: and *note Formatted Input::. -- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t N_DIGITS, mpf_t OP) Print OP to STREAM, as a string of digits. Return the number of bytes written, or if an error occurred, return 0. The mantissa is prefixed with an `0.' and is in the given BASE, which may vary from 2 to 62 or from -2 to -36. An exponent is then printed, separated by an `e', or if the base is greater than 10 then by an `@'. The exponent is always in decimal. The decimal point follows the current locale, on systems providing `localeconv'. For BASE in the range 2..36, digits and lower-case letters are used; for -2..-36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. Up to N_DIGITS will be printed from the mantissa, except that no more digits than are accurately representable by OP will be printed. N_DIGITS can be 0 to select that accurate maximum. -- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE) Read a string in base BASE from STREAM, and put the read float in ROP. The string is of the form `M@N' or, if the base is 10 or less, alternatively `MeN'. `M' is the mantissa and `N' is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if BASE is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing `localeconv'. The argument BASE may be in the ranges 2 to 36, or -36 to -2. Negative values are used to specify that the exponent is in decimal. Unlike the corresponding `mpz' function, the base will not be determined from the leading characters of the string if BASE is 0. This is so that numbers like `0.23' are not interpreted as octal. Return the number of bytes read, or if an error occurred, return 0.  File: gmp.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions 7.8 Miscellaneous Functions =========================== -- Function: void mpf_ceil (mpf_t ROP, mpf_t OP) -- Function: void mpf_floor (mpf_t ROP, mpf_t OP) -- Function: void mpf_trunc (mpf_t ROP, mpf_t OP) Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the next higher integer, `mpf_floor' to the next lower, and `mpf_trunc' to the integer towards zero. -- Function: int mpf_integer_p (mpf_t OP) Return non-zero if OP is an integer. -- Function: int mpf_fits_ulong_p (mpf_t OP) -- Function: int mpf_fits_slong_p (mpf_t OP) -- Function: int mpf_fits_uint_p (mpf_t OP) -- Function: int mpf_fits_sint_p (mpf_t OP) -- Function: int mpf_fits_ushort_p (mpf_t OP) -- Function: int mpf_fits_sshort_p (mpf_t OP) Return non-zero if OP would fit in the respective C data type, when truncated to an integer. -- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE, mp_bitcnt_t NBITS) Generate a uniformly distributed random float in ROP, such that 0 <= ROP < 1, with NBITS significant bits in the mantissa or less if the precision of ROP is smaller. The variable STATE must be initialized by calling one of the `gmp_randinit' functions (*note Random State Initialization::) before invoking this function. -- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t EXP) Generate a random float of at most MAX_SIZE limbs, with long strings of zeros and ones in the binary representation. The exponent of the number is in the interval -EXP to EXP (in limbs). This function is useful for testing functions and algorithms, since these kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when MAX_SIZE is negative.  File: gmp.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top 8 Low-level Functions ********************* This chapter describes low-level GMP functions, used to implement the high-level GMP functions, but also intended for time-critical user code. These functions start with the prefix `mpn_'. The `mpn' functions are designed to be as fast as possible, *not* to provide a coherent calling interface. The different functions have somewhat similar interfaces, but there are variations that make them hard to use. These functions do as little as possible apart from the real multiple precision computation, so that no time is spent on things that not all callers need. A source operand is specified by a pointer to the least significant limb and a limb count. A destination operand is specified by just a pointer. It is the responsibility of the caller to ensure that the destination has enough space for storing the result. With this way of specifying operands, it is possible to perform computations on subranges of an argument, and store the result into a subrange of a destination. A common requirement for all functions is that each source area needs at least one limb. No size argument may be zero. Unless otherwise stated, in-place operations are allowed where source and destination are the same, but not where they only partly overlap. The `mpn' functions are the base for the implementation of the `mpz_', `mpf_', and `mpq_' functions. This example adds the number beginning at S1P and the number beginning at S2P and writes the sum at DESTP. All areas have N limbs. cy = mpn_add_n (destp, s1p, s2p, n) It should be noted that the `mpn' functions make no attempt to identify high or low zero limbs on their operands, or other special forms. On random data such cases will be unlikely and it'd be wasteful for every function to check every time. An application knowing something about its data can take steps to trim or perhaps split its calculations. In the notation used below, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, {S1P, S1N}. -- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Add {S1P, N} and {S2P, N}, and write the N least significant limbs of the result to RP. Return carry, either 0 or 1. This is the lowest-level function for addition. It is the preferred function for addition, since it is written in assembly for most CPUs. For addition of a variable to itself (i.e., S1P equals S2P) use `mpn_lshift' with a count of 1 for optimal speed. -- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Add {S1P, N} and S2LIMB, and write the N least significant limbs of the result to RP. Return carry, either 0 or 1. -- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N) Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant limbs of the result to RP. Return carry, either 0 or 1. This function requires that S1N is greater than or equal to S2N. -- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Subtract {S2P, N} from {S1P, N}, and write the N least significant limbs of the result to RP. Return borrow, either 0 or 1. This is the lowest-level function for subtraction. It is the preferred function for subtraction, since it is written in assembly for most CPUs. -- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Subtract S2LIMB from {S1P, N}, and write the N least significant limbs of the result to RP. Return borrow, either 0 or 1. -- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N) Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least significant limbs of the result to RP. Return borrow, either 0 or 1. This function requires that S1N is greater than or equal to S2N. -- Function: mp_limb_t mpn_neg (mp_limb_t *RP, const mp_limb_t *SP, mp_size_t N) Perform the negation of {SP, N}, and write the result to {RP, N}. Return carry-out. -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to RP. The destination has to have space for 2*N limbs, even if the product's most significant limb is zero. No overlap is permitted between the destination and either source. If the two input operands are the same, use `mpn_sqr'. -- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N) Multiply {S1P, S1N} and {S2P, S2N}, and write the (S1N+S2N)-limb result to RP. Return the most significant limb of the result. The destination has to have space for S1N + S2N limbs, even if the product's most significant limb is zero. No overlap is permitted between the destination and either source. This function requires that S1N is greater than or equal to S2N. -- Function: void mpn_sqr (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N) Compute the square of {S1P, N} and write the 2*N-limb result to RP. The destination has to have space for 2*N limbs, even if the result's most significant limb is zero. No overlap is permitted between the destination and the source. -- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Multiply {S1P, N} by S2LIMB, and write the N least significant limbs of the product to RP. Return the most significant limb of the product. {S1P, N} and {RP, N} are allowed to overlap provided RP <= S1P. This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs. Don't call this function if S2LIMB is a power of 2; use `mpn_lshift' with a count equal to the logarithm of S2LIMB instead, for optimal speed. -- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Multiply {S1P, N} and S2LIMB, and add the N least significant limbs of the product to {RP, N} and write the result to RP. Return the most significant limb of the product, plus carry-out from the addition. This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs. -- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Multiply {S1P, N} and S2LIMB, and subtract the N least significant limbs of the product from {RP, N} and write the result to RP. Return the most significant limb of the product, plus borrow-out from the subtraction. This is a low-level function that is a building block for general multiplication and division as well as other operations in GMP. It is written in assembly for most CPUs. -- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP, mp_size_t DN) Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1} and the remainder at {RP, DN}. The quotient is rounded towards 0. No overlap is permitted between arguments, except that NP might equal RP. The dividend size NN must be greater than or equal to divisor size DN. The most significant limb of the divisor must be non-zero. The QXN operand must be zero. -- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN, mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N) [This function is obsolete. Please call `mpn_tdiv_qr' instead for best performance.] Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P, with the exception of the most significant limb, which is returned. The remainder replaces the dividend at RS2P; it will be S3N limbs long (i.e., as many limbs as the divisor). In addition to an integer quotient, QXN fraction limbs are developed, and stored after the integral limbs. For most usages, QXN will be zero. It is required that RS2N is greater than or equal to S3N. It is required that the most significant bit of the divisor is set. If the quotient is not needed, pass RS2P + S3N as R1P. Aside from that special case, no overlap between arguments is permitted. Return the most significant limb of the quotient, either 0 or 1. The area at R1P needs to be RS2N - S3N + QXN limbs large. -- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN, mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB) -- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB) Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P. Return the remainder. The integer quotient is written to {R1P+QXN, S2N} and in addition QXN fraction limbs are developed and written to {R1P, QXN}. Either or both S2N and QXN can be zero. For most usages, QXN will be zero. `mpn_divmod_1' exists for upward source compatibility and is simply a macro calling `mpn_divrem_1' with a QXN of 0. The areas at R1P and S2P have to be identical or completely separate, not partially overlapping. -- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N) [This function is obsolete. Please call `mpn_tdiv_qr' instead for best performance.] -- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP, mp_size_t N) -- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP, mp_size_t N, mp_limb_t CARRY) Divide {SP, N} by 3, expecting it to divide exactly, and writing the result to {RP, N}. If 3 divides exactly, the return value is zero and the result is the quotient. If not, the return value is non-zero and the result won't be anything useful. `mpn_divexact_by3c' takes an initial carry parameter, which can be the return value from a previous call, so a large calculation can be done piece by piece from low to high. `mpn_divexact_by3' is simply a macro calling `mpn_divexact_by3c' with a 0 carry parameter. These routines use a multiply-by-inverse and will be faster than `mpn_divrem_1' on CPUs with fast multiplication but slow division. The source a, result q, size n, initial carry i, and return value c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return c is always 0, 1 or 2, and the initial carry i must also be 0, 1 or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3. When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b == 1 mod 3 (when `mp_bits_per_limb' is even, which is always so currently). -- Function: mp_limb_t mpn_mod_1 (const mp_limb_t *S1P, mp_size_t S1N, mp_limb_t S2LIMB) Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be zero. -- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP, mp_size_t N, unsigned int COUNT) Shift {SP, N} left by COUNT bits, and write the result to {RP, N}. The bits shifted out at the left are returned in the least significant COUNT bits of the return value (the rest of the return value is zero). COUNT must be in the range 1 to mp_bits_per_limb-1. The regions {SP, N} and {RP, N} may overlap, provided RP >= SP. This function is written in assembly for most CPUs. -- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP, mp_size_t N, unsigned int COUNT) Shift {SP, N} right by COUNT bits, and write the result to {RP, N}. The bits shifted out at the right are returned in the most significant COUNT bits of the return value (the rest of the return value is zero). COUNT must be in the range 1 to mp_bits_per_limb-1. The regions {SP, N} and {RP, N} may overlap, provided RP <= SP. This function is written in assembly for most CPUs. -- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Compare {S1P, N} and {S2P, N} and return a positive value if S1 > S2, 0 if they are equal, or a negative value if S1 < S2. -- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *XP, mp_size_t XN, mp_limb_t *YP, mp_size_t YN) Set {RP, RETVAL} to the greatest common divisor of {XP, XN} and {YP, YN}. The result can be up to YN limbs, the return value is the actual number produced. Both source operands are destroyed. It is required that XN >= YN > 0, and the most significant limb of {YP, YN} must be non-zero. No overlap is permitted between {XP, XN} and {YP, YN}. -- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *XP, mp_size_t XN, mp_limb_t YLIMB) Return the greatest common divisor of {XP, XN} and YLIMB. Both operands must be non-zero. -- Function: mp_size_t mpn_gcdext (mp_limb_t *GP, mp_limb_t *SP, mp_size_t *SN, mp_limb_t *UP, mp_size_t UN, mp_limb_t *VP, mp_size_t VN) Let U be defined by {UP, UN} and let V be defined by {VP, VN}. Compute the greatest common divisor G of U and V. Compute a cofactor S such that G = US + VT. The second cofactor T is not computed but can easily be obtained from (G - U*S) / V (the division will be exact). It is required that UN >= VN > 0, and the most significant limb of {VP, VN} must be non-zero. S satisfies S = 1 or abs(S) < V / (2 G). S = 0 if and only if V divides U (i.e., G = V). Store G at GP and let the return value define its limb count. Store S at SP and let |*SN| define its limb count. S can be negative; when this happens *SN will be negative. The area at GP should have room for VN limbs and the area at SP should have room for VN+1 limbs. Both source operands are destroyed. Compatibility notes: GMP 4.3.0 and 4.3.1 defined S less strictly. Earlier as well as later GMP releases define S as described here. GMP releases before GMP 4.3.0 required additional space for both input and output areas. More precisely, the areas {UP, UN+1} and {VP, VN+1} were destroyed (i.e. the operands plus an extra limb past the end of each), and the areas pointed to by GP and SP should each have room for UN+1 limbs. -- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P, const mp_limb_t *SP, mp_size_t N) Compute the square root of {SP, N} and put the result at {R1P, ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space for N limbs, but the return value indicates how many are produced. The most significant limb of {SP, N} must be non-zero. The areas {R1P, ceil(N/2)} and {SP, N} must be completely separate. The areas {R2P, N} and {SP, N} must be either identical or completely separate. If the remainder is not wanted then R2P can be `NULL', and in this case the return value is zero or non-zero according to whether the remainder would have been zero or non-zero. A return value of zero indicates a perfect square. See also `mpn_perfect_square_p'. -- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE, mp_limb_t *S1P, mp_size_t S1N) Convert {S1P, S1N} to a raw unsigned char array at STR in base BASE, and return the number of characters produced. There may be leading zeros in the string. The string is not in ASCII; to convert it to printable format, add the ASCII codes for `0' or `A', depending on the base and range. BASE can vary from 2 to 256. The most significant limb of the input {S1P, S1N} must be non-zero. The input {S1P, S1N} is clobbered, except when BASE is a power of 2, in which case it's unchanged. The area at STR has to have space for the largest possible number represented by a S1N long limb array, plus one extra character. -- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char *STR, size_t STRSIZE, int BASE) Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP. STR[0] is the most significant byte and STR[STRSIZE-1] is the least significant. Each byte should be a value in the range 0 to BASE-1, not an ASCII character. BASE can vary from 2 to 256. The return value is the number of limbs written to RP. If the most significant input byte is non-zero then the high limb at RP will be non-zero, and only that exact number of limbs will be required there. If the most significant input byte is zero then there may be high zero limbs written to RP and included in the return value. STRSIZE must be at least 1, and no overlap is permitted between {STR,STRSIZE} and the result at RP. -- Function: mp_bitcnt_t mpn_scan0 (const mp_limb_t *S1P, mp_bitcnt_t BIT) Scan S1P from bit position BIT for the next clear bit. It is required that there be a clear bit within the area at S1P at or beyond bit position BIT, so that the function has something to return. -- Function: mp_bitcnt_t mpn_scan1 (const mp_limb_t *S1P, mp_bitcnt_t BIT) Scan S1P from bit position BIT for the next set bit. It is required that there be a set bit within the area at S1P at or beyond bit position BIT, so that the function has something to return. -- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N) -- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N) Generate a random number of length R1N and store it at R1P. The most significant limb is always non-zero. `mpn_random' generates uniformly distributed limb data, `mpn_random2' generates long strings of zeros and ones in the binary representation. `mpn_random2' is intended for testing the correctness of the `mpn' routines. -- Function: mp_bitcnt_t mpn_popcount (const mp_limb_t *S1P, mp_size_t N) Count the number of set bits in {S1P, N}. -- Function: mp_bitcnt_t mpn_hamdist (const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Compute the hamming distance between {S1P, N} and {S2P, N}, which is the number of bit positions where the two operands have different bit values. -- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t N) Return non-zero iff {S1P, N} is a perfect square. The most significant limb of the input {S1P, N} must be non-zero. -- Function: void mpn_and_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical and of {S1P, N} and {S2P, N}, and write the result to {RP, N}. -- Function: void mpn_ior_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N}, and write the result to {RP, N}. -- Function: void mpn_xor_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N}, and write the result to {RP, N}. -- Function: void mpn_andn_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical and of {S1P, N} and the bitwise complement of {S2P, N}, and write the result to {RP, N}. -- Function: void mpn_iorn_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical inclusive or of {S1P, N} and the bitwise complement of {S2P, N}, and write the result to {RP, N}. -- Function: void mpn_nand_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical and of {S1P, N} and {S2P, N}, and write the bitwise complement of the result to {RP, N}. -- Function: void mpn_nior_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N}, and write the bitwise complement of the result to {RP, N}. -- Function: void mpn_xnor_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N}, and write the bitwise complement of the result to {RP, N}. -- Function: void mpn_com (mp_limb_t *RP, const mp_limb_t *SP, mp_size_t N) Perform the bitwise complement of {SP, N}, and write the result to {RP, N}. -- Function: void mpn_copyi (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N) Copy from {S1P, N} to {RP, N}, increasingly. -- Function: void mpn_copyd (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N) Copy from {S1P, N} to {RP, N}, decreasingly. -- Function: void mpn_zero (mp_limb_t *RP, mp_size_t N) Zero {RP, N}. 8.1 Nails ========= *Everything in this section is highly experimental and may disappear or be subject to incompatible changes in a future version of GMP.* Nails are an experimental feature whereby a few bits are left unused at the top of each `mp_limb_t'. This can significantly improve carry handling on some processors. All the `mpn' functions accepting limb data will expect the nail bits to be zero on entry, and will return data with the nails similarly all zero. This applies both to limb vectors and to single limb arguments. Nails can be enabled by configuring with `--enable-nails'. By default the number of bits will be chosen according to what suits the host processor, but a particular number can be selected with `--enable-nails=N'. At the mpn level, a nail build is neither source nor binary compatible with a non-nail build, strictly speaking. But programs acting on limbs only through the mpn functions are likely to work equally well with either build, and judicious use of the definitions below should make any program compatible with either build, at the source level. For the higher level routines, meaning `mpz' etc, a nail build should be fully source and binary compatible with a non-nail build. -- Macro: GMP_NAIL_BITS -- Macro: GMP_NUMB_BITS -- Macro: GMP_LIMB_BITS `GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are not in use. `GMP_NUMB_BITS' is the number of data bits in a limb. `GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In all cases GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS -- Macro: GMP_NAIL_MASK -- Macro: GMP_NUMB_MASK Bit masks for the nail and number parts of a limb. `GMP_NAIL_MASK' is 0 when nails are not in use. `GMP_NAIL_MASK' is not often needed, since the nail part can be obtained with `x >> GMP_NUMB_BITS', and that means one less large constant, which can help various RISC chips. -- Macro: GMP_NUMB_MAX The maximum value that can be stored in the number part of a limb. This is the same as `GMP_NUMB_MASK', but can be used for clarity when doing comparisons rather than bit-wise operations. The term "nails" comes from finger or toe nails, which are at the ends of a limb (arm or leg). "numb" is short for number, but is also how the developers felt after trying for a long time to come up with sensible names for these things. In the future (the distant future most likely) a non-zero nail might be permitted, giving non-unique representations for numbers in a limb vector. This would help vector processors since carries would only ever need to propagate one or two limbs.  File: gmp.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top 9 Random Number Functions ************************* Sequences of pseudo-random numbers in GMP are generated using a variable of type `gmp_randstate_t', which holds an algorithm selection and a current state. Such a variable must be initialized by a call to one of the `gmp_randinit' functions, and can be seeded with one of the `gmp_randseed' functions. The functions actually generating random numbers are described in *note Integer Random Numbers::, and *note Miscellaneous Float Functions::. The older style random number functions don't accept a `gmp_randstate_t' parameter but instead share a global variable of that type. They use a default algorithm and are currently not seeded (though perhaps that will change in the future). The new functions accepting a `gmp_randstate_t' are recommended for applications that care about randomness. * Menu: * Random State Initialization:: * Random State Seeding:: * Random State Miscellaneous::  File: gmp.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions 9.1 Random State Initialization =============================== -- Function: void gmp_randinit_default (gmp_randstate_t STATE) Initialize STATE with a default algorithm. This will be a compromise between speed and randomness, and is recommended for applications with no special requirements. Currently this is `gmp_randinit_mt'. -- Function: void gmp_randinit_mt (gmp_randstate_t STATE) Initialize STATE for a Mersenne Twister algorithm. This algorithm is fast and has good randomness properties. -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t A, unsigned long C, mp_bitcnt_t M2EXP) Initialize STATE with a linear congruential algorithm X = (A*X + C) mod 2^M2EXP. The low bits of X in this algorithm are not very random. The least significant bit will have a period no more than 2, and the second bit no more than 4, etc. For this reason only the high half of each X is actually used. When a random number of more than M2EXP/2 bits is to be generated, multiple iterations of the recurrence are used and the results concatenated. -- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE, mp_bitcnt_t SIZE) Initialize STATE for a linear congruential algorithm as per `gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table, chosen so that SIZE bits (or more) of each X will be used, i.e. M2EXP/2 >= SIZE. If successful the return value is non-zero. If SIZE is bigger than the table data provides then the return value is zero. The maximum SIZE currently supported is 128. -- Function: void gmp_randinit_set (gmp_randstate_t ROP, gmp_randstate_t OP) Initialize ROP with a copy of the algorithm and state from OP. -- Function: void gmp_randinit (gmp_randstate_t STATE, gmp_randalg_t ALG, ...) *This function is obsolete.* Initialize STATE with an algorithm selected by ALG. The only choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size' described above. A third parameter of type `unsigned long' is required, this is the SIZE for that function. `GMP_RAND_ALG_DEFAULT' or 0 are the same as `GMP_RAND_ALG_LC'. `gmp_randinit' sets bits in the global variable `gmp_errno' to indicate an error. `GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is unsupported, or `GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter is too big. It may be noted this error reporting is not thread safe (a good reason to use `gmp_randinit_lc_2exp_size' instead). -- Function: void gmp_randclear (gmp_randstate_t STATE) Free all memory occupied by STATE.  File: gmp.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions 9.2 Random State Seeding ======================== -- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED) -- Function: void gmp_randseed_ui (gmp_randstate_t STATE, unsigned long int SEED) Set an initial seed value into STATE. The size of a seed determines how many different sequences of random numbers that it's possible to generate. The "quality" of the seed is the randomness of a given seed compared to the previous seed used, and this affects the randomness of separate number sequences. The method for choosing a seed is critical if the generated numbers are to be used for important applications, such as generating cryptographic keys. Traditionally the system time has been used to seed, but care needs to be taken with this. If an application seeds often and the resolution of the system clock is low, then the same sequence of numbers might be repeated. Also, the system time is quite easy to guess, so if unpredictability is required then it should definitely not be the only source for the seed value. On some systems there's a special device `/dev/random' which provides random data better suited for use as a seed.  File: gmp.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions 9.3 Random State Miscellaneous ============================== -- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE, unsigned long N) Return a uniformly distributed random number of N bits, i.e. in the range 0 to 2^N-1 inclusive. N must be less than or equal to the number of bits in an `unsigned long'. -- Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE, unsigned long N) Return a uniformly distributed random number in the range 0 to N-1, inclusive.  File: gmp.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top 10 Formatted Output ******************* * Menu: * Formatted Output Strings:: * Formatted Output Functions:: * C++ Formatted Output::  File: gmp.info, Node: Formatted Output Strings, Next: Formatted Output Functions, Prev: Formatted Output, Up: Formatted Output 10.1 Format Strings =================== `gmp_printf' and friends accept format strings similar to the standard C `printf' (*note Formatted Output: (libc)Formatted Output.). A format specification is of the form % [flags] [width] [.[precision]] [type] conv GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t' respectively, `M' for `mp_limb_t', and `N' for an `mp_limb_t' array. `Z', `Q', `M' and `N' behave like integers. `Q' will print a `/' and a denominator, if needed. `F' behaves like a float. For example, mpz_t z; gmp_printf ("%s is an mpz %Zd\n", "here", z); mpq_t q; gmp_printf ("a hex rational: %#40Qx\n", q); mpf_t f; int n; gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n); mp_limb_t l; gmp_printf ("limb %Mu\n", l); const mp_limb_t *ptr; mp_size_t size; gmp_printf ("limb array %Nx\n", ptr, size); For `N' the limbs are expected least significant first, as per the `mpn' functions (*note Low-level Functions::). A negative size can be given to print the value as a negative. All the standard C `printf' types behave the same as the C library `printf', and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to `printf' and only the GMP extensions handled directly. The flags accepted are as follows. GLIBC style ' is only for the standard C types (not the GMP types), and only if the C library supports it. 0 pad with zeros (rather than spaces) # show the base with `0x', `0X' or `0' + always show a sign (space) show a space or a `-' sign ' group digits, GLIBC style (not GMP types) The optional width and precision can be given as a number within the format string, or as a `*' to take an extra parameter of type `int', the same as the standard `printf'. The standard types accepted are as follows. `h' and `l' are portable, the rest will depend on the compiler (or include files) for the type and the C library for the output. h short hh char j intmax_t or uintmax_t l long or wchar_t ll long long L long double q quad_t or u_quad_t t ptrdiff_t z size_t The GMP types are F mpf_t, float conversions Q mpq_t, integer conversions M mp_limb_t, integer conversions N mp_limb_t array, integer conversions Z mpz_t, integer conversions The conversions accepted are as follows. `a' and `A' are always supported for `mpf_t' but depend on the C library for standard C float types. `m' and `p' depend on the C library. a A hex floats, C99 style c character d decimal integer e E scientific format float f fixed point float i same as d g G fixed or scientific float m `strerror' string, GLIBC style n store characters written so far o octal integer p pointer s string u unsigned integer x X hex integer `o', `x' and `X' are unsigned for the standard C types, but for types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z', `Q' and `N'. `M' is a proxy for the C library `l' or `L', according to the size of `mp_limb_t'. Unsigned conversions will be usual, but a signed conversion can be used and will interpret the value as a twos complement negative. `n' can be used with any type, even the GMP types. Other types or conversions that might be accepted by the C library `printf' cannot be used through `gmp_printf', this includes for instance extensions registered with GLIBC `register_printf_function'. Also currently there's no support for POSIX `$' style numbered arguments (perhaps this will be added in the future). The precision field has its usual meaning for integer `Z' and float `F' types, but is currently undefined for `Q' and should not be used with that. `mpf_t' conversions only ever generate as many digits as can be accurately represented by the operand, the same as `mpf_get_str' does. Zeros will be used if necessary to pad to the requested precision. This happens even for an `f' conversion of an `mpf_t' which is an integer, for instance 2^1024 in an `mpf_t' of 128 bits precision will only produce about 40 digits, then pad with zeros to the decimal point. An empty precision field like `%.Fe' or `%.Ff' can be used to specifically request just the significant digits. Without any dot and thus no precision field, a precision value of 6 will be used. Note that these rules mean that `%Ff', `%.Ff', and `%.0Ff' will all be different. The decimal point character (or string) is taken from the current locale settings on systems which provide `localeconv' (*note Locales and Internationalization: (libc)Locales.). The C library will normally do the same for standard float output. The format string is only interpreted as plain `char's, multibyte characters are not recognised. Perhaps this will change in the future.  File: gmp.info, Node: Formatted Output Functions, Next: C++ Formatted Output, Prev: Formatted Output Strings, Up: Formatted Output 10.2 Functions ============== Each of the following functions is similar to the corresponding C library function. The basic `printf' forms take a variable argument list. The `vprintf' forms take an argument pointer, see *note Variadic Functions: (libc)Variadic Functions, or `man 3 va_start'. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. The file based functions `gmp_printf' and `gmp_fprintf' will return -1 to indicate a write error. Output is not "atomic", so partial output may be produced if a write error occurs. All the functions can return -1 if the C library `printf' variant in use returns -1, but this shouldn't normally occur. -- Function: int gmp_printf (const char *FMT, ...) -- Function: int gmp_vprintf (const char *FMT, va_list AP) Print to the standard output `stdout'. Return the number of characters written, or -1 if an error occurred. -- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...) -- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP) Print to the stream FP. Return the number of characters written, or -1 if an error occurred. -- Function: int gmp_sprintf (char *BUF, const char *FMT, ...) -- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP) Form a null-terminated string in BUF. Return the number of characters written, excluding the terminating null. No overlap is permitted between the space at BUF and the string FMT. These functions are not recommended, since there's no protection against exceeding the space available at BUF. -- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char *FMT, ...) -- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char *FMT, va_list AP) Form a null-terminated string in BUF. No more than SIZE bytes will be written. To get the full output, SIZE must be enough for the string and null-terminator. The return value is the total number of characters which ought to have been produced, excluding the terminating null. If RETVAL >= SIZE then the actual output has been truncated to the first SIZE-1 characters, and a null appended. No overlap is permitted between the region {BUF,SIZE} and the FMT string. Notice the return value is in ISO C99 `snprintf' style. This is so even if the C library `vsnprintf' is the older GLIBC 2.0.x style. -- Function: int gmp_asprintf (char **PP, const char *FMT, ...) -- Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP) Form a null-terminated string in a block of memory obtained from the current memory allocation function (*note Custom Allocation::). The block will be the size of the string and null-terminator. The address of the block in stored to *PP. The return value is the number of characters produced, excluding the null-terminator. Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1 if there's no more memory available, it lets the current allocation function handle that. -- Function: int gmp_obstack_printf (struct obstack *OB, const char *FMT, ...) -- Function: int gmp_obstack_vprintf (struct obstack *OB, const char *FMT, va_list AP) Append to the current object in OB. The return value is the number of characters written. A null-terminator is not written. FMT cannot be within the current object in OB, since that object might move as it grows. These functions are available only when the C library provides the obstack feature, which probably means only on GNU systems, see *note Obstacks: (libc)Obstacks.  File: gmp.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output 10.3 C++ Formatted Output ========================= The following functions are provided in `libgmpxx' (*note Headers and Libraries::), which is built if C++ support is enabled (*note Build Options::). Prototypes are available from `'. -- Function: ostream& operator<< (ostream& STREAM, mpz_t OP) Print OP to STREAM, using its `ios' formatting settings. `ios::width' is reset to 0 after output, the same as the standard `ostream operator<<' routines do. In hex or octal, OP is printed as a signed number, the same as for decimal. This is unlike the standard `operator<<' routines on `int' etc, which instead give twos complement. -- Function: ostream& operator<< (ostream& STREAM, mpq_t OP) Print OP to STREAM, using its `ios' formatting settings. `ios::width' is reset to 0 after output, the same as the standard `ostream operator<<' routines do. Output will be a fraction like `5/9', or if the denominator is 1 then just a plain integer like `123'. In hex or octal, OP is printed as a signed value, the same as for decimal. If `ios::showbase' is set then a base indicator is shown on both the numerator and denominator (if the denominator is required). -- Function: ostream& operator<< (ostream& STREAM, mpf_t OP) Print OP to STREAM, using its `ios' formatting settings. `ios::width' is reset to 0 after output, the same as the standard `ostream operator<<' routines do. The decimal point follows the standard library float `operator<<', which on recent systems means the `std::locale' imbued on STREAM. Hex and octal are supported, unlike the standard `operator<<' on `double'. The mantissa will be in hex or octal, the exponent will be in decimal. For hex the exponent delimiter is an `@'. This is as per `mpf_out_str'. `ios::showbase' is supported, and will put a base on the mantissa, for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'. This last form is slightly strange, but at least differentiates itself from decimal. These operators mean that GMP types can be printed in the usual C++ way, for example, mpz_t z; int n; ... cout << "iteration " << n << " value " << z << "\n"; But note that `ostream' output (and `istream' input, *note C++ Formatted Input::) is the only overloading available for the GMP types and that for instance using `+' with an `mpz_t' will have unpredictable results. For classes with overloading, see *note C++ Class Interface::.  File: gmp.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top 11 Formatted Input ****************** * Menu: * Formatted Input Strings:: * Formatted Input Functions:: * C++ Formatted Input::  File: gmp.info, Node: Formatted Input Strings, Next: Formatted Input Functions, Prev: Formatted Input, Up: Formatted Input 11.1 Formatted Input Strings ============================ `gmp_scanf' and friends accept format strings similar to the standard C `scanf' (*note Formatted Input: (libc)Formatted Input.). A format specification is of the form % [flags] [width] [type] conv GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t' respectively. `Z' and `Q' behave like integers. `Q' will read a `/' and a denominator, if present. `F' behaves like a float. GMP variables don't require an `&' when passed to `gmp_scanf', since they're already "call-by-reference". For example, /* to read say "a(5) = 1234" */ int n; mpz_t z; gmp_scanf ("a(%d) = %Zd\n", &n, z); mpq_t q1, q2; gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2); /* to read say "topleft (1.55,-2.66)" */ mpf_t x, y; char buf[32]; gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y); All the standard C `scanf' types behave the same as in the C library `scanf', and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to `scanf' and only the GMP extensions handled directly. The flags accepted are as follows. `a' and `'' will depend on support from the C library, and `'' cannot be used with GMP types. * read but don't store a allocate a buffer (string conversions) ' grouped digits, GLIBC style (not GMP types) The standard types accepted are as follows. `h' and `l' are portable, the rest will depend on the compiler (or include files) for the type and the C library for the input. h short hh char j intmax_t or uintmax_t l long int, double or wchar_t ll long long L long double q quad_t or u_quad_t t ptrdiff_t z size_t The GMP types are F mpf_t, float conversions Q mpq_t, integer conversions Z mpz_t, integer conversions The conversions accepted are as follows. `p' and `[' will depend on support from the C library, the rest are standard. c character or characters d decimal integer e E f g G float i integer with base indicator n characters read so far o octal integer p pointer s string of non-whitespace characters u decimal integer x X hex integer [ string of characters in a set `e', `E', `f', `g' and `G' are identical, they all read either fixed point or scientific format, and either upper or lower case `e' for the exponent in scientific format. C99 style hex float format (`printf %a', *note Formatted Output Strings::) is always accepted for `mpf_t', but for the standard float types it will depend on the C library. `x' and `X' are identical, both accept both upper and lower case hexadecimal. `o', `u', `x' and `X' all read positive or negative values. For the standard C types these are described as "unsigned" conversions, but that merely affects certain overflow handling, negatives are still allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of Integers.). For GMP types there are no overflows, so `d' and `u' are identical. `Q' type reads the numerator and (optional) denominator as given. If the value might not be in canonical form then `mpq_canonicalize' must be called before using it in any calculations (*note Rational Number Functions::). `Qi' will read a base specification separately for the numerator and denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11' would be 16/17. `n' can be used with any of the types above, even the GMP types. `*' to suppress assignment is allowed, though in that case it would do nothing at all. Other conversions or types that might be accepted by the C library `scanf' cannot be used through `gmp_scanf'. Whitespace is read and discarded before a field, except for `c' and `[' conversions. For float conversions, the decimal point character (or string) expected is taken from the current locale settings on systems which provide `localeconv' (*note Locales and Internationalization: (libc)Locales.). The C library will normally do the same for standard float input. The format string is only interpreted as plain `char's, multibyte characters are not recognised. Perhaps this will change in the future.  File: gmp.info, Node: Formatted Input Functions, Next: C++ Formatted Input, Prev: Formatted Input Strings, Up: Formatted Input 11.2 Formatted Input Functions ============================== Each of the following functions is similar to the corresponding C library function. The plain `scanf' forms take a variable argument list. The `vscanf' forms take an argument pointer, see *note Variadic Functions: (libc)Variadic Functions, or `man 3 va_start'. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. No overlap is permitted between the FMT string and any of the results produced. -- Function: int gmp_scanf (const char *FMT, ...) -- Function: int gmp_vscanf (const char *FMT, va_list AP) Read from the standard input `stdin'. -- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...) -- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP) Read from the stream FP. -- Function: int gmp_sscanf (const char *S, const char *FMT, ...) -- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list AP) Read from a null-terminated string S. The return value from each of these functions is the same as the standard C99 `scanf', namely the number of fields successfully parsed and stored. `%n' fields and fields read but suppressed by `*' don't count towards the return value. If end of input (or a file error) is reached before a character for a field or a literal, and if no previous non-suppressed fields have matched, then the return value is `EOF' instead of 0. A whitespace character in the format string is only an optional match and doesn't induce an `EOF' in this fashion. Leading whitespace read and discarded for a field don't count as characters for that field. For the GMP types, input parsing follows C99 rules, namely one character of lookahead is used and characters are read while they continue to meet the format requirements. If this doesn't provide a complete number then the function terminates, with that field not stored nor counted towards the return value. For instance with `mpf_t' an input `1.23e-XYZ' would be read up to the `X' and that character pushed back since it's not a digit. The string `1.23e-' would then be considered invalid since an `e' must be followed by at least one digit. For the standard C types, in the current implementation GMP calls the C library `scanf' functions, which might have looser rules about what constitutes a valid input. Note that `gmp_sscanf' is the same as `gmp_fscanf' and only does one character of lookahead when parsing. Although clearly it could look at its entire input, it is deliberately made identical to `gmp_fscanf', the same way C99 `sscanf' is the same as `fscanf'.  File: gmp.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input 11.3 C++ Formatted Input ======================== The following functions are provided in `libgmpxx' (*note Headers and Libraries::), which is built only if C++ support is enabled (*note Build Options::). Prototypes are available from `'. -- Function: istream& operator>> (istream& STREAM, mpz_t ROP) Read ROP from STREAM, using its `ios' formatting settings. -- Function: istream& operator>> (istream& STREAM, mpq_t ROP) An integer like `123' will be read, or a fraction like `5/9'. No whitespace is allowed around the `/'. If the fraction is not in canonical form then `mpq_canonicalize' must be called (*note Rational Number Functions::) before operating on it. As per integer input, an `0' or `0x' base indicator is read when none of `ios::dec', `ios::oct' or `ios::hex' are set. This is done separately for numerator and denominator, so that for instance `0x10/11' is 16/11 and `0x10/0x11' is 16/17. -- Function: istream& operator>> (istream& STREAM, mpf_t ROP) Read ROP from STREAM, using its `ios' formatting settings. Hex or octal floats are not supported, but might be in the future, or perhaps it's best to accept only what the standard float `operator>>' does. Note that digit grouping specified by the `istream' locale is currently not accepted. Perhaps this will change in the future. These operators mean that GMP types can be read in the usual C++ way, for example, mpz_t z; ... cin >> z; But note that `istream' input (and `ostream' output, *note C++ Formatted Output::) is the only overloading available for the GMP types and that for instance using `+' with an `mpz_t' will have unpredictable results. For classes with overloading, see *note C++ Class Interface::.  File: gmp.info, Node: C++ Class Interface, Next: Custom Allocation, Prev: Formatted Input, Up: Top 12 C++ Class Interface ********************** This chapter describes the C++ class based interface to GMP. All GMP C language types and functions can be used in C++ programs, since `gmp.h' has `extern "C"' qualifiers, but the class interface offers overloaded functions and operators which may be more convenient. Due to the implementation of this interface, a reasonably recent C++ compiler is required, one supporting namespaces, partial specialization of templates and member templates. For GCC this means version 2.91 or later. *Everything described in this chapter is to be considered preliminary and might be subject to incompatible changes if some unforeseen difficulty reveals itself.* * Menu: * C++ Interface General:: * C++ Interface Integers:: * C++ Interface Rationals:: * C++ Interface Floats:: * C++ Interface Random Numbers:: * C++ Interface Limitations::  File: gmp.info, Node: C++ Interface General, Next: C++ Interface Integers, Prev: C++ Class Interface, Up: C++ Class Interface 12.1 C++ Interface General ========================== All the C++ classes and functions are available with #include Programs should be linked with the `libgmpxx' and `libgmp' libraries. For example, g++ mycxxprog.cc -lgmpxx -lgmp The classes defined are -- Class: mpz_class -- Class: mpq_class -- Class: mpf_class The standard operators and various standard functions are overloaded to allow arithmetic with these classes. For example, int main (void) { mpz_class a, b, c; a = 1234; b = "-5678"; c = a+b; cout << "sum is " << c << "\n"; cout << "absolute value is " << abs(c) << "\n"; return 0; } An important feature of the implementation is that an expression like `a=b+c' results in a single call to the corresponding `mpz_add', without using a temporary for the `b+c' part. Expressions which by their nature imply intermediate values, like `a=b*c+d*e', still use temporaries though. The classes can be freely intermixed in expressions, as can the classes and the standard types `long', `unsigned long' and `double'. Smaller types like `int' or `float' can also be intermixed, since C++ will promote them. Note that `bool' is not accepted directly, but must be explicitly cast to an `int' first. This is because C++ will automatically convert any pointer to a `bool', so if GMP accepted `bool' it would make all sorts of invalid class and pointer combinations compile but almost certainly not do anything sensible. Conversions back from the classes to standard C++ types aren't done automatically, instead member functions like `get_si' are provided (see the following sections for details). Also there are no automatic conversions from the classes to the corresponding GMP C types, instead a reference to the underlying C object can be obtained with the following functions, -- Function: mpz_t mpz_class::get_mpz_t () -- Function: mpq_t mpq_class::get_mpq_t () -- Function: mpf_t mpf_class::get_mpf_t () These can be used to call a C function which doesn't have a C++ class interface. For example to set `a' to the GCD of `b' and `c', mpz_class a, b, c; ... mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t()); In the other direction, a class can be initialized from the corresponding GMP C type, or assigned to if an explicit constructor is used. In both cases this makes a copy of the value, it doesn't create any sort of association. For example, mpz_t z; // ... init and calculate z ... mpz_class x(z); mpz_class y; y = mpz_class (z); There are no namespace setups in `gmpxx.h', all types and functions are simply put into the global namespace. This is what `gmp.h' has done in the past, and continues to do for compatibility. The extras provided by `gmpxx.h' follow GMP naming conventions and are unlikely to clash with anything.  File: gmp.info, Node: C++ Interface Integers, Next: C++ Interface Rationals, Prev: C++ Interface General, Up: C++ Class Interface 12.2 C++ Interface Integers =========================== -- Function: mpz_class::mpz_class (type N) Construct an `mpz_class'. All the standard C++ types may be used, except `long long' and `long double', and all the GMP C++ classes can be used, although conversions from `mpq_class' and `mpf_class' are `explicit'. Any necessary conversion follows the corresponding C function, for example `double' follows `mpz_set_d' (*note Assigning Integers::). -- Function: explicit mpz_class::mpz_class (mpz_t Z) Construct an `mpz_class' from an `mpz_t'. The value in Z is copied into the new `mpz_class', there won't be any permanent association between it and Z. -- Function: explicit mpz_class::mpz_class (const char *S, int BASE = 0) -- Function: explicit mpz_class::mpz_class (const string& S, int BASE = 0) Construct an `mpz_class' converted from a string using `mpz_set_str' (*note Assigning Integers::). If the string is not a valid integer, an `std::invalid_argument' exception is thrown. The same applies to `operator='. -- Function: mpz_class operator"" _mpz (const char *STR) With C++11 compilers, integers can be constructed with the syntax `123_mpz' which is equivalent to `mpz_class("123")'. -- Function: mpz_class operator/ (mpz_class A, mpz_class D) -- Function: mpz_class operator% (mpz_class A, mpz_class D) Divisions involving `mpz_class' round towards zero, as per the `mpz_tdiv_q' and `mpz_tdiv_r' functions (*note Integer Division::). This is the same as the C99 `/' and `%' operators. The `mpz_fdiv...' or `mpz_cdiv...' functions can always be called directly if desired. For example, mpz_class q, a, d; ... mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t()); -- Function: mpz_class abs (mpz_class OP) -- Function: int cmp (mpz_class OP1, type OP2) -- Function: int cmp (type OP1, mpz_class OP2) -- Function: bool mpz_class::fits_sint_p (void) -- Function: bool mpz_class::fits_slong_p (void) -- Function: bool mpz_class::fits_sshort_p (void) -- Function: bool mpz_class::fits_uint_p (void) -- Function: bool mpz_class::fits_ulong_p (void) -- Function: bool mpz_class::fits_ushort_p (void) -- Function: double mpz_class::get_d (void) -- Function: long mpz_class::get_si (void) -- Function: string mpz_class::get_str (int BASE = 10) -- Function: unsigned long mpz_class::get_ui (void) -- Function: int mpz_class::set_str (const char *STR, int BASE) -- Function: int mpz_class::set_str (const string& STR, int BASE) -- Function: int sgn (mpz_class OP) -- Function: mpz_class sqrt (mpz_class OP) -- Function: void mpz_class::swap (mpz_class& OP) -- Function: void swap (mpz_class& OP1, mpz_class& OP2) These functions provide a C++ class interface to the corresponding GMP C routines. `cmp' can be used with any of the classes or the standard C++ types, except `long long' and `long double'. Overloaded operators for combinations of `mpz_class' and `double' are provided for completeness, but it should be noted that if the given `double' is not an integer then the way any rounding is done is currently unspecified. The rounding might take place at the start, in the middle, or at the end of the operation, and it might change in the future. Conversions between `mpz_class' and `double', however, are defined to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'. And comparisons are always made exactly, as per `mpz_cmp_d'.  File: gmp.info, Node: C++ Interface Rationals, Next: C++ Interface Floats, Prev: C++ Interface Integers, Up: C++ Class Interface 12.3 C++ Interface Rationals ============================ In all the following constructors, if a fraction is given then it should be in canonical form, or if not then `mpq_class::canonicalize' called. -- Function: mpq_class::mpq_class (type OP) -- Function: mpq_class::mpq_class (integer NUM, integer DEN) Construct an `mpq_class'. The initial value can be a single value of any type (conversion from `mpf_class' is `explicit'), or a pair of integers (`mpz_class' or standard C++ integer types) representing a fraction, except that `long long' and `long double' are not supported. For example, mpq_class q (99); mpq_class q (1.75); mpq_class q (1, 3); -- Function: explicit mpq_class::mpq_class (mpq_t Q) Construct an `mpq_class' from an `mpq_t'. The value in Q is copied into the new `mpq_class', there won't be any permanent association between it and Q. -- Function: explicit mpq_class::mpq_class (const char *S, int BASE = 0) -- Function: explicit mpq_class::mpq_class (const string& S, int BASE = 0) Construct an `mpq_class' converted from a string using `mpq_set_str' (*note Initializing Rationals::). If the string is not a valid rational, an `std::invalid_argument' exception is thrown. The same applies to `operator='. -- Function: mpq_class operator"" _mpq (const char *STR) With C++11 compilers, integral rationals can be constructed with the syntax `123_mpq' which is equivalent to `mpq_class(123_mpz)'. Other rationals can be built as `-1_mpq/2' or `0xb_mpq/123456_mpz'. -- Function: void mpq_class::canonicalize () Put an `mpq_class' into canonical form, as per *note Rational Number Functions::. All arithmetic operators require their operands in canonical form, and will return results in canonical form. -- Function: mpq_class abs (mpq_class OP) -- Function: int cmp (mpq_class OP1, type OP2) -- Function: int cmp (type OP1, mpq_class OP2) -- Function: double mpq_class::get_d (void) -- Function: string mpq_class::get_str (int BASE = 10) -- Function: int mpq_class::set_str (const char *STR, int BASE) -- Function: int mpq_class::set_str (const string& STR, int BASE) -- Function: int sgn (mpq_class OP) -- Function: void mpq_class::swap (mpq_class& OP) -- Function: void swap (mpq_class& OP1, mpq_class& OP2) These functions provide a C++ class interface to the corresponding GMP C routines. `cmp' can be used with any of the classes or the standard C++ types, except `long long' and `long double'. -- Function: mpz_class& mpq_class::get_num () -- Function: mpz_class& mpq_class::get_den () Get a reference to an `mpz_class' which is the numerator or denominator of an `mpq_class'. This can be used both for read and write access. If the object returned is modified, it modifies the original `mpq_class'. If direct manipulation might produce a non-canonical value, then `mpq_class::canonicalize' must be called before further operations. -- Function: mpz_t mpq_class::get_num_mpz_t () -- Function: mpz_t mpq_class::get_den_mpz_t () Get a reference to the underlying `mpz_t' numerator or denominator of an `mpq_class'. This can be passed to C functions expecting an `mpz_t'. Any modifications made to the `mpz_t' will modify the original `mpq_class'. If direct manipulation might produce a non-canonical value, then `mpq_class::canonicalize' must be called before further operations. -- Function: istream& operator>> (istream& STREAM, mpq_class& ROP); Read ROP from STREAM, using its `ios' formatting settings, the same as `mpq_t operator>>' (*note C++ Formatted Input::). If the ROP read might not be in canonical form then `mpq_class::canonicalize' must be called.  File: gmp.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface 12.4 C++ Interface Floats ========================= When an expression requires the use of temporary intermediate `mpf_class' values, like `f=g*h+x*y', those temporaries will have the same precision as the destination `f'. Explicit constructors can be used if this doesn't suit. -- Function: mpf_class::mpf_class (type OP) -- Function: mpf_class::mpf_class (type OP, mp_bitcnt_t PREC) Construct an `mpf_class'. Any standard C++ type can be used, except `long long' and `long double', and any of the GMP C++ classes can be used. If PREC is given, the initial precision is that value, in bits. If PREC is not given, then the initial precision is determined by the type of OP given. An `mpz_class', `mpq_class', or C++ builtin type will give the default `mpf' precision (*note Initializing Floats::). An `mpf_class' or expression will give the precision of that value. The precision of a binary expression is the higher of the two operands. mpf_class f(1.5); // default precision mpf_class f(1.5, 500); // 500 bits (at least) mpf_class f(x); // precision of x mpf_class f(abs(x)); // precision of x mpf_class f(-g, 1000); // 1000 bits (at least) mpf_class f(x+y); // greater of precisions of x and y -- Function: explicit mpf_class::mpf_class (mpf_t F) -- Function: mpf_class::mpf_class (mpf_t F, mp_bitcnt_t PREC) Construct an `mpf_class' from an `mpf_t'. The value in F is copied into the new `mpf_class', there won't be any permanent association between it and F. If PREC is given, the initial precision is that value, in bits. If PREC is not given, then the initial precision is that of F. -- Function: explicit mpf_class::mpf_class (const char *S) -- Function: mpf_class::mpf_class (const char *S, mp_bitcnt_t PREC, int BASE = 0) -- Function: explicit mpf_class::mpf_class (const string& S) -- Function: mpf_class::mpf_class (const string& S, mp_bitcnt_t PREC, int BASE = 0) Construct an `mpf_class' converted from a string using `mpf_set_str' (*note Assigning Floats::). If PREC is given, the initial precision is that value, in bits. If not, the default `mpf' precision (*note Initializing Floats::) is used. If the string is not a valid float, an `std::invalid_argument' exception is thrown. The same applies to `operator='. -- Function: mpf_class operator"" _mpf (const char *STR) With C++11 compilers, floats can be constructed with the syntax `1.23e-1_mpf' which is equivalent to `mpf_class("1.23e-1")'. -- Function: mpf_class& mpf_class::operator= (type OP) Convert and store the given OP value to an `mpf_class' object. The same types are accepted as for the constructors above. Note that `operator=' only stores a new value, it doesn't copy or change the precision of the destination, instead the value is truncated if necessary. This is the same as `mpf_set' etc. Note in particular this means for `mpf_class' a copy constructor is not the same as a default constructor plus assignment. mpf_class x (y); // x created with precision of y mpf_class x; // x created with default precision x = y; // value truncated to that precision Applications using templated code may need to be careful about the assumptions the code makes in this area, when working with `mpf_class' values of various different or non-default precisions. For instance implementations of the standard `complex' template have been seen in both styles above, though of course `complex' is normally only actually specified for use with the builtin float types. -- Function: mpf_class abs (mpf_class OP) -- Function: mpf_class ceil (mpf_class OP) -- Function: int cmp (mpf_class OP1, type OP2) -- Function: int cmp (type OP1, mpf_class OP2) -- Function: bool mpf_class::fits_sint_p (void) -- Function: bool mpf_class::fits_slong_p (void) -- Function: bool mpf_class::fits_sshort_p (void) -- Function: bool mpf_class::fits_uint_p (void) -- Function: bool mpf_class::fits_ulong_p (void) -- Function: bool mpf_class::fits_ushort_p (void) -- Function: mpf_class floor (mpf_class OP) -- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2) -- Function: double mpf_class::get_d (void) -- Function: long mpf_class::get_si (void) -- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10, size_t DIGITS = 0) -- Function: unsigned long mpf_class::get_ui (void) -- Function: int mpf_class::set_str (const char *STR, int BASE) -- Function: int mpf_class::set_str (const string& STR, int BASE) -- Function: int sgn (mpf_class OP) -- Function: mpf_class sqrt (mpf_class OP) -- Function: void mpf_class::swap (mpf_class& OP) -- Function: void swap (mpf_class& OP1, mpf_class& OP2) -- Function: mpf_class trunc (mpf_class OP) These functions provide a C++ class interface to the corresponding GMP C routines. `cmp' can be used with any of the classes or the standard C++ types, except `long long' and `long double'. The accuracy provided by `hypot' is not currently guaranteed. -- Function: mp_bitcnt_t mpf_class::get_prec () -- Function: void mpf_class::set_prec (mp_bitcnt_t PREC) -- Function: void mpf_class::set_prec_raw (mp_bitcnt_t PREC) Get or set the current precision of an `mpf_class'. The restrictions described for `mpf_set_prec_raw' (*note Initializing Floats::) apply to `mpf_class::set_prec_raw'. Note in particular that the `mpf_class' must be restored to it's allocated precision before being destroyed. This must be done by application code, there's no automatic mechanism for it.  File: gmp.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface 12.5 C++ Interface Random Numbers ================================= -- Class: gmp_randclass The C++ class interface to the GMP random number functions uses `gmp_randclass' to hold an algorithm selection and current state, as per `gmp_randstate_t'. -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT) (gmp_randstate_t, ...), ...) Construct a `gmp_randclass', using a call to the given RANDINIT function (*note Random State Initialization::). The arguments expected are the same as RANDINIT, but with `mpz_class' instead of `mpz_t'. For example, gmp_randclass r1 (gmp_randinit_default); gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32); gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp); gmp_randclass r4 (gmp_randinit_mt); `gmp_randinit_lc_2exp_size' will fail if the size requested is too big, an `std::length_error' exception is thrown in that case. -- Function: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...) Construct a `gmp_randclass' using the same parameters as `gmp_randinit' (*note Random State Initialization::). This function is obsolete and the above RANDINIT style should be preferred. -- Function: void gmp_randclass::seed (unsigned long int S) -- Function: void gmp_randclass::seed (mpz_class S) Seed a random number generator. See *note Random Number Functions::, for how to choose a good seed. -- Function: mpz_class gmp_randclass::get_z_bits (mp_bitcnt_t BITS) -- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS) Generate a random integer with a specified number of bits. -- Function: mpz_class gmp_randclass::get_z_range (mpz_class N) Generate a random integer in the range 0 to N-1 inclusive. -- Function: mpf_class gmp_randclass::get_f () -- Function: mpf_class gmp_randclass::get_f (mp_bitcnt_t PREC) Generate a random float F in the range 0 <= F < 1. F will be to PREC bits precision, or if PREC is not given then to the precision of the destination. For example, gmp_randclass r; ... mpf_class f (0, 512); // 512 bits precision f = r.get_f(); // random number, 512 bits  File: gmp.info, Node: C++ Interface Limitations, Prev: C++ Interface Random Numbers, Up: C++ Class Interface 12.6 C++ Interface Limitations ============================== `mpq_class' and Templated Reading A generic piece of template code probably won't know that `mpq_class' requires a `canonicalize' call if inputs read with `operator>>' might be non-canonical. This can lead to incorrect results. `operator>>' behaves as it does for reasons of efficiency. A canonicalize can be quite time consuming on large operands, and is best avoided if it's not necessary. But this potential difficulty reduces the usefulness of `mpq_class'. Perhaps a mechanism to tell `operator>>' what to do will be adopted in the future, maybe a preprocessor define, a global flag, or an `ios' flag pressed into service. Or maybe, at the risk of inconsistency, the `mpq_class' `operator>>' could canonicalize and leave `mpq_t' `operator>>' not doing so, for use on those occasions when that's acceptable. Send feedback or alternate ideas to . Subclassing Subclassing the GMP C++ classes works, but is not currently recommended. Expressions involving subclasses resolve correctly (or seem to), but in normal C++ fashion the subclass doesn't inherit constructors and assignments. There's many of those in the GMP classes, and a good way to reestablish them in a subclass is not yet provided. Templated Expressions A subtle difficulty exists when using expressions together with application-defined template functions. Consider the following, with `T' intended to be some numeric type, template T fun (const T &, const T &); When used with, say, plain `mpz_class' variables, it works fine: `T' is resolved as `mpz_class'. mpz_class f(1), g(2); fun (f, g); // Good But when one of the arguments is an expression, it doesn't work. mpz_class f(1), g(2), h(3); fun (f, g+h); // Bad This is because `g+h' ends up being a certain expression template type internal to `gmpxx.h', which the C++ template resolution rules are unable to automatically convert to `mpz_class'. The workaround is simply to add an explicit cast. mpz_class f(1), g(2), h(3); fun (f, mpz_class(g+h)); // Good Similarly, within `fun' it may be necessary to cast an expression to type `T' when calling a templated `fun2'. template void fun (T f, T g) { fun2 (f, f+g); // Bad } template void fun (T f, T g) { fun2 (f, T(f+g)); // Good }  File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: C++ Class Interface, Up: Top 13 Custom Allocation ******************** By default GMP uses `malloc', `realloc' and `free' for memory allocation, and if they fail GMP prints a message to the standard error output and terminates the program. Alternate functions can be specified, to allocate memory in a different way or to have a different error action on running out of memory. -- Function: void mp_set_memory_functions ( void *(*ALLOC_FUNC_PTR) (size_t), void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t), void (*FREE_FUNC_PTR) (void *, size_t)) Replace the current allocation functions from the arguments. If an argument is `NULL', the corresponding default function is used. These functions will be used for all memory allocation done by GMP, apart from temporary space from `alloca' if that function is available and GMP is configured to use it (*note Build Options::). *Be sure to call `mp_set_memory_functions' only when there are no active GMP objects allocated using the previous memory functions! Usually that means calling it before any other GMP function.* The functions supplied should fit the following declarations: -- Function: void * allocate_function (size_t ALLOC_SIZE) Return a pointer to newly allocated space with at least ALLOC_SIZE bytes. -- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE, size_t NEW_SIZE) Resize a previously allocated block PTR of OLD_SIZE bytes to be NEW_SIZE bytes. The block may be moved if necessary or if desired, and in that case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to the new location. The return value is a pointer to the resized block, that being the new location if moved or just PTR if not. PTR is never `NULL', it's always a previously allocated block. NEW_SIZE may be bigger or smaller than OLD_SIZE. -- Function: void free_function (void *PTR, size_t SIZE) De-allocate the space pointed to by PTR. PTR is never `NULL', it's always a previously allocated block of SIZE bytes. A "byte" here means the unit used by the `sizeof' operator. The REALLOCATE_FUNCTION parameter OLD_SIZE and the FREE_FUNCTION parameter SIZE are passed for convenience, but of course they can be ignored if not needed by an implementation. The default functions using `malloc' and friends for instance don't use them. No error return is allowed from any of these functions, if they return then they must have performed the specified operation. In particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't return `NULL'. Getting a different fatal error action is a good use for custom allocation functions, for example giving a graphical dialog rather than the default print to `stderr'. How much is possible when genuinely out of memory is another question though. There's currently no defined way for the allocation functions to recover from an error such as out of memory, they must terminate program execution. A `longjmp' or throwing a C++ exception will have undefined results. This may change in the future. GMP may use allocated blocks to hold pointers to other allocated blocks. This will limit the assumptions a conservative garbage collection scheme can make. Since the default GMP allocation uses `malloc' and friends, those functions will be linked in even if the first thing a program does is an `mp_set_memory_functions'. It's necessary to change the GMP sources if this is a problem. -- Function: void mp_get_memory_functions ( void *(**ALLOC_FUNC_PTR) (size_t), void *(**REALLOC_FUNC_PTR) (void *, size_t, size_t), void (**FREE_FUNC_PTR) (void *, size_t)) Get the current allocation functions, storing function pointers to the locations given by the arguments. If an argument is `NULL', that function pointer is not stored. For example, to get just the current free function, void (*freefunc) (void *, size_t); mp_get_memory_functions (NULL, NULL, &freefunc);  File: gmp.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top 14 Language Bindings ******************** The following packages and projects offer access to GMP from languages other than C, though perhaps with varying levels of functionality and efficiency. C++ * GMP C++ class interface, *note C++ Class Interface:: Straightforward interface, expression templates to eliminate temporaries. * ALP `http://www-sop.inria.fr/saga/logiciels/ALP/' Linear algebra and polynomials using templates. * Arithmos `http://cant.ua.ac.be/old/arithmos/' Rationals with infinities and square roots. * CLN `http://www.ginac.de/CLN/' High level classes for arithmetic. * LiDIA `http://www.cdc.informatik.tu-darmstadt.de/TI/LiDIA/' A C++ library for computational number theory. * Linbox `http://www.linalg.org/' Sparse vectors and matrices. * NTL `http://www.shoup.net/ntl/' A C++ number theory library. Eiffel * Eiffelroom `http://www.eiffelroom.org/node/442' Fortran * Omni F77 `http://phase.hpcc.jp/Omni/home.html' Arbitrary precision floats. Haskell * Glasgow Haskell Compiler `http://www.haskell.org/ghc/' Java * Kaffe `http://www.kaffe.org/' * Kissme `http://kissme.sourceforge.net/' Lisp * GNU Common Lisp `http://www.gnu.org/software/gcl/gcl.html' * Librep `http://librep.sourceforge.net/' * XEmacs (21.5.18 beta and up) `http://www.xemacs.org' Optional big integers, rationals and floats using GMP. M4 * GNU m4 betas `http://www.seindal.dk/rene/gnu/' Optionally provides an arbitrary precision `mpeval'. ML * MLton compiler `http://mlton.org/' Objective Caml * MLGMP `http://www.di.ens.fr/~monniaux/programmes.html.en' * Numerix `http://pauillac.inria.fr/~quercia/' Optionally using GMP. Oz * Mozart `http://www.mozart-oz.org/' Pascal * GNU Pascal Compiler `http://www.gnu-pascal.de/' GMP unit. * Numerix `http://pauillac.inria.fr/~quercia/' For Free Pascal, optionally using GMP. Perl * GMP module, see `demos/perl' in the GMP sources (*note Demonstration Programs::). * Math::GMP `http://www.cpan.org/' Compatible with Math::BigInt, but not as many functions as the GMP module above. * Math::BigInt::GMP `http://www.cpan.org/' Plug Math::GMP into normal Math::BigInt operations. Pike * mpz module in the standard distribution, `http://pike.ida.liu.se/' Prolog * SWI Prolog `http://www.swi-prolog.org/' Arbitrary precision floats. Python * GMPY `http://code.google.com/p/gmpy/' Ruby * http://rubygems.org/gems/gmp Scheme * GNU Guile (upcoming 1.8) `http://www.gnu.org/software/guile/guile.html' * RScheme `http://www.rscheme.org/' * STklos `http://www.stklos.org/' Smalltalk * GNU Smalltalk `http://www.smalltalk.org/versions/GNUSmalltalk.html' Other * Axiom `http://savannah.nongnu.org/projects/axiom' Computer algebra using GCL. * DrGenius `http://drgenius.seul.org/' Geometry system and mathematical programming language. * GiNaC `http://www.ginac.de/' C++ computer algebra using CLN. * GOO `http://www.googoogaga.org/' Dynamic object oriented language. * Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html' Macsyma computer algebra using GCL. * Q `http://q-lang.sourceforge.net/' Equational programming system. * Regina `http://regina.sourceforge.net/' Topological calculator. * Yacas `yacas.sourceforge.net' Yet another computer algebra system.  File: gmp.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top 15 Algorithms ************* This chapter is an introduction to some of the algorithms used for various GMP operations. The code is likely to be hard to understand without knowing something about the algorithms. Some GMP internals are mentioned, but applications that expect to be compatible with future GMP releases should take care to use only the documented functions. * Menu: * Multiplication Algorithms:: * Division Algorithms:: * Greatest Common Divisor Algorithms:: * Powering Algorithms:: * Root Extraction Algorithms:: * Radix Conversion Algorithms:: * Other Algorithms:: * Assembly Coding::  File: gmp.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms 15.1 Multiplication =================== NxN limb multiplications and squares are done using one of seven algorithms, as the size N increases. Algorithm Threshold Basecase (none) Karatsuba `MUL_TOOM22_THRESHOLD' Toom-3 `MUL_TOOM33_THRESHOLD' Toom-4 `MUL_TOOM44_THRESHOLD' Toom-6.5 `MUL_TOOM6H_THRESHOLD' Toom-8.5 `MUL_TOOM8H_THRESHOLD' FFT `MUL_FFT_THRESHOLD' Similarly for squaring, with the `SQR' thresholds. NxM multiplications of operands with different sizes above `MUL_TOOM22_THRESHOLD' are currently done by special Toom-inspired algorithms or directly with FFT, depending on operand size (*note Unbalanced Multiplication::). * Menu: * Basecase Multiplication:: * Karatsuba Multiplication:: * Toom 3-Way Multiplication:: * Toom 4-Way Multiplication:: * Higher degree Toom'n'half:: * FFT Multiplication:: * Other Multiplication:: * Unbalanced Multiplication::  File: gmp.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms 15.1.1 Basecase Multiplication ------------------------------ Basecase NxM multiplication is a straightforward rectangular set of cross-products, the same as long multiplication done by hand and for that reason sometimes known as the schoolbook or grammar school method. This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M (*note References::), and the `mpn/generic/mul_basecase.c' code. Assembly implementations of `mpn_mul_basecase' are essentially the same as the generic C code, but have all the usual assembly tricks and obscurities introduced for speed. A square can be done in roughly half the time of a multiply, by using the fact that the cross products above and below the diagonal are the same. A triangle of products below the diagonal is formed, doubled (left shift by one bit), and then the products on the diagonal added. This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembly implementations take essentially the same approach. u0 u1 u2 u3 u4 +---+---+---+---+---+ u0 | d | | | | | +---+---+---+---+---+ u1 | | d | | | | +---+---+---+---+---+ u2 | | | d | | | +---+---+---+---+---+ u3 | | | | d | | +---+---+---+---+---+ u4 | | | | | d | +---+---+---+---+---+ In practice squaring isn't a full 2x faster than multiplying, it's usually around 1.5x. Less than 1.5x probably indicates `mpn_sqr_basecase' wants improving on that CPU. On some CPUs `mpn_mul_basecase' can be faster than the generic C `mpn_sqr_basecase' on some small sizes. `SQR_BASECASE_THRESHOLD' is the size at which to use `mpn_sqr_basecase', this will be zero if that routine should be used always.  File: gmp.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms 15.1.2 Karatsuba Multiplication ------------------------------- The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks. A brief description is given here. The inputs x and y are treated as each split into two parts of equal length (or the most significant part one limb shorter if N is odd). high low +----------+----------+ | x1 | x0 | +----------+----------+ +----------+----------+ | y1 | y0 | +----------+----------+ Let b be the power of 2 where the split occurs, i.e. if x0 is k limbs (y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and y=y1*b+y0, and the following holds, x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0 This formula means doing only three multiplies of (N/2)x(N/2) limbs, whereas a basecase multiply of NxN limbs is equivalent to four multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the positions where the three products must be added. high low +--------+--------+ +--------+--------+ | x1*y1 | | x0*y0 | +--------+--------+ +--------+--------+ +--------+--------+ add | x1*y1 | +--------+--------+ +--------+--------+ add | x0*y0 | +--------+--------+ +--------+--------+ sub | (x1-x0)*(y1-y0) | +--------+--------+ The term (x1-x0)*(y1-y0) is best calculated as an absolute value, and the sign used to choose to add or subtract. Notice the sum high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb additions, rather than 6*k, but in GMP extra function call overheads outweigh the saving. Squaring is similar to multiplying, but with x=y the formula reduces to an equivalent with three squares, x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2 The final result is accumulated from those three squares the same way as for the three multiplies above. The middle term (x1-x0)^2 is now always positive. A similar formula for both multiplying and squaring can be constructed with a middle term (x1+x0)*(y1+y0). But those sums can exceed k limbs, leading to more carry handling and additions than the form above. Karatsuba multiplication is asymptotically an O(N^1.585) algorithm, the exponent being log(3)/log(2), representing 3 multiplies each 1/2 the size of the inputs. This is a big improvement over the basecase multiply at O(N^2) and the advantage soon overcomes the extra additions Karatsuba performs. `MUL_TOOM22_THRESHOLD' can be as little as 10 limbs. The `SQR' threshold is usually about twice the `MUL'. The basecase algorithm will take a time of the form M(N) = a*N^2 + b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4 for a means per-crossproduct speedups in the basecase code will increase the threshold since they benefit M(N) more than K(N). And conversely the 3/2 for b means linear style speedups of b will increase the threshold since they benefit K(N) more than M(N). The latter can be seen for instance when adding an optimized `mpn_sqr_diagonal' to `mpn_sqr_basecase'. Of course all speedups reduce total time, and in that sense the algorithm thresholds are merely of academic interest.  File: gmp.info, Node: Toom 3-Way Multiplication, Next: Toom 4-Way Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms 15.1.3 Toom 3-Way Multiplication -------------------------------- The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom and FFT algorithms. A description of Toom can be found in Knuth section 4.3.3, with an example 3-way calculation after Theorem A. The 3-way form used in GMP is described here. The operands are each considered split into 3 pieces of equal length (or the most significant part 1 or 2 limbs shorter than the other two). high low +----------+----------+----------+ | x2 | x1 | x0 | +----------+----------+----------+ +----------+----------+----------+ | y2 | y1 | y0 | +----------+----------+----------+ These parts are treated as the coefficients of two polynomials X(t) = x2*t^2 + x1*t + x0 Y(t) = y2*t^2 + y1*t + y0 Let b equal the power of 2 which is the size of the x0, x1, y0 and y1 pieces, i.e. if they're k limbs each then b=2^(k*mp_bits_per_limb). With this x=X(b) and y=Y(b). Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0 The w[i] are going to be determined, and when they are they'll give the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The coefficients will be roughly b^2 each, and the final W(b) will be an addition like, high low +-------+-------+ | w4 | +-------+-------+ +--------+-------+ | w3 | +--------+-------+ +--------+-------+ | w2 | +--------+-------+ +--------+-------+ | w1 | +--------+-------+ +-------+-------+ | w0 | +-------+-------+ The w[i] coefficients could be formed by a simple set of cross products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but this would need all nine x[i]*y[j] for i,j=0,1,2, and would be equivalent merely to a basecase multiply. Instead the following approach is used. X(t) and Y(t) are evaluated and multiplied at 5 points, giving values of W(t) at those points. In GMP the following points are used, Point Value t=0 x0 * y0, which gives w0 immediately t=1 (x2+x1+x0) * (y2+y1+y0) t=-1 (x2-x1+x0) * (y2-y1+y0) t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0) t=inf x2 * y2, which gives w4 immediately At t=-1 the values can be negative and that's handled using the absolute values and tracking the sign separately. At t=inf the value is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but it's much easier to think of as simply x2*y2 giving w4 immediately (much like x0*y0 at t=0 gives w0 immediately). Each of the points substituted into W(t)=w4*t^4+...+w0 gives a linear combination of the w[i] coefficients, and the value of those combinations has just been calculated. W(0) = w0 W(1) = w4 + w3 + w2 + w1 + w0 W(-1) = w4 - w3 + w2 - w1 + w0 W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0 W(inf) = w4 This is a set of five equations in five unknowns, and some elementary linear algebra quickly isolates each w[i]. This involves adding or subtracting one W(t) value from another, and a couple of divisions by powers of 2 and one division by 3, the latter using the special `mpn_divexact_by3' (*note Exact Division::). The conversion of W(t) values to the coefficients is interpolation. A polynomial of degree 4 like W(t) is uniquely determined by values known at 5 different points. The points are arbitrary and can be chosen to make the linear equations come out with a convenient set of steps for quickly isolating the w[i]. Squaring follows the same procedure as multiplication, but there's only one X(t) and it's evaluated at the 5 points, and those values squared to give values of W(t). The interpolation is then identical, and in fact the same `toom_interpolate_5pts' subroutine is used for both squaring and multiplying. Toom-3 is asymptotically O(N^1.465), the exponent being log(5)/log(3), representing 5 recursive multiplies of 1/3 the original size each. This is an improvement over Karatsuba at O(N^1.585), though Toom does more work in the evaluation and interpolation and so it only realizes its advantage above a certain size. Near the crossover between Toom-3 and Karatsuba there's generally a range of sizes where the difference between the two is small. `MUL_TOOM33_THRESHOLD' is a somewhat arbitrary point in that range and successive runs of the tune program can give different values due to small variations in measuring. A graph of time versus size for the two shows the effect, see `tune/README'. At the fairly small sizes where the Toom-3 thresholds occur it's worth remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be expected to make accurate predictions, due of course to the big influence of all sorts of overheads, and the fact that only a few recursions of each are being performed. Even at large sizes there's a good chance machine dependent effects like cache architecture will mean actual performance deviates from what might be predicted. The formula given for the Karatsuba algorithm (*note Karatsuba Multiplication::) has an equivalent for Toom-3 involving only five multiplies, but this would be complicated and unenlightening. An alternate view of Toom-3 can be found in Zuras (*note References::), using a vector to represent the x and y splits and a matrix multiplication for the evaluation and interpolation stages. The matrix inverses are not meant to be actually used, and they have elements with values much greater than in fact arise in the interpolation steps. The diagram shown for the 3-way is attractive, but again doesn't have to be implemented that way and for example with a bit of rearrangement just one division by 6 can be done.  File: gmp.info, Node: Toom 4-Way Multiplication, Next: Higher degree Toom'n'half, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms 15.1.4 Toom 4-Way Multiplication -------------------------------- Karatsuba and Toom-3 split the operands into 2 and 3 coefficients, respectively. Toom-4 analogously splits the operands into 4 coefficients. Using the notation from the section on Toom-3 multiplication, we form two polynomials: X(t) = x3*t^3 + x2*t^2 + x1*t + x0 Y(t) = y3*t^3 + y2*t^2 + y1*t + y0 X(t) and Y(t) are evaluated and multiplied at 7 points, giving values of W(t) at those points. In GMP the following points are used, Point Value t=0 x0 * y0, which gives w0 immediately t=1/2 (x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0) t=-1/2 (-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0) t=1 (x3+x2+x1+x0) * (y3+y2+y1+y0) t=-1 (-x3+x2-x1+x0) * (-y3+y2-y1+y0) t=2 (8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0) t=inf x3 * y3, which gives w6 immediately The number of additions and subtractions for Toom-4 is much larger than for Toom-3. But several subexpressions occur multiple times, for example x2+x0, occurs for both t=1 and t=-1. Toom-4 is asymptotically O(N^1.404), the exponent being log(7)/log(4), representing 7 recursive multiplies of 1/4 the original size each.  File: gmp.info, Node: Higher degree Toom'n'half, Next: FFT Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms 15.1.5 Higher degree Toom'n'half -------------------------------- The Toom algorithms described above (*note Toom 3-Way Multiplication::, *note Toom 4-Way Multiplication::) generalizes to split into an arbitrary number of pieces. In general a split of two equally long operands into r pieces leads to evaluations and pointwise multiplications done at 2*r-1 points. To fully exploit symmetries it would be better to have a multiple of 4 points, that's why for higher degree Toom'n'half is used. Toom'n'half means that the existence of one more piece is considered for a single operand. It can be virtual, i.e. zero, or real, when the two operand are not exactly balanced. By chosing an even r, Toom-r+1/2 requires 2r points, a multiple of four. The four-plets of points inlcude 0, inf, +1, -1 and +-2^i, +-2^-i . Each of them giving shortcuts for the evaluation phase and for some steps in the interpolation phase. Further tricks are used to reduce the memory footprint of the whole multiplication algorithm to a memory buffer equanl in size to the result of the product. Current GMP uses both Toom-6'n'half and Toom-8'n'half.  File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Higher degree Toom'n'half, Up: Multiplication Algorithms 15.1.6 FFT Multiplication ------------------------- At large to very large sizes a Fermat style FFT multiplication is used, following Schönhage and Strassen (*note References::). Descriptions of FFTs in various forms can be found in many textbooks, for instance Knuth section 4.3.3 part C or Lipson chapter IX. A brief description of the form used in GMP is given here. The multiplication done is x*y mod 2^N+1, for a given N. A full product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x and y with high zero limbs. The modular product is the native form for the algorithm, so padding to get a full product is unavoidable. The algorithm follows a split, evaluate, pointwise multiply, interpolate and combine similar to that described above for Karatsuba and Toom-3. A k parameter controls the split, with an FFT-k splitting into 2^k pieces of M=N/2^k bits each. N must be a multiple of (2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding bit shifts in the split and combine stages. The evaluations, pointwise multiplications, and interpolation, are all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of 2^k and of `mp_bits_per_limb'. The results of interpolation will be the following negacyclic convolution of the input pieces, and the choice of N' ensures these sums aren't truncated. --- \ b w[n] = / (-1) * x[i] * y[j] --- i+j==b*2^k+n b=0,1 The points used for the evaluation are g^i for i=0 to 2^k-1 where g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces necessary cancellations at the interpolation stage, and it's also a power of 2 so the fast Fourier transforms used for the evaluation and interpolation do only shifts, adds and negations. The pointwise multiplications are done modulo 2^N'+1 and either recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or basecase), whichever is optimal at the size N'. The interpolation is an inverse fast Fourier transform. The resulting set of sums of x[i]*y[j] are added at appropriate offsets to give the final result. Squaring is the same, but x is the only input so it's one transform at the evaluate stage and the pointwise multiplies are squares. The interpolation is the same. For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm, the exponent representing 2^k recursed modular multiplies each 1/2^(k-1) the size of the original. Each successive k is an asymptotic improvement, but overheads mean each is only faster at bigger and bigger sizes. In the code, `MUL_FFT_TABLE' and `SQR_FFT_TABLE' are the thresholds where each k is used. Each new k effectively swaps some multiplying for some shifts, adds and overheads. A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply plus a subtraction, so an FFT and Toom-3 etc can be compared directly. A k=4 FFT at O(N^1.333) can be expected to be the first faster than Toom-3 at O(N^1.465). In practice this is what's found, with `MUL_FFT_MODF_THRESHOLD' and `SQR_FFT_MODF_THRESHOLD' being between 300 and 1000 limbs, depending on the CPU. So far it's been found that only very large FFTs recurse into pointwise multiplies above these sizes. When an FFT is to give a full product, the change of N to 2N doesn't alter the theoretical complexity for a given k, but for the purposes of considering where an FFT might be first used it can be assumed that the FFT is recursing into a normal multiply and that on that basis it's doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs, making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the first FFT faster than Toom-3. In practice `MUL_FFT_THRESHOLD' and `SQR_FFT_THRESHOLD' have been found to be in the k=8 range, somewhere between 3000 and 10000 limbs. The way N is split into 2^k pieces and then 2M+k+3 is rounded up to a multiple of 2^k and `mp_bits_per_limb' means that when 2^k>=mp_bits_per_limb the effective N is a multiple of 2^(2k-1) bits. The +k+3 means some values of N just under such a multiple will be rounded to the next. The complexity calculations above assume that a favourable size is used, meaning one which isn't padded through rounding, and it's also assumed that the extra +k+3 bits are negligible at typical FFT sizes. The practical effect of the 2^(2k-1) constraint is to introduce a step-effect into measured speeds. For example k=8 will round N up to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb groups of sizes for which `mpn_mul_n' runs at the same speed. Or for k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice it's been found each k is used at quite small multiples of its size constraint and so the step effect is quite noticeable in a time versus size graph. The threshold determinations currently measure at the mid-points of size steps, but this is sub-optimal since at the start of a new step it can happen that it's better to go back to the previous k for a while. Something more sophisticated for `MUL_FFT_TABLE' and `SQR_FFT_TABLE' will be needed.  File: gmp.info, Node: Other Multiplication, Next: Unbalanced Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms 15.1.7 Other Multiplication --------------------------- The Toom algorithms described above (*note Toom 3-Way Multiplication::, *note Toom 4-Way Multiplication::) generalizes to split into an arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C. This is not currently used. The notes here are merely for interest. In general a split into r+1 pieces is made, and evaluations and pointwise multiplications done at 2*r+1 points. A 4-way split does 7 pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise multiplications count towards big-O complexity, but the time spent in the evaluate and interpolate stages grows with r and has a significant practical impact, with the asymptotic advantage of each r realized only at bigger and bigger sizes. The overheads grow as O(N*r), whereas in an r=2^k FFT they grow only as O(N*log(r)). Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4 uses -r,...,0,...,r and the latter saves some small multiplies in the evaluate stage (or rather trades them for additions), and has a further saving of nearly half the interpolate steps. The idea is to separate odd and even final coefficients and then perform algorithm C steps C7 and C8 on them separately. The divisors at step C7 become j^2 and the multipliers at C8 become 2*t*j-j^2. Splitting odd and even parts through positive and negative points can be thought of as using -1 as a square root of unity. If a 4th root of unity was available then a further split and speedup would be possible, but no such root exists for plain integers. Going to complex integers with i=sqrt(-1) doesn't help, essentially because in Cartesian form it takes three real multiplies to do a complex multiply. The existence of 2^k'th roots of unity in a suitable ring or field lets the fast Fourier transform keep splitting and get to O(N*log(r)). Floating point FFTs use complex numbers approximating Nth roots of unity. Some processors have special support for such FFTs. But these are not used in GMP since it's very difficult to guarantee an exact result (to some number of bits). An occasional difference of 1 in the last bit might not matter to a typical signal processing algorithm, but is of course of vital importance to GMP.  File: gmp.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms 15.1.8 Unbalanced Multiplication -------------------------------- Multiplication of operands with different sizes, both below `MUL_TOOM22_THRESHOLD' are done with plain schoolbook multiplication (*note Basecase Multiplication::). For really large operands, we invoke FFT directly. For operands between these sizes, we use Toom inspired algorithms suggested by Alberto Zanoni and Marco Bodrato. The idea is to split the operands into polynomials of different degree. GMP currently splits the smaller operand onto 2 coefficients, i.e., a polynomial of degree 1, but the larger operand can be split into 2, 3, or 4 coefficients, i.e., a polynomial of degree 1 to 3.  File: gmp.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms 15.2 Division Algorithms ======================== * Menu: * Single Limb Division:: * Basecase Division:: * Divide and Conquer Division:: * Block-Wise Barrett Division:: * Exact Division:: * Exact Remainder:: * Small Quotient Division::  File: gmp.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms 15.2.1 Single Limb Division --------------------------- Nx1 division is implemented using repeated 2x1 divisions from high to low, either with a hardware divide instruction or a multiplication by inverse, whichever is best on a given CPU. The multiply by inverse follows "Improved division by invariant integers" by Möller and Granlund (*note References::) and is implemented as `udiv_qrnnd_preinv' in `gmp-impl.h'. The idea is to have a fixed-point approximation to 1/d (see `invert_limb') and then multiply by the high limb (plus one bit) of the dividend to get a quotient q. With d normalized (high bit set), q is no more than 1 too small. Subtracting q*d from the dividend gives a remainder, and reveals whether q or q-1 is correct. The result is a division done with two multiplications and four or five arithmetic operations. On CPUs with low latency multipliers this can be much faster than a hardware divide, though the cost of calculating the inverse at the start may mean it's only better on inputs bigger than say 4 or 5 limbs. When a divisor must be normalized, either for the generic C `__udiv_qrnnd_c' or the multiply by inverse, the division performed is actually a*2^k by d*2^k where a is the dividend and k is the power necessary to have the high bit of d*2^k set. The bit shifts for the dividend are usually accomplished "on the fly" meaning by extracting the appropriate bits at each step. Done this way the quotient limbs come out aligned ready to store. When only the remainder is wanted, an alternative is to take the dividend limbs unshifted and calculate r = a mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can help on CPUs with poor bit shifts or few registers. The multiply by inverse can be done two limbs at a time. The calculation is basically the same, but the inverse is two limbs and the divisor treated as if padded with a low zero limb. This means more work, since the inverse will need a 2x2 multiply, but the four 1x1s to do that are independent and can therefore be done partly or wholly in parallel. Likewise for a 2x1 calculating q*d. The net effect is to process two limbs with roughly the same two multiplies worth of latency that one limb at a time gives. This extends to 3 or 4 limbs at a time, though the extra work to apply the inverse will almost certainly soon reach the limits of multiplier throughput. A similar approach in reverse can be taken to process just half a limb at a time if the divisor is only a half limb. In this case the 1x1 multiply for the inverse effectively becomes two (1/2)x1 for each limb, which can be a saving on CPUs with a fast half limb multiply, or in fact if the only multiply is a half limb, and especially if it's not pipelined.  File: gmp.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms 15.2.2 Basecase Division ------------------------ Basecase NxM division is like long division done by hand, but in base 2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D, and `mpn/generic/sb_divrem_mn.c'. Briefly stated, while the dividend remains larger than the divisor, a high quotient limb is formed and the Nx1 product q*d subtracted at the top end of the dividend. With a normalized divisor (most significant bit set), each quotient limb can be formed with a 2x1 division and a 1x1 multiplication plus some subtractions. The 2x1 division is by the high limb of the divisor and is done either with a hardware divide or a multiply by inverse (the same as in *note Single Limb Division::) whichever is faster. Such a quotient is sometimes one too big, requiring an addback of the divisor, but that happens rarely. With Q=N-M being the number of quotient limbs, this is an O(Q*M) algorithm and will run at a speed similar to a basecase QxM multiplication, differing in fact only in the extra multiply and divide for each of the Q quotient limbs.  File: gmp.info, Node: Divide and Conquer Division, Next: Block-Wise Barrett Division, Prev: Basecase Division, Up: Division Algorithms 15.2.3 Divide and Conquer Division ---------------------------------- For divisors larger than `DC_DIV_QR_THRESHOLD', division is done by dividing. Or to be precise by a recursive divide and conquer algorithm based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler (*note References::). The algorithm consists essentially of recognising that a 2NxN division can be done with the basecase division algorithm (*note Basecase Division::), but using N/2 limbs as a base, not just a single limb. This way the multiplications that arise are (N/2)x(N/2) and can take advantage of Karatsuba and higher multiplication algorithms (*note Multiplication Algorithms::). The two "digits" of the quotient are formed by recursive Nx(N/2) divisions. If the (N/2)x(N/2) multiplies are done with a basecase multiplication then the work is about the same as a basecase division, but with more function call overheads and with some subtractions separated from the multiplies. These overheads mean that it's only when N/2 is above `MUL_TOOM22_THRESHOLD' that divide and conquer is of use. `DC_DIV_QR_THRESHOLD' is based on the divisor size N, so it will be somewhere above twice `MUL_TOOM22_THRESHOLD', but how much above depends on the CPU. An optimized `mpn_mul_basecase' can lower `DC_DIV_QR_THRESHOLD' a little by offering a ready-made advantage over repeated `mpn_submul_1' calls. Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is the time for an NxN multiplication done with FFTs. The actual time is a sum over multiplications of the recursed sizes, as can be seen near the end of section 2.2 of Burnikel and Ziegler. For example, within the Toom-3 range, divide and conquer is 2.63*M(N). With higher algorithms the M(N) term improves and the multiplier tends to log(N). In practice, at moderate to large sizes, a 2NxN division is about 2 to 4 times slower than an NxN multiplication.  File: gmp.info, Node: Block-Wise Barrett Division, Next: Exact Division, Prev: Divide and Conquer Division, Up: Division Algorithms 15.2.4 Block-Wise Barrett Division ---------------------------------- For the largest divisions, a block-wise Barrett division algorithm is used. Here, the divisor is inverted to a precision determined by the relative size of the dividend and divisor. Blocks of quotient limbs are then generated by multiplying blocks from the dividend by the inverse. Our block-wise algorithm computes a smaller inverse than in the plain Barrett algorithm. For a 2n/n division, the inverse will be just ceil(n/2) limbs.  File: gmp.info, Node: Exact Division, Next: Exact Remainder, Prev: Block-Wise Barrett Division, Up: Division Algorithms 15.2.5 Exact Division --------------------- A so-called exact division is when the dividend is known to be an exact multiple of the divisor. Jebelean's exact division algorithm uses this knowledge to make some significant optimizations (*note References::). The idea can be illustrated in decimal for example with 368154 divided by 543. Because the low digit of the dividend is 4, the low digit of the quotient must be 8. This is arrived at from 4*7 mod 10, using the fact 7 is the modular inverse of 3 (the low digit of the divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from the dividend leaving 363810. Notice the low digit has become zero. The procedure is repeated at the second digit, with the next quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving 325800. And finally at the third digit with quotient digit 6 (8*7 mod 10), subtracting 6*543=3258 leaving 0. So the quotient is 678. Notice however that the multiplies and subtractions don't need to extend past the low three digits of the dividend, since that's enough to determine the three quotient digits. For the last quotient digit no subtraction is needed at all. On a 2NxN division like this one, only about half the work of a normal basecase division is necessary. For an NxM exact division producing Q=N-M quotient limbs, the saving over a normal basecase division is in two parts. Firstly, each of the Q quotient limbs needs only one multiply, not a 2x1 divide and multiply. Secondly, the crossproducts are reduced when Q>M to Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2. Notice the savings are complementary. If Q is big then many divisions are saved, or if Q is small then the crossproducts reduce to a small number. The modular inverse used is calculated efficiently by `binvert_limb' in `gmp-impl.h'. This does four multiplies for a 32-bit limb, or six for a 64-bit limb. `tune/modlinv.c' has some alternate implementations that might suit processors better at bit twiddling than multiplying. The sub-quadratic exact division described by Jebelean in "Exact Division with Karatsuba Complexity" is not currently implemented. It uses a rearrangement similar to the divide and conquer for normal division (*note Divide and Conquer Division::), but operating from low to high. A further possibility not currently implemented is "Bidirectional Exact Integer Division" by Krandick and Jebelean which forms quotient limbs from both the high and low ends of the dividend, and can halve once more the number of crossproducts needed in a 2NxN division. A special case exact division by 3 exists in `mpn_divexact_by3', supporting Toom-3 multiplication and `mpq' canonicalizations. It forms quotient digits with a multiply by the modular inverse of 3 (which is `0xAA..AAB') and uses two comparisons to determine a borrow for the next limb. The multiplications don't need to be on the dependent chain, as long as the effect of the borrows is applied, which can help chips with pipelined multipliers.  File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms 15.2.6 Exact Remainder ---------------------- If the exact division algorithm is done with a full subtraction at each stage and the dividend isn't a multiple of the divisor, then low zero limbs are produced but with a remainder in the high limbs. For dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this remainder r is of the form a = q*d + r*b^n n represents the number of zero limbs produced by the subtractions, that being the number of limbs produced for q. r will be in the range 0<=rb*r+u2 condition appropriately relaxed.  File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms 15.3 Greatest Common Divisor ============================ * Menu: * Binary GCD:: * Lehmer's Algorithm:: * Subquadratic GCD:: * Extended GCD:: * Jacobi Symbol::  File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms 15.3.1 Binary GCD ----------------- At small sizes GMP uses an O(N^2) binary style GCD. This is described in many textbooks, for example Knuth section 4.5.2 algorithm B. It simply consists of successively reducing odd operands a and b using a,b = abs(a-b),min(a,b) strip factors of 2 from a The Euclidean GCD algorithm, as per Knuth algorithms E and A, repeatedly computes the quotient q = floor(a/b) and replaces a,b by v, u - q v. The binary algorithm has so far been found to be faster than the Euclidean algorithm everywhere. One reason the binary method does well is that the implied quotient at each step is usually small, so often only one or two subtractions are needed to get the same effect as a division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth section 4.5.3 Theorem E. When the implied quotient is large, meaning b is much smaller than a, then a division is worthwhile. This is the basis for the initial a mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1 and 1x1 cases). But after that initial reduction, big quotients occur too rarely to make it worth checking for them. The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as described above. For two N-bit operands, the algorithm takes about 0.68 iterations per bit. For optimum performance some attention needs to be paid to the way the factors of 2 are stripped from a. Firstly it may be noted that in twos complement the number of low zero bits on a-b is the same as b-a, so counting or testing can begin on a-b without waiting for abs(a-b) to be determined. A loop stripping low zero bits tends not to branch predict well, since the condition is data dependent. But on average there's only a few low zeros, so an option is to strip one or two bits arithmetically then loop for more (as done for AMD K6). Or use a lookup table to get a count for several bits then loop for more (as done for AMD K7). An alternative approach is to keep just one of a or b odd and iterate a,b = abs(a-b), min(a,b) a = a/2 if even b = b/2 if even This requires about 1.25 iterations per bit, but stripping of a single bit at each step avoids any branching. Repeating the bit strip reduces to about 0.9 iterations per bit, which may be a worthwhile tradeoff. Generally with the above approaches a speed of perhaps 6 cycles per bit can be achieved, which is still not terribly fast with for instance a 64-bit GCD taking nearly 400 cycles. It's this sort of time which means it's not usually advantageous to combine a set of divisibility tests into a GCD. Currently, the binary algorithm is used for GCD only when N < 3.  File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms 15.3.2 Lehmer's algorithm ------------------------- Lehmer's improvement of the Euclidean algorithms is based on the observation that the initial part of the quotient sequence depends only on the most significant parts of the inputs. The variant of Lehmer's algorithm used in GMP splits off the most significant two limbs, as suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean (*note References::). The quotients of two double-limb inputs are collected as a 2 by 2 matrix with single-limb elements. This is done by the function `mpn_hgcd2'. The resulting matrix is applied to the inputs using `mpn_mul_1' and `mpn_submul_1'. Each iteration usually reduces the inputs by almost one limb. In the rare case of a large quotient, no progress can be made by examining just the most significant two limbs, and the quotient is computed using plain division. The resulting algorithm is asymptotically O(N^2), just as the Euclidean algorithm and the binary algorithm. The quadratic part of the work are the calls to `mpn_mul_1' and `mpn_submul_1'. For small sizes, the linear work is also significant. There are roughly N calls to the `mpn_hgcd2' function. This function uses a couple of important optimizations: * It uses the same relaxed notion of correctness as `mpn_hgcd' (see next section). This means that when called with the most significant two limbs of two large numbers, the returned matrix does not always correspond exactly to the initial quotient sequence for the two large numbers; the final quotient may sometimes be one off. * It takes advantage of the fact the quotients are usually small. The division operator is not used, since the corresponding assembler instruction is very slow on most architectures. (This code could probably be improved further, it uses many branches that are unfriendly to prediction). * It switches from double-limb calculations to single-limb calculations half-way through, when the input numbers have been reduced in size from two limbs to one and a half.  File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms 15.3.3 Subquadratic GCD ----------------------- For inputs larger than `GCD_DC_THRESHOLD', GCD is computed via the HGCD (Half GCD) function, as a generalization to Lehmer's algorithm. Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1. Then HGCD(a,b) returns a transformation matrix T with non-negative elements, and reduced numbers (c;d) = T^-1 (a;b). The reduced numbers c,d must be larger than S limbs, while their difference abs(c-d) must fit in S limbs. The matrix elements will also be of size roughly N/2. The HGCD base case uses Lehmer's algorithm, but with the above stop condition that returns reduced numbers and the corresponding transformation matrix half-way through. For inputs larger than `HGCD_THRESHOLD', HGCD is computed recursively, using the divide and conquer algorithm in "On Schönhage's algorithm and subquadratic integer GCD computation" by Möller (*note References::). The recursive algorithm consists of these main steps. * Call HGCD recursively, on the most significant N/2 limbs. Apply the resulting matrix T_1 to the full numbers, reducing them to a size just above 3N/2. * Perform a small number of division or subtraction steps to reduce the numbers to size below 3N/2. This is essential mainly for the unlikely case of large quotients. * Call HGCD recursively, on the most significant N/2 limbs of the reduced numbers. Apply the resulting matrix T_2 to the full numbers, reducing them to a size just above N/2. * Compute T = T_1 T_2. * Perform a small number of division and subtraction steps to satisfy the requirements, and return. GCD is then implemented as a loop around HGCD, similarly to Lehmer's algorithm. Where Lehmer repeatedly chops off the top two limbs, calls `mpn_hgcd2', and applies the resulting matrix to the full numbers, the subquadratic GCD chops off the most significant third of the limbs (the proportion is a tuning parameter, and 1/3 seems to be more efficient than, e.g, 1/2), calls `mpn_hgcd', and applies the resulting matrix. Once the input numbers are reduced to size below `GCD_DC_THRESHOLD', Lehmer's algorithm is used for the rest of the work. The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)), where M(N) is the time for multiplying two N-limb numbers.  File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms 15.3.4 Extended GCD ------------------- The extended GCD function, or GCDEXT, calculates gcd(a,b) and also cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used for plain GCD are extended to handle this case. The binary algorithm is used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is implemented as a loop around HGCD, but with more book-keeping to keep track of the cofactors. This gives the same asymptotic running time as for GCD and HGCD, O(M(N)*log(N)) One difference to plain GCD is that while the inputs a and b are reduced as the algorithm proceeds, the cofactors x and y grow in size. This makes the tuning of the chopping-point more difficult. The current code chops off the most significant half of the inputs for the call to HGCD in the first iteration, and the most significant two thirds for the remaining calls. This strategy could surely be improved. Also the stop condition for the loop, where Lehmer's algorithm is invoked once the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be improved by taking into account the current size of the cofactors.  Local Variables: coding: iso-8859-1 End: gmp-doc-5.1.2/doc/Makefile.am0000644000175000000620000000167012146435154014665 0ustar stevestaff## Process this file with automake to generate Makefile.in # Copyright 2003 Free Software Foundation, Inc. # # This file is part of the GNU MP Library. # # The GNU MP Library is free software; you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation; either version 3 of the License, or (at your # option) any later version. # # The GNU MP Library is distributed in the hope that it will be useful, but # WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY # or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public # License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. 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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being ``A GNU Manual'', and with the Back-Cover Texts being ``You have freedom to copy and modify this GNU Manual, like GNU software''. A copy of the license is included in @ref{GNU Free Documentation License}. @end copying @c Note the @ref above must be on one line, a line break in an @ref within @c @copying will bomb in recent texinfo.tex (eg. 2004-04-07.08 which comes @c with texinfo 4.7), with messages about missing @endcsname. @c Texinfo version 4.2 or up will be needed to process this file. @c @c The version number and edition number are taken from version.texi provided @c by automake (note that it's regenerated only if you configure with @c --enable-maintainer-mode). @c @c Notes discussing the present version number of GMP in relation to previous @c ones (for instance in the "Compatibility" section) must be updated at @c manually though. @c @c @cindex entries have been made for function categories and programming @c topics. The "mpn" section is not included in this, because a beginner @c looking for "GCD" or something is only going to be confused by pointers to @c low level routines. @c @c @cindex entries are present for processors and systems when there's @c particular notes concerning them, but not just for everything GMP @c supports. @c @c Index entries for files use @code rather than @file, @samp or @option, @c since the latter come out with quotes in TeX, which are nice in the text @c but don't look so good in index columns. @c @c Tex: @c @c A suitable texinfo.tex is supplied, a newer one should work equally well. @c @c HTML: @c @c Nothing special is done for links to external manuals, they just come out @c in the usual makeinfo style, eg. "../libc/Locales.html". If you have @c local copies of such manuals then this is a good thing, if not then you @c may want to search-and-replace to some online source. @c @dircategory GNU libraries @direntry * gmp: (gmp). GNU Multiple Precision Arithmetic Library. @end direntry @c html @documentdescription How to install and use the GNU multiple precision arithmetic library, version @value{VERSION}. @end documentdescription @c smallbook @finalout @setchapternewpage on @ifnottex @node Top, Copying, (dir), (dir) @top GNU MP @end ifnottex @iftex @titlepage @title GNU MP @subtitle The GNU Multiple Precision Arithmetic Library @subtitle Edition @value{EDITION} @subtitle @value{UPDATED} @author by Torbj@"orn Granlund and the GMP development team @c @email{tg@@gmplib.org} @c Include the Distribution inside the titlepage so @c that headings are turned off. @tex \global\parindent=0pt \global\parskip=8pt \global\baselineskip=13pt @end tex @page @vskip 0pt plus 1filll @end iftex @insertcopying @ifnottex @sp 1 @end ifnottex @iftex @end titlepage @headings double @end iftex @c Don't bother with contents for html, the menus seem adequate. @ifnothtml @contents @end ifnothtml @menu * Copying:: GMP Copying Conditions (LGPL). * Introduction to GMP:: Brief introduction to GNU MP. * Installing GMP:: How to configure and compile the GMP library. * GMP Basics:: What every GMP user should know. * Reporting Bugs:: How to usefully report bugs. * Integer Functions:: Functions for arithmetic on signed integers. * Rational Number Functions:: Functions for arithmetic on rational numbers. * Floating-point Functions:: Functions for arithmetic on floats. * Low-level Functions:: Fast functions for natural numbers. * Random Number Functions:: Functions for generating random numbers. * Formatted Output:: @code{printf} style output. * Formatted Input:: @code{scanf} style input. * C++ Class Interface:: Class wrappers around GMP types. * Custom Allocation:: How to customize the internal allocation. * Language Bindings:: Using GMP from other languages. * Algorithms:: What happens behind the scenes. * Internals:: How values are represented behind the scenes. * Contributors:: Who brings you this library? * References:: Some useful papers and books to read. * GNU Free Documentation License:: * Concept Index:: * Function Index:: @end menu @c @m{T,N} is $T$ in tex or @math{N} otherwise. This is an easy way to give @c different forms for math in tex and info. Commas in N or T don't work, @c but @C{} can be used instead. \, works in info but not in tex. @iftex @macro m {T,N} @tex$\T\$@end tex @end macro @end iftex @ifnottex @macro m {T,N} @math{\N\} @end macro @end ifnottex @macro C {} , @end macro @c @ms{V,N} is $V_N$ in tex or just vn otherwise. This suits simple @c subscripts like @ms{x,0}. @iftex @macro ms {V,N} @tex$\V\_{\N\}$@end tex @end macro @end iftex @ifnottex @macro ms {V,N} \V\\N\ @end macro @end ifnottex @c @nicode{S} is plain S in info, or @code{S} elsewhere. This can be used @c when the quotes that @code{} gives in info aren't wanted, but the @c fontification in tex or html is wanted. Doesn't work as @nicode{'\\0'} @c though (gives two backslashes in tex). @ifinfo @macro nicode {S} \S\ @end macro @end ifinfo @ifnotinfo @macro nicode {S} @code{\S\} @end macro @end ifnotinfo @c @nisamp{S} is plain S in info, or @samp{S} elsewhere. This can be used @c when the quotes that @samp{} gives in info aren't wanted, but the @c fontification in tex or html is wanted. @ifinfo @macro nisamp {S} \S\ @end macro @end ifinfo @ifnotinfo @macro nisamp {S} @samp{\S\} @end macro @end ifnotinfo @c Usage: @GMPtimes{} @c Give either \times or the word "times". @tex \gdef\GMPtimes{\times} @end tex @ifnottex @macro GMPtimes times @end macro @end ifnottex @c Usage: @GMPmultiply{} @c Give * in info, or nothing in tex. @tex \gdef\GMPmultiply{} @end tex @ifnottex @macro GMPmultiply * @end macro @end ifnottex @c Usage: @GMPabs{x} @c Give either |x| in tex, or abs(x) in info or html. @tex \gdef\GMPabs#1{|#1|} @end tex @ifnottex @macro GMPabs {X} @abs{}(\X\) @end macro @end ifnottex @c Usage: @GMPfloor{x} @c Give either \lfloor x\rfloor in tex, or floor(x) in info or html. @tex \gdef\GMPfloor#1{\lfloor #1\rfloor} @end tex @ifnottex @macro GMPfloor {X} floor(\X\) @end macro @end ifnottex @c Usage: @GMPceil{x} @c Give either \lceil x\rceil in tex, or ceil(x) in info or html. @tex \gdef\GMPceil#1{\lceil #1 \rceil} @end tex @ifnottex @macro GMPceil {X} ceil(\X\) @end macro @end ifnottex @c Math operators already available in tex, made available in info too. @c For example @bmod{} can be used in both tex and info. @ifnottex @macro bmod mod @end macro @macro gcd gcd @end macro @macro ge >= @end macro @macro le <= @end macro @macro log log @end macro @macro min min @end macro @macro leftarrow <- @end macro @macro rightarrow -> @end macro @end ifnottex @c New math operators. @c @abs{} can be used in both tex and info, or just \abs in tex. @tex \gdef\abs{\mathop{\rm abs}} @end tex @ifnottex @macro abs abs @end macro @end ifnottex @c @cross{} is a \times symbol in tex, or an "x" in info. In tex it works @c inside or outside $ $. @tex \gdef\cross{\ifmmode\times\else$\times$\fi} @end tex @ifnottex @macro cross x @end macro @end ifnottex @c @times{} made available as a "*" in info and html (already works in tex). @ifnottex @macro times * @end macro @end ifnottex @c Usage: @W{text} @c Like @w{} but working in math mode too. @tex \gdef\W#1{\ifmmode{#1}\else\w{#1}\fi} @end tex @ifnottex @macro W {S} @w{\S\} @end macro @end ifnottex @c Usage: \GMPdisplay{text} @c Put the given text in an @display style indent, but without turning off @c paragraph reflow etc. @tex \gdef\GMPdisplay#1{% \noindent \advance\leftskip by \lispnarrowing #1\par} @end tex @c Usage: \GMPhat @c A new \hat that will work in math mode, unlike the texinfo redefined @c version. @tex \gdef\GMPhat{\mathaccent"705E} @end tex @c Usage: \GMPraise{text} @c For use in a $ $ math expression as an alternative to "^". This is good @c for @code{} in an exponent, since there seems to be no superscript font @c for that. @tex \gdef\GMPraise#1{\mskip0.5\thinmuskip\hbox{\raise0.8ex\hbox{#1}}} @end tex @c Usage: @texlinebreak{} @c A line break as per @*, but only in tex. @iftex @macro texlinebreak @* @end macro @end iftex @ifnottex @macro texlinebreak @end macro @end ifnottex @c Usage: @maybepagebreak @c Allow tex to insert a page break, if it feels the urge. @c Normally blocks of @deftypefun/funx are kept together, which can lead to @c some poor page break positioning if it's a big block, like the sets of @c division functions etc. @tex \gdef\maybepagebreak{\penalty0} @end tex @ifnottex @macro maybepagebreak @end macro @end ifnottex @c Usage: @GMPreftop{info,title} @c Usage: @GMPpxreftop{info,title} @c @c Like @ref{} and @pxref{}, but designed for a reference to the top of a @c document, not a particular section. The TeX output for plain @ref insists @c on printing a particular section, GMPreftop gives just the title. @c @c The texinfo manual recommends putting a likely section name in references @c like this, eg. "Introduction", but it seems better to just give the title. @c @iftex @macro GMPreftop{info,title} @i{\title\} @end macro @macro GMPpxreftop{info,title} see @i{\title\} @end macro @end iftex @c @ifnottex @macro GMPreftop{info,title} @ref{Top,\title\,\title\,\info\,\title\} @end macro @macro GMPpxreftop{info,title} @pxref{Top,\title\,\title\,\info\,\title\} @end macro @end ifnottex @node Copying, Introduction to GMP, Top, Top @comment node-name, next, previous, up @unnumbered GNU MP Copying Conditions @cindex Copying conditions @cindex Conditions for copying GNU MP @cindex License conditions This library is @dfn{free}; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you.@refill Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things.@refill To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MP library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.@refill Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MP library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.@refill The precise conditions of the license for the GNU MP library are found in the Lesser General Public License version 3 that accompanies the source code, see @file{COPYING.LIB}. Certain demonstration programs are provided under the terms of the plain General Public License version 3, see @file{COPYING}. @node Introduction to GMP, Installing GMP, Copying, Top @comment node-name, next, previous, up @chapter Introduction to GNU MP @cindex Introduction GNU MP is a portable library written in C for arbitrary precision arithmetic on integers, rational numbers, and floating-point numbers. It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types. Many applications use just a few hundred bits of precision; but some applications may need thousands or even millions of bits. GMP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum. The speed of GMP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance). There is assembly code for these CPUs: @cindex CPU types ARM, DEC Alpha 21064, 21164, and 21264, AMD 29000, AMD K6, K6-2, Athlon, and Athlon64, Hitachi SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and Pyramid AP/XP. @cindex Home page @cindex Web page @noindent For up-to-date information on GMP, please see the GMP web pages at @display @uref{http://gmplib.org/} @end display @cindex Latest version of GMP @cindex Anonymous FTP of latest version @cindex FTP of latest version @noindent The latest version of the library is available at @display @uref{ftp://ftp.gnu.org/gnu/gmp/} @end display Many sites around the world mirror @samp{ftp.gnu.org}, please use a mirror near you, see @uref{http://www.gnu.org/order/ftp.html} for a full list. @cindex Mailing lists There are three public mailing lists of interest. One for release announcements, one for general questions and discussions about usage of the GMP library and one for bug reports. For more information, see @display @uref{http://gmplib.org/mailman/listinfo/}. @end display The proper place for bug reports is @email{gmp-bugs@@gmplib.org}. See @ref{Reporting Bugs} for information about reporting bugs. @sp 1 @section How to use this Manual @cindex About this manual Everyone should read @ref{GMP Basics}. If you need to install the library yourself, then read @ref{Installing GMP}. If you have a system with multiple ABIs, then read @ref{ABI and ISA}, for the compiler options that must be used on applications. The rest of the manual can be used for later reference, although it is probably a good idea to glance through it. @node Installing GMP, GMP Basics, Introduction to GMP, Top @comment node-name, next, previous, up @chapter Installing GMP @cindex Installing GMP @cindex Configuring GMP @cindex Building GMP GMP has an autoconf/automake/libtool based configuration system. On a Unix-like system a basic build can be done with @example ./configure make @end example @noindent Some self-tests can be run with @example make check @end example @noindent And you can install (under @file{/usr/local} by default) with @example make install @end example If you experience problems, please report them to @email{gmp-bugs@@gmplib.org}. See @ref{Reporting Bugs}, for information on what to include in useful bug reports. @menu * Build Options:: * ABI and ISA:: * Notes for Package Builds:: * Notes for Particular Systems:: * Known Build Problems:: * Performance optimization:: @end menu @node Build Options, ABI and ISA, Installing GMP, Installing GMP @section Build Options @cindex Build options All the usual autoconf configure options are available, run @samp{./configure --help} for a summary. The file @file{INSTALL.autoconf} has some generic installation information too. @table @asis @item Tools @cindex Non-Unix systems @samp{configure} requires various Unix-like tools. See @ref{Notes for Particular Systems}, for some options on non-Unix systems. It might be possible to build without the help of @samp{configure}, certainly all the code is there, but unfortunately you'll be on your own. @item Build Directory @cindex Build directory To compile in a separate build directory, @command{cd} to that directory, and prefix the configure command with the path to the GMP source directory. For example @example cd /my/build/dir /my/sources/gmp-@value{VERSION}/configure @end example Not all @samp{make} programs have the necessary features (@code{VPATH}) to support this. In particular, SunOS and Slowaris @command{make} have bugs that make them unable to build in a separate directory. Use GNU @command{make} instead. @item @option{--prefix} and @option{--exec-prefix} @cindex Prefix @cindex Exec prefix @cindex Install prefix @cindex @code{--prefix} @cindex @code{--exec-prefix} The @option{--prefix} option can be used in the normal way to direct GMP to install under a particular tree. The default is @samp{/usr/local}. @option{--exec-prefix} can be used to direct architecture-dependent files like @file{libgmp.a} to a different location. This can be used to share architecture-independent parts like the documentation, but separate the dependent parts. Note however that @file{gmp.h} and @file{mp.h} are architecture-dependent since they encode certain aspects of @file{libgmp}, so it will be necessary to ensure both @file{$prefix/include} and @file{$exec_prefix/include} are available to the compiler. @item @option{--disable-shared}, @option{--disable-static} @cindex @code{--disable-shared} @cindex @code{--disable-static} By default both shared and static libraries are built (where possible), but one or other can be disabled. Shared libraries result in smaller executables and permit code sharing between separate running processes, but on some CPUs are slightly slower, having a small cost on each function call. @item Native Compilation, @option{--build=CPU-VENDOR-OS} @cindex Native compilation @cindex Build system @cindex @code{--build} For normal native compilation, the system can be specified with @samp{--build}. By default @samp{./configure} uses the output from running @samp{./config.guess}. On some systems @samp{./config.guess} can determine the exact CPU type, on others it will be necessary to give it explicitly. For example, @example ./configure --build=ultrasparc-sun-solaris2.7 @end example In all cases the @samp{OS} part is important, since it controls how libtool generates shared libraries. Running @samp{./config.guess} is the simplest way to see what it should be, if you don't know already. @item Cross Compilation, @option{--host=CPU-VENDOR-OS} @cindex Cross compiling @cindex Host system @cindex @code{--host} When cross-compiling, the system used for compiling is given by @samp{--build} and the system where the library will run is given by @samp{--host}. For example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries, @example ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu @end example Compiler tools are sought first with the host system type as a prefix. For example @command{m68k-mac-linux-gnu-ranlib} is tried, then plain @command{ranlib}. This makes it possible for a set of cross-compiling tools to co-exist with native tools. The prefix is the argument to @samp{--host}, and this can be an alias, such as @samp{m68k-linux}. But note that tools don't have to be setup this way, it's enough to just have a @env{PATH} with a suitable cross-compiling @command{cc} etc. Compiling for a different CPU in the same family as the build system is a form of cross-compilation, though very possibly this would merely be special options on a native compiler. In any case @samp{./configure} avoids depending on being able to run code on the build system, which is important when creating binaries for a newer CPU since they very possibly won't run on the build system. In all cases the compiler must be able to produce an executable (of whatever format) from a standard C @code{main}. Although only object files will go to make up @file{libgmp}, @samp{./configure} uses linking tests for various purposes, such as determining what functions are available on the host system. Currently a warning is given unless an explicit @samp{--build} is used when cross-compiling, because it may not be possible to correctly guess the build system type if the @env{PATH} has only a cross-compiling @command{cc}. Note that the @samp{--target} option is not appropriate for GMP@. It's for use when building compiler tools, with @samp{--host} being where they will run, and @samp{--target} what they'll produce code for. Ordinary programs or libraries like GMP are only interested in the @samp{--host} part, being where they'll run. (Some past versions of GMP used @samp{--target} incorrectly.) @item CPU types @cindex CPU types In general, if you want a library that runs as fast as possible, you should configure GMP for the exact CPU type your system uses. However, this may mean the binaries won't run on older members of the family, and might run slower on other members, older or newer. The best idea is always to build GMP for the exact machine type you intend to run it on. The following CPUs have specific support. See @file{configure.in} for details of what code and compiler options they select. @itemize @bullet @c Keep this formatting, it's easy to read and it can be grepped to @c automatically test that CPUs listed get through ./config.sub @item Alpha: @nisamp{alpha}, @nisamp{alphaev5}, @nisamp{alphaev56}, @nisamp{alphapca56}, @nisamp{alphapca57}, @nisamp{alphaev6}, @nisamp{alphaev67}, @nisamp{alphaev68} @nisamp{alphaev7} @item Cray: @nisamp{c90}, @nisamp{j90}, @nisamp{t90}, @nisamp{sv1} @item HPPA: @nisamp{hppa1.0}, @nisamp{hppa1.1}, @nisamp{hppa2.0}, @nisamp{hppa2.0n}, @nisamp{hppa2.0w}, @nisamp{hppa64} @item IA-64: @nisamp{ia64}, @nisamp{itanium}, @nisamp{itanium2} @item MIPS: @nisamp{mips}, @nisamp{mips3}, @nisamp{mips64} @item Motorola: @nisamp{m68k}, @nisamp{m68000}, @nisamp{m68010}, @nisamp{m68020}, @nisamp{m68030}, @nisamp{m68040}, @nisamp{m68060}, @nisamp{m68302}, @nisamp{m68360}, @nisamp{m88k}, @nisamp{m88110} @item POWER: @nisamp{power}, @nisamp{power1}, @nisamp{power2}, @nisamp{power2sc} @item PowerPC: @nisamp{powerpc}, @nisamp{powerpc64}, @nisamp{powerpc401}, @nisamp{powerpc403}, @nisamp{powerpc405}, @nisamp{powerpc505}, @nisamp{powerpc601}, @nisamp{powerpc602}, @nisamp{powerpc603}, @nisamp{powerpc603e}, @nisamp{powerpc604}, @nisamp{powerpc604e}, @nisamp{powerpc620}, @nisamp{powerpc630}, @nisamp{powerpc740}, @nisamp{powerpc7400}, @nisamp{powerpc7450}, @nisamp{powerpc750}, @nisamp{powerpc801}, @nisamp{powerpc821}, @nisamp{powerpc823}, @nisamp{powerpc860}, @nisamp{powerpc970} @item SPARC: @nisamp{sparc}, @nisamp{sparcv8}, @nisamp{microsparc}, @nisamp{supersparc}, @nisamp{sparcv9}, @nisamp{ultrasparc}, @nisamp{ultrasparc2}, @nisamp{ultrasparc2i}, @nisamp{ultrasparc3}, @nisamp{sparc64} @item x86 family: @nisamp{i386}, @nisamp{i486}, @nisamp{i586}, @nisamp{pentium}, @nisamp{pentiummmx}, @nisamp{pentiumpro}, @nisamp{pentium2}, @nisamp{pentium3}, @nisamp{pentium4}, @nisamp{k6}, @nisamp{k62}, @nisamp{k63}, @nisamp{athlon}, @nisamp{amd64}, @nisamp{viac3}, @nisamp{viac32} @item Other: @nisamp{a29k}, @nisamp{arm}, @nisamp{clipper}, @nisamp{i960}, @nisamp{ns32k}, @nisamp{pyramid}, @nisamp{sh}, @nisamp{sh2}, @nisamp{vax}, @nisamp{z8k} @end itemize CPUs not listed will use generic C code. @item Generic C Build @cindex Generic C If some of the assembly code causes problems, or if otherwise desired, the generic C code can be selected with the configure @option{--disable-assembly}. Note that this will run quite slowly, but it should be portable and should at least make it possible to get something running if all else fails. @item Fat binary, @option{--enable-fat} @cindex Fat binary @cindex @option{--enable-fat} Using @option{--enable-fat} selects a ``fat binary'' build on x86, where optimized low level subroutines are chosen at runtime according to the CPU detected. This means more code, but gives good performance on all x86 chips. (This option might become available for more architectures in the future.) @item @option{ABI} @cindex ABI On some systems GMP supports multiple ABIs (application binary interfaces), meaning data type sizes and calling conventions. By default GMP chooses the best ABI available, but a particular ABI can be selected. For example @example ./configure --host=mips64-sgi-irix6 ABI=n32 @end example See @ref{ABI and ISA}, for the available choices on relevant CPUs, and what applications need to do. @item @option{CC}, @option{CFLAGS} @cindex C compiler @cindex @code{CC} @cindex @code{CFLAGS} By default the C compiler used is chosen from among some likely candidates, with @command{gcc} normally preferred if it's present. The usual @samp{CC=whatever} can be passed to @samp{./configure} to choose something different. For various systems, default compiler flags are set based on the CPU and compiler. The usual @samp{CFLAGS="-whatever"} can be passed to @samp{./configure} to use something different or to set good flags for systems GMP doesn't otherwise know. The @samp{CC} and @samp{CFLAGS} used are printed during @samp{./configure}, and can be found in each generated @file{Makefile}. This is the easiest way to check the defaults when considering changing or adding something. Note that when @samp{CC} and @samp{CFLAGS} are specified on a system supporting multiple ABIs it's important to give an explicit @samp{ABI=whatever}, since GMP can't determine the ABI just from the flags and won't be able to select the correct assembly code. If just @samp{CC} is selected then normal default @samp{CFLAGS} for that compiler will be used (if GMP recognises it). For example @samp{CC=gcc} can be used to force the use of GCC, with default flags (and default ABI). @item @option{CPPFLAGS} @cindex @code{CPPFLAGS} Any flags like @samp{-D} defines or @samp{-I} includes required by the preprocessor should be set in @samp{CPPFLAGS} rather than @samp{CFLAGS}. Compiling is done with both @samp{CPPFLAGS} and @samp{CFLAGS}, but preprocessing uses just @samp{CPPFLAGS}. This distinction is because most preprocessors won't accept all the flags the compiler does. Preprocessing is done separately in some configure tests. @item @option{CC_FOR_BUILD} @cindex @code{CC_FOR_BUILD} Some build-time programs are compiled and run to generate host-specific data tables. @samp{CC_FOR_BUILD} is the compiler used for this. It doesn't need to be in any particular ABI or mode, it merely needs to generate executables that can run. The default is to try the selected @samp{CC} and some likely candidates such as @samp{cc} and @samp{gcc}, looking for something that works. No flags are used with @samp{CC_FOR_BUILD} because a simple invocation like @samp{cc foo.c} should be enough. If some particular options are required they can be included as for instance @samp{CC_FOR_BUILD="cc -whatever"}. @item C++ Support, @option{--enable-cxx} @cindex C++ support @cindex @code{--enable-cxx} C++ support in GMP can be enabled with @samp{--enable-cxx}, in which case a C++ compiler will be required. As a convenience @samp{--enable-cxx=detect} can be used to enable C++ support only if a compiler can be found. The C++ support consists of a library @file{libgmpxx.la} and header file @file{gmpxx.h} (@pxref{Headers and Libraries}). A separate @file{libgmpxx.la} has been adopted rather than having C++ objects within @file{libgmp.la} in order to ensure dynamic linked C programs aren't bloated by a dependency on the C++ standard library, and to avoid any chance that the C++ compiler could be required when linking plain C programs. @file{libgmpxx.la} will use certain internals from @file{libgmp.la} and can only be expected to work with @file{libgmp.la} from the same GMP version. Future changes to the relevant internals will be accompanied by renaming, so a mismatch will cause unresolved symbols rather than perhaps mysterious misbehaviour. In general @file{libgmpxx.la} will be usable only with the C++ compiler that built it, since name mangling and runtime support are usually incompatible between different compilers. @item @option{CXX}, @option{CXXFLAGS} @cindex C++ compiler @cindex @code{CXX} @cindex @code{CXXFLAGS} When C++ support is enabled, the C++ compiler and its flags can be set with variables @samp{CXX} and @samp{CXXFLAGS} in the usual way. The default for @samp{CXX} is the first compiler that works from a list of likely candidates, with @command{g++} normally preferred when available. The default for @samp{CXXFLAGS} is to try @samp{CFLAGS}, @samp{CFLAGS} without @samp{-g}, then for @command{g++} either @samp{-g -O2} or @samp{-O2}, or for other compilers @samp{-g} or nothing. Trying @samp{CFLAGS} this way is convenient when using @samp{gcc} and @samp{g++} together, since the flags for @samp{gcc} will usually suit @samp{g++}. It's important that the C and C++ compilers match, meaning their startup and runtime support routines are compatible and that they generate code in the same ABI (if there's a choice of ABIs on the system). @samp{./configure} isn't currently able to check these things very well itself, so for that reason @samp{--disable-cxx} is the default, to avoid a build failure due to a compiler mismatch. Perhaps this will change in the future. Incidentally, it's normally not good enough to set @samp{CXX} to the same as @samp{CC}. Although @command{gcc} for instance recognises @file{foo.cc} as C++ code, only @command{g++} will invoke the linker the right way when building an executable or shared library from C++ object files. @item Temporary Memory, @option{--enable-alloca=} @cindex Temporary memory @cindex Stack overflow @cindex @code{alloca} @cindex @code{--enable-alloca} GMP allocates temporary workspace using one of the following three methods, which can be selected with for instance @samp{--enable-alloca=malloc-reentrant}. @itemize @bullet @item @samp{alloca} - C library or compiler builtin. @item @samp{malloc-reentrant} - the heap, in a re-entrant fashion. @item @samp{malloc-notreentrant} - the heap, with global variables. @end itemize For convenience, the following choices are also available. @samp{--disable-alloca} is the same as @samp{no}. @itemize @bullet @item @samp{yes} - a synonym for @samp{alloca}. @item @samp{no} - a synonym for @samp{malloc-reentrant}. @item @samp{reentrant} - @code{alloca} if available, otherwise @samp{malloc-reentrant}. This is the default. @item @samp{notreentrant} - @code{alloca} if available, otherwise @samp{malloc-notreentrant}. @end itemize @code{alloca} is reentrant and fast, and is recommended. It actually allocates just small blocks on the stack; larger ones use malloc-reentrant. @samp{malloc-reentrant} is, as the name suggests, reentrant and thread safe, but @samp{malloc-notreentrant} is faster and should be used if reentrancy is not required. The two malloc methods in fact use the memory allocation functions selected by @code{mp_set_memory_functions}, these being @code{malloc} and friends by default. @xref{Custom Allocation}. An additional choice @samp{--enable-alloca=debug} is available, to help when debugging memory related problems (@pxref{Debugging}). @item FFT Multiplication, @option{--disable-fft} @cindex FFT multiplication @cindex @code{--disable-fft} By default multiplications are done using Karatsuba, 3-way Toom, higher degree Toom, and Fermat FFT@. The FFT is only used on large to very large operands and can be disabled to save code size if desired. @item Assertion Checking, @option{--enable-assert} @cindex Assertion checking @cindex @code{--enable-assert} This option enables some consistency checking within the library. This can be of use while debugging, @pxref{Debugging}. @item Execution Profiling, @option{--enable-profiling=prof/gprof/instrument} @cindex Execution profiling @cindex @code{--enable-profiling} Enable profiling support, in one of various styles, @pxref{Profiling}. @item @option{MPN_PATH} @cindex @code{MPN_PATH} Various assembly versions of each mpn subroutines are provided. For a given CPU, a search is made though a path to choose a version of each. For example @samp{sparcv8} has @example MPN_PATH="sparc32/v8 sparc32 generic" @end example which means look first for v8 code, then plain sparc32 (which is v7), and finally fall back on generic C@. Knowledgeable users with special requirements can specify a different path. Normally this is completely unnecessary. @item Documentation @cindex Documentation formats @cindex Texinfo The source for the document you're now reading is @file{doc/gmp.texi}, in Texinfo format, see @GMPreftop{texinfo, Texinfo}. @cindex Postscript @cindex DVI @cindex PDF Info format @samp{doc/gmp.info} is included in the distribution. The usual automake targets are available to make PostScript, DVI, PDF and HTML (these will require various @TeX{} and Texinfo tools). @cindex DocBook @cindex XML DocBook and XML can be generated by the Texinfo @command{makeinfo} program too, see @ref{makeinfo options,, Options for @command{makeinfo}, texinfo, Texinfo}. Some supplementary notes can also be found in the @file{doc} subdirectory. @end table @need 2000 @node ABI and ISA, Notes for Package Builds, Build Options, Installing GMP @section ABI and ISA @cindex ABI @cindex Application Binary Interface @cindex ISA @cindex Instruction Set Architecture ABI (Application Binary Interface) refers to the calling conventions between functions, meaning what registers are used and what sizes the various C data types are. ISA (Instruction Set Architecture) refers to the instructions and registers a CPU has available. Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the latter for compatibility with older CPUs in the family. GMP supports some CPUs like this in both ABIs. In fact within GMP @samp{ABI} means a combination of chip ABI, plus how GMP chooses to use it. For example in some 32-bit ABIs, GMP may support a limb as either a 32-bit @code{long} or a 64-bit @code{long long}. By default GMP chooses the best ABI available for a given system, and this generally gives significantly greater speed. But an ABI can be chosen explicitly to make GMP compatible with other libraries, or particular application requirements. For example, @example ./configure ABI=32 @end example In all cases it's vital that all object code used in a given program is compiled for the same ABI. Usually a limb is implemented as a @code{long}. When a @code{long long} limb is used this is encoded in the generated @file{gmp.h}. This is convenient for applications, but it does mean that @file{gmp.h} will vary, and can't be just copied around. @file{gmp.h} remains compiler independent though, since all compilers for a particular ABI will be expected to use the same limb type. Currently no attempt is made to follow whatever conventions a system has for installing library or header files built for a particular ABI@. This will probably only matter when installing multiple builds of GMP, and it might be as simple as configuring with a special @samp{libdir}, or it might require more than that. Note that builds for different ABIs need to done separately, with a fresh @command{./configure} and @command{make} each. @sp 1 @table @asis @need 1000 @item AMD64 (@samp{x86_64}) @cindex AMD64 On AMD64 systems supporting both 32-bit and 64-bit modes for applications, the following ABI choices are available. @table @asis @item @samp{ABI=64} The 64-bit ABI uses 64-bit limbs and pointers and makes full use of the chip architecture. This is the default. Applications will usually not need special compiler flags, but for reference the option is @example gcc -m64 @end example @item @samp{ABI=32} The 32-bit ABI is the usual i386 conventions. This will be slower, and is not recommended except for inter-operating with other code not yet 64-bit capable. Applications must be compiled with @example gcc -m32 @end example (In GCC 2.95 and earlier there's no @samp{-m32} option, it's the only mode.) @end table @sp 1 @need 1000 @item HPPA 2.0 (@samp{hppa2.0*}, @samp{hppa64}) @cindex HPPA @cindex HP-UX @table @asis @item @samp{ABI=2.0w} The 2.0w ABI uses 64-bit limbs and pointers and is available on HP-UX 11 or up. Applications must be compiled with @example gcc [built for 2.0w] cc +DD64 @end example @item @samp{ABI=2.0n} The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal calling conventions, but with 64-bit instructions permitted within functions. GMP uses a 64-bit @code{long long} for a limb. This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or higher. Applications must be compiled with @example gcc [built for 2.0n] cc +DA2.0 +e @end example Note that current versions of GCC (eg.@: 3.2) don't generate 64-bit instructions for @code{long long} operations and so may be slower than for 2.0w. (The GMP assembly code is the same though.) @item @samp{ABI=1.0} HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit HPPA 1.0 ABI@. No special compiler options are needed for applications. @end table All three ABIs are available for CPU types @samp{hppa2.0w}, @samp{hppa2.0} and @samp{hppa64}, but for CPU type @samp{hppa2.0n} only 2.0n or 1.0 are considered. Note that GCC on HP-UX has no options to choose between 2.0n and 2.0w modes, unlike HP @command{cc}. Instead it must be built for one or the other ABI@. GMP will detect how it was built, and skip to the corresponding @samp{ABI}. @sp 1 @need 1500 @item IA-64 under HP-UX (@samp{ia64*-*-hpux*}, @samp{itanium*-*-hpux*}) @cindex IA-64 @cindex HP-UX HP-UX supports two ABIs for IA-64. GMP performance is the same in both. @table @asis @item @samp{ABI=32} In the 32-bit ABI, pointers, @code{int}s and @code{long}s are 32 bits and GMP uses a 64 bit @code{long long} for a limb. Applications can be compiled without any special flags since this ABI is the default in both HP C and GCC, but for reference the flags are @example gcc -milp32 cc +DD32 @end example @item @samp{ABI=64} In the 64-bit ABI, @code{long}s and pointers are 64 bits and GMP uses a @code{long} for a limb. Applications must be compiled with @example gcc -mlp64 cc +DD64 @end example @end table On other IA-64 systems, GNU/Linux for instance, @samp{ABI=64} is the only choice. @sp 1 @need 1000 @item MIPS under IRIX 6 (@samp{mips*-*-irix[6789]}) @cindex MIPS @cindex IRIX IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs o32, n32, and 64. n32 or 64 are recommended, and GMP performance will be the same in each. The default is n32. @table @asis @item @samp{ABI=o32} The o32 ABI is 32-bit pointers and integers, and no 64-bit operations. GMP will be slower than in n32 or 64, this option only exists to support old compilers, eg.@: GCC 2.7.2. Applications can be compiled with no special flags on an old compiler, or on a newer compiler with @example gcc -mabi=32 cc -32 @end example @item @samp{ABI=n32} The n32 ABI is 32-bit pointers and integers, but with a 64-bit limb using a @code{long long}. Applications must be compiled with @example gcc -mabi=n32 cc -n32 @end example @item @samp{ABI=64} The 64-bit ABI is 64-bit pointers and integers. Applications must be compiled with @example gcc -mabi=64 cc -64 @end example @end table Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code. @sp 1 @need 1000 @item PowerPC 64 (@samp{powerpc64}, @samp{powerpc620}, @samp{powerpc630}, @samp{powerpc970}, @samp{power4}, @samp{power5}) @cindex PowerPC @table @asis @item @samp{ABI=mode64} @cindex AIX The AIX 64 ABI uses 64-bit limbs and pointers and is the default on PowerPC 64 @samp{*-*-aix*} systems. Applications must be compiled with @example gcc -maix64 xlc -q64 @end example On 64-bit GNU/Linux, BSD, and Mac OS X/Darwin systems, the applications must be compiled with @example gcc -m64 @end example @item @samp{ABI=mode32} The @samp{mode32} ABI uses a 64-bit @code{long long} limb but with the chip still in 32-bit mode and using 32-bit calling conventions. This is the default for systems where the true 64-bit ABI is unavailable. No special compiler options are typically needed for applications. This ABI is not available under AIX. @item @samp{ABI=32} This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No special compiler options are needed for applications. @end table GMP's speed is greatest for the @samp{mode64} ABI, the @samp{mode32} ABI is 2nd best. In @samp{ABI=32} only the 32-bit ISA is used and this doesn't make full use of a 64-bit chip. @sp 1 @need 1000 @item Sparc V9 (@samp{sparc64}, @samp{sparcv9}, @samp{ultrasparc*}) @cindex Sparc V9 @cindex Solaris @cindex Sun @table @asis @item @samp{ABI=64} The 64-bit V9 ABI is available on the various BSD sparc64 ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7 and up (when the kernel is in 64-bit mode). GCC 3.2 or higher, or Sun @command{cc} is required. On GNU/Linux, depending on the default @command{gcc} mode, applications must be compiled with @example gcc -m64 @end example On Solaris applications must be compiled with @example gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9 cc -xarch=v9 @end example On the BSD sparc64 systems no special options are required, since 64-bits is the only ABI available. @item @samp{ABI=32} For the basic 32-bit ABI, GMP still uses as much of the V9 ISA as it can. In the Sun documentation this combination is known as ``v8plus''. On GNU/Linux, depending on the default @command{gcc} mode, applications may need to be compiled with @example gcc -m32 @end example On Solaris, no special compiler options are required for applications, though using something like the following is recommended. (@command{gcc} 2.8 and earlier only support @samp{-mv8} though.) @example gcc -mv8plus cc -xarch=v8plus @end example @end table GMP speed is greatest in @samp{ABI=64}, so it's the default where available. The speed is partly because there are extra registers available and partly because 64-bits is considered the more important case and has therefore had better code written for it. Don't be confused by the names of the @samp{-m} and @samp{-x} compiler options, they're called @samp{arch} but effectively control both ABI and ISA@. On Solaris 2.6 and earlier, only @samp{ABI=32} is available since the kernel doesn't save all registers. On Solaris 2.7 with the kernel in 32-bit mode, a normal native build will reject @samp{ABI=64} because the resulting executables won't run. @samp{ABI=64} can still be built if desired by making it look like a cross-compile, for example @example ./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64 @end example @end table @need 2000 @node Notes for Package Builds, Notes for Particular Systems, ABI and ISA, Installing GMP @section Notes for Package Builds @cindex Build notes for binary packaging @cindex Packaged builds GMP should present no great difficulties for packaging in a binary distribution. @cindex Libtool versioning @cindex Shared library versioning Libtool is used to build the library and @samp{-version-info} is set appropriately, having started from @samp{3:0:0} in GMP 3.0 (@pxref{Versioning, Library interface versions, Library interface versions, libtool, GNU Libtool}). The GMP 4 series will be upwardly binary compatible in each release and will be upwardly binary compatible with all of the GMP 3 series. Additional function interfaces may be added in each release, so on systems where libtool versioning is not fully checked by the loader an auxiliary mechanism may be needed to express that a dynamic linked application depends on a new enough GMP. An auxiliary mechanism may also be needed to express that @file{libgmpxx.la} (from @option{--enable-cxx}, @pxref{Build Options}) requires @file{libgmp.la} from the same GMP version, since this is not done by the libtool versioning, nor otherwise. A mismatch will result in unresolved symbols from the linker, or perhaps the loader. When building a package for a CPU family, care should be taken to use @samp{--host} (or @samp{--build}) to choose the least common denominator among the CPUs which might use the package. For example this might mean plain @samp{sparc} (meaning V7) for SPARCs. For x86s, @option{--enable-fat} sets things up for a fat binary build, making a runtime selection of optimized low level routines. This is a good choice for packaging to run on a range of x86 chips. Users who care about speed will want GMP built for their exact CPU type, to make best use of the available optimizations. Providing a way to suitably rebuild a package may be useful. This could be as simple as making it possible for a user to omit @samp{--build} (and @samp{--host}) so @samp{./config.guess} will detect the CPU@. But a way to manually specify a @samp{--build} will be wanted for systems where @samp{./config.guess} is inexact. On systems with multiple ABIs, a packaged build will need to decide which among the choices is to be provided, see @ref{ABI and ISA}. A given run of @samp{./configure} etc will only build one ABI@. If a second ABI is also required then a second run of @samp{./configure} etc must be made, starting from a clean directory tree (@samp{make distclean}). As noted under ``ABI and ISA'', currently no attempt is made to follow system conventions for install locations that vary with ABI, such as @file{/usr/lib/sparcv9} for @samp{ABI=64} as opposed to @file{/usr/lib} for @samp{ABI=32}. A package build can override @samp{libdir} and other standard variables as necessary. Note that @file{gmp.h} is a generated file, and will be architecture and ABI dependent. When attempting to install two ABIs simultaneously it will be important that an application compile gets the correct @file{gmp.h} for its desired ABI@. If compiler include paths don't vary with ABI options then it might be necessary to create a @file{/usr/include/gmp.h} which tests preprocessor symbols and chooses the correct actual @file{gmp.h}. @need 2000 @node Notes for Particular Systems, Known Build Problems, Notes for Package Builds, Installing GMP @section Notes for Particular Systems @cindex Build notes for particular systems @cindex Particular systems @cindex Systems @table @asis @c This section is more or less meant for notes about performance or about @c build problems that have been worked around but might leave a user @c scratching their head. Fun with different ABIs on a system belongs in the @c above section. @item AIX 3 and 4 @cindex AIX On systems @samp{*-*-aix[34]*} shared libraries are disabled by default, since some versions of the native @command{ar} fail on the convenience libraries used. A shared build can be attempted with @example ./configure --enable-shared --disable-static @end example Note that the @samp{--disable-static} is necessary because in a shared build libtool makes @file{libgmp.a} a symlink to @file{libgmp.so}, apparently for the benefit of old versions of @command{ld} which only recognise @file{.a}, but unfortunately this is done even if a fully functional @command{ld} is available. @item ARM @cindex ARM On systems @samp{arm*-*-*}, versions of GCC up to and including 2.95.3 have a bug in unsigned division, giving wrong results for some operands. GMP @samp{./configure} will demand GCC 2.95.4 or later. @item Compaq C++ @cindex Compaq C++ Compaq C++ on OSF 5.1 has two flavours of @code{iostream}, a standard one and an old pre-standard one (see @samp{man iostream_intro}). GMP can only use the standard one, which unfortunately is not the default but must be selected by defining @code{__USE_STD_IOSTREAM}. Configure with for instance @example ./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM @end example @item Floating Point Mode @cindex Floating point mode @cindex Hardware floating point mode @cindex Precision of hardware floating point @cindex x87 On some systems, the hardware floating point has a control mode which can set all operations to be done in a particular precision, for instance single, double or extended on x86 systems (x87 floating point). The GMP functions involving a @code{double} cannot be expected to operate to their full precision when the hardware is in single precision mode. Of course this affects all code, including application code, not just GMP. @item MS-DOS and MS Windows @cindex MS-DOS @cindex MS Windows @cindex Windows @cindex Cygwin @cindex DJGPP @cindex MINGW On an MS-DOS system DJGPP can be used to build GMP, and on an MS Windows system Cygwin, DJGPP and MINGW can be used. All three are excellent ports of GCC and the various GNU tools. @display @uref{http://www.cygwin.com/} @uref{http://www.delorie.com/djgpp/} @uref{http://www.mingw.org/} @end display @cindex Interix @cindex Services for Unix Microsoft also publishes an Interix ``Services for Unix'' which can be used to build GMP on Windows (with a normal @samp{./configure}), but it's not free software. @item MS Windows DLLs @cindex DLLs @cindex MS Windows @cindex Windows On systems @samp{*-*-cygwin*}, @samp{*-*-mingw*} and @samp{*-*-pw32*} by default GMP builds only a static library, but a DLL can be built instead using @example ./configure --disable-static --enable-shared @end example Static and DLL libraries can't both be built, since certain export directives in @file{gmp.h} must be different. A MINGW DLL build of GMP can be used with Microsoft C@. Libtool doesn't install a @file{.lib} format import library, but it can be created with MS @command{lib} as follows, and copied to the install directory. Similarly for @file{libmp} and @file{libgmpxx}. @example cd .libs lib /def:libgmp-3.dll.def /out:libgmp-3.lib @end example MINGW uses the C runtime library @samp{msvcrt.dll} for I/O, so applications wanting to use the GMP I/O routines must be compiled with @samp{cl /MD} to do the same. If one of the other C runtime library choices provided by MS C is desired then the suggestion is to use the GMP string functions and confine I/O to the application. @item Motorola 68k CPU Types @cindex 68000 @samp{m68k} is taken to mean 68000. @samp{m68020} or higher will give a performance boost on applicable CPUs. @samp{m68360} can be used for CPU32 series chips. @samp{m68302} can be used for ``Dragonball'' series chips, though this is merely a synonym for @samp{m68000}. @item OpenBSD 2.6 @cindex OpenBSD @command{m4} in this release of OpenBSD has a bug in @code{eval} that makes it unsuitable for @file{.asm} file processing. @samp{./configure} will detect the problem and either abort or choose another m4 in the @env{PATH}. The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4. @item Power CPU Types @cindex Power/PowerPC In GMP, CPU types @samp{power*} and @samp{powerpc*} will each use instructions not available on the other, so it's important to choose the right one for the CPU that will be used. Currently GMP has no assembly code support for using just the common instruction subset. To get executables that run on both, the current suggestion is to use the generic C code (@option{--disable-assembly}), possibly with appropriate compiler options (like @samp{-mcpu=common} for @command{gcc}). CPU @samp{rs6000} (which is not a CPU but a family of workstations) is accepted by @file{config.sub}, but is currently equivalent to @option{--disable-assembly}. @item Sparc CPU Types @cindex Sparc @samp{sparcv8} or @samp{supersparc} on relevant systems will give a significant performance increase over the V7 code selected by plain @samp{sparc}. @item Sparc App Regs @cindex Sparc The GMP assembly code for both 32-bit and 64-bit Sparc clobbers the ``application registers'' @code{g2}, @code{g3} and @code{g4}, the same way that the GCC default @samp{-mapp-regs} does (@pxref{SPARC Options,, SPARC Options, gcc, Using the GNU Compiler Collection (GCC)}). This makes that code unsuitable for use with the special V9 @samp{-mcmodel=embmedany} (which uses @code{g4} as a data segment pointer), and for applications wanting to use those registers for special purposes. In these cases the only suggestion currently is to build GMP with @option{--disable-assembly} to avoid the assembly code. @item SunOS 4 @cindex SunOS @command{/usr/bin/m4} lacks various features needed to process @file{.asm} files, and instead @samp{./configure} will automatically use @command{/usr/5bin/m4}, which we believe is always available (if not then use GNU m4). @item x86 CPU Types @cindex x86 @cindex 80x86 @cindex i386 @samp{i586}, @samp{pentium} or @samp{pentiummmx} code is good for its intended P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II, P-III)@. @samp{i386} is a better choice when making binaries that must run on both. @item x86 MMX and SSE2 Code @cindex MMX @cindex SSE2 If the CPU selected has MMX code but the assembler doesn't support it, a warning is given and non-MMX code is used instead. This will be an inferior build, since the MMX code that's present is there because it's faster than the corresponding plain integer code. The same applies to SSE2. Old versions of @samp{gas} don't support MMX instructions, in particular version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent OpenBSD 3.1 doesn't. Solaris 2.6 and 2.7 @command{as} generate incorrect object code for register to register @code{movq} instructions, and so can't be used for MMX code. Install a recent @command{gas} if MMX code is wanted on these systems. @end table @need 2000 @node Known Build Problems, Performance optimization, Notes for Particular Systems, Installing GMP @section Known Build Problems @cindex Build problems known @c This section is more or less meant for known build problems that are not @c otherwise worked around and require some sort of manual intervention. You might find more up-to-date information at @uref{http://gmplib.org/}. @table @asis @item Compiler link options The version of libtool currently in use rather aggressively strips compiler options when linking a shared library. This will hopefully be relaxed in the future, but for now if this is a problem the suggestion is to create a little script to hide them, and for instance configure with @example ./configure CC=gcc-with-my-options @end example @item DJGPP (@samp{*-*-msdosdjgpp*}) @cindex DJGPP The DJGPP port of @command{bash} 2.03 is unable to run the @samp{configure} script, it exits silently, having died writing a preamble to @file{config.log}. Use @command{bash} 2.04 or higher. @samp{make all} was found to run out of memory during the final @file{libgmp.la} link on one system tested, despite having 64Mb available. Running @samp{make libgmp.la} directly helped, perhaps recursing into the various subdirectories uses up memory. @item GNU binutils @command{strip} prior to 2.12 @cindex Stripped libraries @cindex Binutils @command{strip} @cindex GNU @command{strip} @command{strip} from GNU binutils 2.11 and earlier should not be used on the static libraries @file{libgmp.a} and @file{libmp.a} since it will discard all but the last of multiple archive members with the same name, like the three versions of @file{init.o} in @file{libgmp.a}. Binutils 2.12 or higher can be used successfully. The shared libraries @file{libgmp.so} and @file{libmp.so} are not affected by this and any version of @command{strip} can be used on them. @item @command{make} syntax error @cindex SCO @cindex IRIX On certain versions of SCO OpenServer 5 and IRIX 6.5 the native @command{make} is unable to handle the long dependencies list for @file{libgmp.la}. The symptom is a ``syntax error'' on the following line of the top-level @file{Makefile}. @example libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES) @end example Either use GNU Make, or as a workaround remove @code{$(libgmp_la_DEPENDENCIES)} from that line (which will make the initial build work, but if any recompiling is done @file{libgmp.la} might not be rebuilt). @item MacOS X (@samp{*-*-darwin*}) @cindex MacOS X @cindex Darwin Libtool currently only knows how to create shared libraries on MacOS X using the native @command{cc} (which is a modified GCC), not a plain GCC@. A static-only build should work though (@samp{--disable-shared}). @item NeXT prior to 3.3 @cindex NeXT The system compiler on old versions of NeXT was a massacred and old GCC, even if it called itself @file{cc}. This compiler cannot be used to build GMP, you need to get a real GCC, and install that. (NeXT may have fixed this in release 3.3 of their system.) @item POWER and PowerPC @cindex Power/PowerPC Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP on POWER or PowerPC@. If you want to use GCC for these machines, get GCC 2.7.2.1 (or later). @item Sequent Symmetry @cindex Sequent Symmetry Use the GNU assembler instead of the system assembler, since the latter has serious bugs. @item Solaris 2.6 @cindex Solaris The system @command{sed} prints an error ``Output line too long'' when libtool builds @file{libgmp.la}. This doesn't seem to cause any obvious ill effects, but GNU @command{sed} is recommended, to avoid any doubt. @item Sparc Solaris 2.7 with gcc 2.95.2 in @samp{ABI=32} @cindex Solaris A shared library build of GMP seems to fail in this combination, it builds but then fails the tests, apparently due to some incorrect data relocations within @code{gmp_randinit_lc_2exp_size}. The exact cause is unknown, @samp{--disable-shared} is recommended. @end table @need 2000 @node Performance optimization, , Known Build Problems, Installing GMP @section Performance optimization @cindex Optimizing performance @c At some point, this should perhaps move to a separate chapter on optimizing @c performance. For optimal performance, build GMP for the exact CPU type of the target computer, see @ref{Build Options}. Unlike what is the case for most other programs, the compiler typically doesn't matter much, since GMP uses assembly language for the most critical operation. In particular for long-running GMP applications, and applications demanding extremely large numbers, building and running the @code{tuneup} program in the @file{tune} subdirectory, can be important. For example, @example cd tune make tuneup ./tuneup @end example will generate better contents for the @file{gmp-mparam.h} parameter file. To use the results, put the output in the file indicated in the @samp{Parameters for ...} header. Then recompile from scratch. The @code{tuneup} program takes one useful parameter, @samp{-f NNN}, which instructs the program how long to check FFT multiply parameters. If you're going to use GMP for extremely large numbers, you may want to run @code{tuneup} with a large NNN value. @node GMP Basics, Reporting Bugs, Installing GMP, Top @comment node-name, next, previous, up @chapter GMP Basics @cindex Basics @strong{Using functions, macros, data types, etc.@: not documented in this manual is strongly discouraged. If you do so your application is guaranteed to be incompatible with future versions of GMP.} @menu * Headers and Libraries:: * Nomenclature and Types:: * Function Classes:: * Variable Conventions:: * Parameter Conventions:: * Memory Management:: * Reentrancy:: * Useful Macros and Constants:: * Compatibility with older versions:: * Demonstration Programs:: * Efficiency:: * Debugging:: * Profiling:: * Autoconf:: * Emacs:: @end menu @node Headers and Libraries, Nomenclature and Types, GMP Basics, GMP Basics @section Headers and Libraries @cindex Headers @cindex @file{gmp.h} @cindex Include files @cindex @code{#include} All declarations needed to use GMP are collected in the include file @file{gmp.h}. It is designed to work with both C and C++ compilers. @example #include @end example @cindex @code{stdio.h} Note however that prototypes for GMP functions with @code{FILE *} parameters are only provided if @code{} is included too. @example #include #include @end example @cindex @code{stdarg.h} Likewise @code{} (or @code{}) is required for prototypes with @code{va_list} parameters, such as @code{gmp_vprintf}. And @code{} for prototypes with @code{struct obstack} parameters, such as @code{gmp_obstack_printf}, when available. @cindex Libraries @cindex Linking @cindex @code{libgmp} All programs using GMP must link against the @file{libgmp} library. On a typical Unix-like system this can be done with @samp{-lgmp}, for example @example gcc myprogram.c -lgmp @end example @cindex @code{libgmpxx} GMP C++ functions are in a separate @file{libgmpxx} library. This is built and installed if C++ support has been enabled (@pxref{Build Options}). For example, @example g++ mycxxprog.cc -lgmpxx -lgmp @end example @cindex Libtool GMP is built using Libtool and an application can use that to link if desired, @GMPpxreftop{libtool, GNU Libtool}. If GMP has been installed to a non-standard location then it may be necessary to use @samp{-I} and @samp{-L} compiler options to point to the right directories, and some sort of run-time path for a shared library. @node Nomenclature and Types, Function Classes, Headers and Libraries, GMP Basics @section Nomenclature and Types @cindex Nomenclature @cindex Types @cindex Integer @tindex @code{mpz_t} In this manual, @dfn{integer} usually means a multiple precision integer, as defined by the GMP library. The C data type for such integers is @code{mpz_t}. Here are some examples of how to declare such integers: @example mpz_t sum; struct foo @{ mpz_t x, y; @}; mpz_t vec[20]; @end example @cindex Rational number @tindex @code{mpq_t} @dfn{Rational number} means a multiple precision fraction. The C data type for these fractions is @code{mpq_t}. For example: @example mpq_t quotient; @end example @cindex Floating-point number @tindex @code{mpf_t} @dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision mantissa with a limited precision exponent. The C data type for such objects is @code{mpf_t}. For example: @example mpf_t fp; @end example @tindex @code{mp_exp_t} The floating point functions accept and return exponents in the C type @code{mp_exp_t}. Currently this is usually a @code{long}, but on some systems it's an @code{int} for efficiency. @cindex Limb @tindex @code{mp_limb_t} A @dfn{limb} means the part of a multi-precision number that fits in a single machine word. (We chose this word because a limb of the human body is analogous to a digit, only larger, and containing several digits.) Normally a limb is 32 or 64 bits. The C data type for a limb is @code{mp_limb_t}. @tindex @code{mp_size_t} Counts of limbs of a multi-precision number represented in the C type @code{mp_size_t}. Currently this is normally a @code{long}, but on some systems it's an @code{int} for efficiency, and on some systems it will be @code{long long} in the future. @tindex @code{mp_bitcnt_t} Counts of bits of a multi-precision number are represented in the C type @code{mp_bitcnt_t}. Currently this is always an @code{unsigned long}, but on some systems it will be an @code{unsigned long long} in the future. @cindex Random state @tindex @code{gmp_randstate_t} @dfn{Random state} means an algorithm selection and current state data. The C data type for such objects is @code{gmp_randstate_t}. For example: @example gmp_randstate_t rstate; @end example Also, in general @code{mp_bitcnt_t} is used for bit counts and ranges, and @code{size_t} is used for byte or character counts. @node Function Classes, Variable Conventions, Nomenclature and Types, GMP Basics @section Function Classes @cindex Function classes There are six classes of functions in the GMP library: @enumerate @item Functions for signed integer arithmetic, with names beginning with @code{mpz_}. The associated type is @code{mpz_t}. There are about 150 functions in this class. (@pxref{Integer Functions}) @item Functions for rational number arithmetic, with names beginning with @code{mpq_}. The associated type is @code{mpq_t}. There are about 40 functions in this class, but the integer functions can be used for arithmetic on the numerator and denominator separately. (@pxref{Rational Number Functions}) @item Functions for floating-point arithmetic, with names beginning with @code{mpf_}. The associated type is @code{mpf_t}. There are about 60 functions is this class. (@pxref{Floating-point Functions}) @item Fast low-level functions that operate on natural numbers. These are used by the functions in the preceding groups, and you can also call them directly from very time-critical user programs. These functions' names begin with @code{mpn_}. The associated type is array of @code{mp_limb_t}. There are about 30 (hard-to-use) functions in this class. (@pxref{Low-level Functions}) @item Miscellaneous functions. Functions for setting up custom allocation and functions for generating random numbers. (@pxref{Custom Allocation}, and @pxref{Random Number Functions}) @end enumerate @node Variable Conventions, Parameter Conventions, Function Classes, GMP Basics @section Variable Conventions @cindex Variable conventions @cindex Conventions for variables GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last. GMP lets you use the same variable for both input and output in one call. For example, the main function for integer multiplication, @code{mpz_mul}, can be used to square @code{x} and put the result back in @code{x} with @example mpz_mul (x, x, x); @end example Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details. A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid excessive initializing and clearing. In general, initialize near the start of a function and clear near the end. For example, @example void foo (void) @{ mpz_t n; int i; mpz_init (n); for (i = 1; i < 100; i++) @{ mpz_mul (n, @dots{}); mpz_fdiv_q (n, @dots{}); @dots{} @} mpz_clear (n); @} @end example @node Parameter Conventions, Memory Management, Variable Conventions, GMP Basics @section Parameter Conventions @cindex Parameter conventions @cindex Conventions for parameters When a GMP variable is used as a function parameter, it's effectively a call-by-reference, meaning if the function stores a value there it will change the original in the caller. Parameters which are input-only can be designated @code{const} to provoke a compiler error or warning on attempting to modify them. When a function is going to return a GMP result, it should designate a parameter that it sets, like the library functions do. More than one value can be returned by having more than one output parameter, again like the library functions. A @code{return} of an @code{mpz_t} etc doesn't return the object, only a pointer, and this is almost certainly not what's wanted. Here's an example accepting an @code{mpz_t} parameter, doing a calculation, and storing the result to the indicated parameter. @example void foo (mpz_t result, const mpz_t param, unsigned long n) @{ unsigned long i; mpz_mul_ui (result, param, n); for (i = 1; i < n; i++) mpz_add_ui (result, result, i*7); @} int main (void) @{ mpz_t r, n; mpz_init (r); mpz_init_set_str (n, "123456", 0); foo (r, n, 20L); gmp_printf ("%Zd\n", r); return 0; @} @end example @code{foo} works even if the mainline passes the same variable for @code{param} and @code{result}, just like the library functions. But sometimes it's tricky to make that work, and an application might not want to bother supporting that sort of thing. For interest, the GMP types @code{mpz_t} etc are implemented as one-element arrays of certain structures. This is why declaring a variable creates an object with the fields GMP needs, but then using it as a parameter passes a pointer to the object. Note that the actual fields in each @code{mpz_t} etc are for internal use only and should not be accessed directly by code that expects to be compatible with future GMP releases. @need 1000 @node Memory Management, Reentrancy, Parameter Conventions, GMP Basics @section Memory Management @cindex Memory management The GMP types like @code{mpz_t} are small, containing only a couple of sizes, and pointers to allocated data. Once a variable is initialized, GMP takes care of all space allocation. Additional space is allocated whenever a variable doesn't have enough. @code{mpz_t} and @code{mpq_t} variables never reduce their allocated space. Normally this is the best policy, since it avoids frequent reallocation. Applications that need to return memory to the heap at some particular point can use @code{mpz_realloc2}, or clear variables no longer needed. @code{mpf_t} variables, in the current implementation, use a fixed amount of space, determined by the chosen precision and allocated at initialization, so their size doesn't change. All memory is allocated using @code{malloc} and friends by default, but this can be changed, see @ref{Custom Allocation}. Temporary memory on the stack is also used (via @code{alloca}), but this can be changed at build-time if desired, see @ref{Build Options}. @node Reentrancy, Useful Macros and Constants, Memory Management, GMP Basics @section Reentrancy @cindex Reentrancy @cindex Thread safety @cindex Multi-threading @noindent GMP is reentrant and thread-safe, with some exceptions: @itemize @bullet @item If configured with @option{--enable-alloca=malloc-notreentrant} (or with @option{--enable-alloca=notreentrant} when @code{alloca} is not available), then naturally GMP is not reentrant. @item @code{mpf_set_default_prec} and @code{mpf_init} use a global variable for the selected precision. @code{mpf_init2} can be used instead, and in the C++ interface an explicit precision to the @code{mpf_class} constructor. @item @code{mpz_random} and the other old random number functions use a global random state and are hence not reentrant. The newer random number functions that accept a @code{gmp_randstate_t} parameter can be used instead. @item @code{gmp_randinit} (obsolete) returns an error indication through a global variable, which is not thread safe. Applications are advised to use @code{gmp_randinit_default} or @code{gmp_randinit_lc_2exp} instead. @item @code{mp_set_memory_functions} uses global variables to store the selected memory allocation functions. @item If the memory allocation functions set by a call to @code{mp_set_memory_functions} (or @code{malloc} and friends by default) are not reentrant, then GMP will not be reentrant either. @item If the standard I/O functions such as @code{fwrite} are not reentrant then the GMP I/O functions using them will not be reentrant either. @item It's safe for two threads to read from the same GMP variable simultaneously, but it's not safe for one to read while the another might be writing, nor for two threads to write simultaneously. It's not safe for two threads to generate a random number from the same @code{gmp_randstate_t} simultaneously, since this involves an update of that variable. @end itemize @need 2000 @node Useful Macros and Constants, Compatibility with older versions, Reentrancy, GMP Basics @section Useful Macros and Constants @cindex Useful macros and constants @cindex Constants @deftypevr {Global Constant} {const int} mp_bits_per_limb @findex mp_bits_per_limb @cindex Bits per limb @cindex Limb size The number of bits per limb. @end deftypevr @defmac __GNU_MP_VERSION @defmacx __GNU_MP_VERSION_MINOR @defmacx __GNU_MP_VERSION_PATCHLEVEL @cindex Version number @cindex GMP version number The major and minor GMP version, and patch level, respectively, as integers. For GMP i.j, these numbers will be i, j, and 0, respectively. For GMP i.j.k, these numbers will be i, j, and k, respectively. @end defmac @deftypevr {Global Constant} {const char * const} gmp_version @findex gmp_version The GMP version number, as a null-terminated string, in the form ``i.j.k''. This release is @nicode{"@value{VERSION}"}. Note that the format ``i.j'' was used, before version 4.3.0, when k was zero. @end deftypevr @defmac __GMP_CC @defmacx __GMP_CFLAGS The compiler and compiler flags, respectively, used when compiling GMP, as strings. @end defmac @node Compatibility with older versions, Demonstration Programs, Useful Macros and Constants, GMP Basics @section Compatibility with older versions @cindex Compatibility with older versions @cindex Past GMP versions @cindex Upward compatibility This version of GMP is upwardly binary compatible with all 5.x, 4.x, and 3.x versions, and upwardly compatible at the source level with all 2.x versions, with the following exceptions. @itemize @bullet @item @code{mpn_gcd} had its source arguments swapped as of GMP 3.0, for consistency with other @code{mpn} functions. @item @code{mpf_get_prec} counted precision slightly differently in GMP 3.0 and 3.0.1, but in 3.1 reverted to the 2.x style. @item @code{mpn_bdivmod}, documented as preliminary in GMP 4, has been removed. @end itemize There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 5. Please see the GMP 2 manual for details. @c @item Integer division functions round the result differently. The obsolete @c functions (@code{mpz_div}, @code{mpz_divmod}, @code{mpz_mdiv}, @c @code{mpz_mdivmod}, etc) now all use floor rounding (i.e., they round the @c quotient towards @c @ifinfo @c @minus{}infinity). @c @end ifinfo @c @iftex @c @tex @c $-\infty$). @c @end tex @c @end iftex @c There are a lot of functions for integer division, giving the user better @c control over the rounding. @c @item The function @code{mpz_mod} now compute the true @strong{mod} function. @c @item The functions @code{mpz_powm} and @code{mpz_powm_ui} now use @c @strong{mod} for reduction. @c @item The assignment functions for rational numbers do no longer canonicalize @c their results. In the case a non-canonical result could arise from an @c assignment, the user need to insert an explicit call to @c @code{mpq_canonicalize}. This change was made for efficiency. @c @item Output generated by @code{mpz_out_raw} in this release cannot be read @c by @code{mpz_inp_raw} in previous releases. This change was made for making @c the file format truly portable between machines with different word sizes. @c @item Several @code{mpn} functions have changed. But they were intentionally @c undocumented in previous releases. @c @item The functions @code{mpz_cmp_ui}, @code{mpz_cmp_si}, and @code{mpq_cmp_ui} @c are now implemented as macros, and thereby sometimes evaluate their @c arguments multiple times. @c @item The functions @code{mpz_pow_ui} and @code{mpz_ui_pow_ui} now yield 1 @c for 0^0. (In version 1, they yielded 0.) @c In version 1 of the library, @code{mpq_set_den} handled negative @c denominators by copying the sign to the numerator. That is no longer done. @c Pure assignment functions do not canonicalize the assigned variable. It is @c the responsibility of the user to canonicalize the assigned variable before @c any arithmetic operations are performed on that variable. @c Note that this is an incompatible change from version 1 of the library. @c @end enumerate @need 1000 @node Demonstration Programs, Efficiency, Compatibility with older versions, GMP Basics @section Demonstration programs @cindex Demonstration programs @cindex Example programs @cindex Sample programs The @file{demos} subdirectory has some sample programs using GMP@. These aren't built or installed, but there's a @file{Makefile} with rules for them. For instance, @example make pexpr ./pexpr 68^975+10 @end example @noindent The following programs are provided @itemize @bullet @item @cindex Expression parsing demo @cindex Parsing expressions demo @samp{pexpr} is an expression evaluator, the program used on the GMP web page. @item @cindex Expression parsing demo @cindex Parsing expressions demo The @samp{calc} subdirectory has a similar but simpler evaluator using @command{lex} and @command{yacc}. @item @cindex Expression parsing demo @cindex Parsing expressions demo The @samp{expr} subdirectory is yet another expression evaluator, a library designed for ease of use within a C program. See @file{demos/expr/README} for more information. @item @cindex Factorization demo @samp{factorize} is a Pollard-Rho factorization program. @item @samp{isprime} is a command-line interface to the @code{mpz_probab_prime_p} function. @item @samp{primes} counts or lists primes in an interval, using a sieve. @item @samp{qcn} is an example use of @code{mpz_kronecker_ui} to estimate quadratic class numbers. @item @cindex @code{perl} @cindex GMP Perl module @cindex Perl module The @samp{perl} subdirectory is a comprehensive perl interface to GMP@. See @file{demos/perl/INSTALL} for more information. Documentation is in POD format in @file{demos/perl/GMP.pm}. @end itemize As an aside, consideration has been given at various times to some sort of expression evaluation within the main GMP library. Going beyond something minimal quickly leads to matters like user-defined functions, looping, fixnums for control variables, etc, which are considered outside the scope of GMP (much closer to language interpreters or compilers, @xref{Language Bindings}.) Something simple for program input convenience may yet be a possibility, a combination of the @file{expr} demo and the @file{pexpr} tree back-end perhaps. But for now the above evaluators are offered as illustrations. @need 1000 @node Efficiency, Debugging, Demonstration Programs, GMP Basics @section Efficiency @cindex Efficiency @table @asis @item Small Operands @cindex Small operands On small operands, the time for function call overheads and memory allocation can be significant in comparison to actual calculation. This is unavoidable in a general purpose variable precision library, although GMP attempts to be as efficient as it can on both large and small operands. @item Static Linking @cindex Static linking On some CPUs, in particular the x86s, the static @file{libgmp.a} should be used for maximum speed, since the PIC code in the shared @file{libgmp.so} will have a small overhead on each function call and global data address. For many programs this will be insignificant, but for long calculations there's a gain to be had. @item Initializing and Clearing @cindex Initializing and clearing Avoid excessive initializing and clearing of variables, since this can be quite time consuming, especially in comparison to otherwise fast operations like addition. A language interpreter might want to keep a free list or stack of initialized variables ready for use. It should be possible to integrate something like that with a garbage collector too. @item Reallocations @cindex Reallocations An @code{mpz_t} or @code{mpq_t} variable used to hold successively increasing values will have its memory repeatedly @code{realloc}ed, which could be quite slow or could fragment memory, depending on the C library. If an application can estimate the final size then @code{mpz_init2} or @code{mpz_realloc2} can be called to allocate the necessary space from the beginning (@pxref{Initializing Integers}). It doesn't matter if a size set with @code{mpz_init2} or @code{mpz_realloc2} is too small, since all functions will do a further reallocation if necessary. Badly overestimating memory required will waste space though. @item @code{2exp} Functions @cindex @code{2exp} functions It's up to an application to call functions like @code{mpz_mul_2exp} when appropriate. General purpose functions like @code{mpz_mul} make no attempt to identify powers of two or other special forms, because such inputs will usually be very rare and testing every time would be wasteful. @item @code{ui} and @code{si} Functions @cindex @code{ui} and @code{si} functions The @code{ui} functions and the small number of @code{si} functions exist for convenience and should be used where applicable. But if for example an @code{mpz_t} contains a value that fits in an @code{unsigned long} there's no need extract it and call a @code{ui} function, just use the regular @code{mpz} function. @item In-Place Operations @cindex In-place operations @code{mpz_abs}, @code{mpq_abs}, @code{mpf_abs}, @code{mpz_neg}, @code{mpq_neg} and @code{mpf_neg} are fast when used for in-place operations like @code{mpz_abs(x,x)}, since in the current implementation only a single field of @code{x} needs changing. On suitable compilers (GCC for instance) this is inlined too. @code{mpz_add_ui}, @code{mpz_sub_ui}, @code{mpf_add_ui} and @code{mpf_sub_ui} benefit from an in-place operation like @code{mpz_add_ui(x,x,y)}, since usually only one or two limbs of @code{x} will need to be changed. The same applies to the full precision @code{mpz_add} etc if @code{y} is small. If @code{y} is big then cache locality may be helped, but that's all. @code{mpz_mul} is currently the opposite, a separate destination is slightly better. A call like @code{mpz_mul(x,x,y)} will, unless @code{y} is only one limb, make a temporary copy of @code{x} before forming the result. Normally that copying will only be a tiny fraction of the time for the multiply, so this is not a particularly important consideration. @code{mpz_set}, @code{mpq_set}, @code{mpq_set_num}, @code{mpf_set}, etc, make no attempt to recognise a copy of something to itself, so a call like @code{mpz_set(x,x)} will be wasteful. Naturally that would never be written deliberately, but if it might arise from two pointers to the same object then a test to avoid it might be desirable. @example if (x != y) mpz_set (x, y); @end example Note that it's never worth introducing extra @code{mpz_set} calls just to get in-place operations. If a result should go to a particular variable then just direct it there and let GMP take care of data movement. @item Divisibility Testing (Small Integers) @cindex Divisibility testing @code{mpz_divisible_ui_p} and @code{mpz_congruent_ui_p} are the best functions for testing whether an @code{mpz_t} is divisible by an individual small integer. They use an algorithm which is faster than @code{mpz_tdiv_ui}, but which gives no useful information about the actual remainder, only whether it's zero (or a particular value). However when testing divisibility by several small integers, it's best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo @math{23@times{}29@times{}31 = 20677} and then test that. The division functions like @code{mpz_tdiv_q_ui} which give a quotient as well as a remainder are generally a little slower than the remainder-only functions like @code{mpz_tdiv_ui}. If the quotient is only rarely wanted then it's probably best to just take a remainder and then go back and calculate the quotient if and when it's wanted (@code{mpz_divexact_ui} can be used if the remainder is zero). @item Rational Arithmetic @cindex Rational arithmetic The @code{mpq} functions operate on @code{mpq_t} values with no common factors in the numerator and denominator. Common factors are checked-for and cast out as necessary. In general, cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations. However, applications that know something about the factorization of the values they're working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it's enough to check for factors of it in the denominator instead of doing a full GCD@. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end. The @code{mpq_numref} and @code{mpq_denref} macros give access to the numerator and denominator to do things outside the scope of the supplied @code{mpq} functions. @xref{Applying Integer Functions}. The canonical form for rationals allows mixed-type @code{mpq_t} and integer additions or subtractions to be done directly with multiples of the denominator. This will be somewhat faster than @code{mpq_add}. For example, @example /* mpq increment */ mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q)); /* mpq += unsigned long */ mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL); /* mpq -= mpz */ mpz_submul (mpq_numref(q), mpq_denref(q), z); @end example @item Number Sequences @cindex Number sequences Functions like @code{mpz_fac_ui}, @code{mpz_fib_ui} and @code{mpz_bin_uiui} are designed for calculating isolated values. If a range of values is wanted it's probably best to call to get a starting point and iterate from there. @item Text Input/Output @cindex Text input/output Hexadecimal or octal are suggested for input or output in text form. Power-of-2 bases like these can be converted much more efficiently than other bases, like decimal. For big numbers there's usually nothing of particular interest to be seen in the digits, so the base doesn't matter much. Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers. @c Reference: Knuth volume 2 section 4.1, page 184 of second edition. :-) @end table @node Debugging, Profiling, Efficiency, GMP Basics @section Debugging @cindex Debugging @table @asis @item Stack Overflow @cindex Stack overflow @cindex Segmentation violation @cindex Bus error Depending on the system, a segmentation violation or bus error might be the only indication of stack overflow. See @samp{--enable-alloca} choices in @ref{Build Options}, for how to address this. In new enough versions of GCC, @samp{-fstack-check} may be able to ensure an overflow is recognised by the system before too much damage is done, or @samp{-fstack-limit-symbol} or @samp{-fstack-limit-register} may be able to add checking if the system itself doesn't do any (@pxref{Code Gen Options,, Options for Code Generation, gcc, Using the GNU Compiler Collection (GCC)}). These options must be added to the @samp{CFLAGS} used in the GMP build (@pxref{Build Options}), adding them just to an application will have no effect. Note also they're a slowdown, adding overhead to each function call and each stack allocation. @item Heap Problems @cindex Heap problems @cindex Malloc problems The most likely cause of application problems with GMP is heap corruption. Failing to @code{init} GMP variables will have unpredictable effects, and corruption arising elsewhere in a program may well affect GMP@. Initializing GMP variables more than once or failing to clear them will cause memory leaks. @cindex Malloc debugger In all such cases a @code{malloc} debugger is recommended. On a GNU or BSD system the standard C library @code{malloc} has some diagnostic facilities, see @ref{Allocation Debugging,, Allocation Debugging, libc, The GNU C Library Reference Manual}, or @samp{man 3 malloc}. Other possibilities, in no particular order, include @display @uref{http://www.inf.ethz.ch/personal/biere/projects/ccmalloc/} @uref{http://dmalloc.com/} @uref{http://www.perens.com/FreeSoftware/} @ (electric fence) @uref{http://packages.debian.org/stable/devel/fda} @uref{http://www.gnupdate.org/components/leakbug/} @uref{http://people.redhat.com/~otaylor/memprof/} @uref{http://www.cbmamiga.demon.co.uk/mpatrol/} @end display The GMP default allocation routines in @file{memory.c} also have a simple sentinel scheme which can be enabled with @code{#define DEBUG} in that file. This is mainly designed for detecting buffer overruns during GMP development, but might find other uses. @item Stack Backtraces @cindex Stack backtrace On some systems the compiler options GMP uses by default can interfere with debugging. In particular on x86 and 68k systems @samp{-fomit-frame-pointer} is used and this generally inhibits stack backtracing. Recompiling without such options may help while debugging, though the usual caveats about it potentially moving a memory problem or hiding a compiler bug will apply. @item GDB, the GNU Debugger @cindex GDB @cindex GNU Debugger A sample @file{.gdbinit} is included in the distribution, showing how to call some undocumented dump functions to print GMP variables from within GDB@. Note that these functions shouldn't be used in final application code since they're undocumented and may be subject to incompatible changes in future versions of GMP. @item Source File Paths GMP has multiple source files with the same name, in different directories. For example @file{mpz}, @file{mpq} and @file{mpf} each have an @file{init.c}. If the debugger can't already determine the right one it may help to build with absolute paths on each C file. One way to do that is to use a separate object directory with an absolute path to the source directory. @example cd /my/build/dir /my/source/dir/gmp-@value{VERSION}/configure @end example This works via @code{VPATH}, and might require GNU @command{make}. Alternately it might be possible to change the @code{.c.lo} rules appropriately. @item Assertion Checking @cindex Assertion checking The build option @option{--enable-assert} is available to add some consistency checks to the library (see @ref{Build Options}). These are likely to be of limited value to most applications. Assertion failures are just as likely to indicate memory corruption as a library or compiler bug. Applications using the low-level @code{mpn} functions, however, will benefit from @option{--enable-assert} since it adds checks on the parameters of most such functions, many of which have subtle restrictions on their usage. Note however that only the generic C code has checks, not the assembly code, so @option{--disable-assembly} should be used for maximum checking. @item Temporary Memory Checking The build option @option{--enable-alloca=debug} arranges that each block of temporary memory in GMP is allocated with a separate call to @code{malloc} (or the allocation function set with @code{mp_set_memory_functions}). This can help a malloc debugger detect accesses outside the intended bounds, or detect memory not released. In a normal build, on the other hand, temporary memory is allocated in blocks which GMP divides up for its own use, or may be allocated with a compiler builtin @code{alloca} which will go nowhere near any malloc debugger hooks. @item Maximum Debuggability To summarize the above, a GMP build for maximum debuggability would be @example ./configure --disable-shared --enable-assert \ --enable-alloca=debug --disable-assembly CFLAGS=-g @end example For C++, add @samp{--enable-cxx CXXFLAGS=-g}. @item Checker @cindex Checker @cindex GCC Checker The GCC checker (@uref{http://savannah.nongnu.org/projects/checker/}) can be used with GMP@. It contains a stub library which means GMP applications compiled with checker can use a normal GMP build. A build of GMP with checking within GMP itself can be made. This will run very very slowly. On GNU/Linux for example, @cindex @command{checkergcc} @example ./configure --disable-assembly CC=checkergcc @end example @option{--disable-assembly} must be used, since the GMP assembly code doesn't support the checking scheme. The GMP C++ features cannot be used, since current versions of checker (0.9.9.1) don't yet support the standard C++ library. @item Valgrind @cindex Valgrind The valgrind program (@uref{http://valgrind.org/}) is a memory checker for x86s. It translates and emulates machine instructions to do strong checks for uninitialized data (at the level of individual bits), memory accesses through bad pointers, and memory leaks. Recent versions of Valgrind are getting support for MMX and SSE/SSE2 instructions, for past versions GMP will need to be configured not to use those, i.e.@: for an x86 without them (for instance plain @samp{i486}). GMP's assembly code sometimes promotes a read of the limbs to some larger size, for efficiency. GMP will do this even at the start and end of a multilimb operand, using naturaly aligned operations on the larger type. This may lead to benign reads outside of allocated areas, triggering complants from Valgrind. @item Other Problems Any suspected bug in GMP itself should be isolated to make sure it's not an application problem, see @ref{Reporting Bugs}. @end table @node Profiling, Autoconf, Debugging, GMP Basics @section Profiling @cindex Profiling @cindex Execution profiling @cindex @code{--enable-profiling} Running a program under a profiler is a good way to find where it's spending most time and where improvements can be best sought. The profiling choices for a GMP build are as follows. @table @asis @item @samp{--disable-profiling} The default is to add nothing special for profiling. It should be possible to just compile the mainline of a program with @code{-p} and use @command{prof} to get a profile consisting of timer-based sampling of the program counter. Most of the GMP assembly code has the necessary symbol information. This approach has the advantage of minimizing interference with normal program operation, but on most systems the resolution of the sampling is quite low (10 milliseconds for instance), requiring long runs to get accurate information. @item @samp{--enable-profiling=prof} @cindex @code{prof} Build with support for the system @command{prof}, which means @samp{-p} added to the @samp{CFLAGS}. This provides call counting in addition to program counter sampling, which allows the most frequently called routines to be identified, and an average time spent in each routine to be determined. The x86 assembly code has support for this option, but on other processors the assembly routines will be as if compiled without @samp{-p} and therefore won't appear in the call counts. On some systems, such as GNU/Linux, @samp{-p} in fact means @samp{-pg} and in this case @samp{--enable-profiling=gprof} described below should be used instead. @item @samp{--enable-profiling=gprof} @cindex @code{gprof} Build with support for @command{gprof}, which means @samp{-pg} added to the @samp{CFLAGS}. This provides call graph construction in addition to call counting and program counter sampling, which makes it possible to count calls coming from different locations. For example the number of calls to @code{mpn_mul} from @code{mpz_mul} versus the number from @code{mpf_mul}. The program counter sampling is still flat though, so only a total time in @code{mpn_mul} would be accumulated, not a separate amount for each call site. The x86 assembly code has support for this option, but on other processors the assembly routines will be as if compiled without @samp{-pg} and therefore not be included in the call counts. On x86 and m68k systems @samp{-pg} and @samp{-fomit-frame-pointer} are incompatible, so the latter is omitted from the default flags in that case, which might result in poorer code generation. Incidentally, it should be possible to use the @command{gprof} program with a plain @samp{--enable-profiling=prof} build. But in that case only the @samp{gprof -p} flat profile and call counts can be expected to be valid, not the @samp{gprof -q} call graph. @item @samp{--enable-profiling=instrument} @cindex @code{-finstrument-functions} @cindex @code{instrument-functions} Build with the GCC option @samp{-finstrument-functions} added to the @samp{CFLAGS} (@pxref{Code Gen Options,, Options for Code Generation, gcc, Using the GNU Compiler Collection (GCC)}). This inserts special instrumenting calls at the start and end of each function, allowing exact timing and full call graph construction. This instrumenting is not normally a standard system feature and will require support from an external library, such as @cindex FunctionCheck @cindex fnccheck @display @uref{http://sourceforge.net/projects/fnccheck/} @end display This should be included in @samp{LIBS} during the GMP configure so that test programs will link. For example, @example ./configure --enable-profiling=instrument LIBS=-lfc @end example On a GNU system the C library provides dummy instrumenting functions, so programs compiled with this option will link. In this case it's only necessary to ensure the correct library is added when linking an application. The x86 assembly code supports this option, but on other processors the assembly routines will be as if compiled without @samp{-finstrument-functions} meaning time spent in them will effectively be attributed to their caller. @end table @node Autoconf, Emacs, Profiling, GMP Basics @section Autoconf @cindex Autoconf Autoconf based applications can easily check whether GMP is installed. The only thing to be noted is that GMP library symbols from version 3 onwards have prefixes like @code{__gmpz}. The following therefore would be a simple test, @cindex @code{AC_CHECK_LIB} @example AC_CHECK_LIB(gmp, __gmpz_init) @end example This just uses the default @code{AC_CHECK_LIB} actions for found or not found, but an application that must have GMP would want to generate an error if not found. For example, @example AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR([GNU MP not found, see http://gmplib.org/])]) @end example If functions added in some particular version of GMP are required, then one of those can be used when checking. For example @code{mpz_mul_si} was added in GMP 3.1, @example AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR( [GNU MP not found, or not 3.1 or up, see http://gmplib.org/])]) @end example An alternative would be to test the version number in @file{gmp.h} using say @code{AC_EGREP_CPP}. That would make it possible to test the exact version, if some particular sub-minor release is known to be necessary. In general it's recommended that applications should simply demand a new enough GMP rather than trying to provide supplements for features not available in past versions. Occasionally an application will need or want to know the size of a type at configuration or preprocessing time, not just with @code{sizeof} in the code. This can be done in the normal way with @code{mp_limb_t} etc, but GMP 4.0 or up is best for this, since prior versions needed certain @samp{-D} defines on systems using a @code{long long} limb. The following would suit Autoconf 2.50 or up, @example AC_CHECK_SIZEOF(mp_limb_t, , [#include ]) @end example @node Emacs, , Autoconf, GMP Basics @section Emacs @cindex Emacs @cindex @code{info-lookup-symbol} @key{C-h C-i} (@code{info-lookup-symbol}) is a good way to find documentation on C functions while editing (@pxref{Info Lookup, , Info Documentation Lookup, emacs, The Emacs Editor}). The GMP manual can be included in such lookups by putting the following in your @file{.emacs}, @c This isn't pretty, but there doesn't seem to be a better way (in emacs @c 21.2 at least). info-lookup->mode-value could be used for the "assoc"s, @c but that function isn't documented, whereas info-lookup-alist is. @c @example (eval-after-load "info-look" '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist)))) (setcar (nthcdr 3 mode-value) (cons '("(gmp)Function Index" nil "^ -.* " "\\>") (nth 3 mode-value))))) @end example @node Reporting Bugs, Integer Functions, GMP Basics, Top @comment node-name, next, previous, up @chapter Reporting Bugs @cindex Reporting bugs @cindex Bug reporting If you think you have found a bug in the GMP library, please investigate it and report it. We have made this library available to you, and it is not too much to ask you to report the bugs you find. Before you report a bug, check it's not already addressed in @ref{Known Build Problems}, or perhaps @ref{Notes for Particular Systems}. You may also want to check @uref{http://gmplib.org/} for patches for this release. Please include the following in any report, @itemize @bullet @item The GMP version number, and if pre-packaged or patched then say so. @item A test program that makes it possible for us to reproduce the bug. Include instructions on how to run the program. @item A description of what is wrong. If the results are incorrect, in what way. If you get a crash, say so. @item If you get a crash, include a stack backtrace from the debugger if it's informative (@samp{where} in @command{gdb}, or @samp{$C} in @command{adb}). @item Please do not send core dumps, executables or @command{strace}s. @item The configuration options you used when building GMP, if any. @item The name of the compiler and its version. For @command{gcc}, get the version with @samp{gcc -v}, otherwise perhaps @samp{what `which cc`}, or similar. @item The output from running @samp{uname -a}. @item The output from running @samp{./config.guess}, and from running @samp{./configfsf.guess} (might be the same). @item If the bug is related to @samp{configure}, then the compressed contents of @file{config.log}. @item If the bug is related to an @file{asm} file not assembling, then the contents of @file{config.m4} and the offending line or lines from the temporary @file{mpn/tmp-.s}. @end itemize Please make an effort to produce a self-contained report, with something definite that can be tested or debugged. Vague queries or piecemeal messages are difficult to act on and don't help the development effort. It is not uncommon that an observed problem is actually due to a bug in the compiler; the GMP code tends to explore interesting corners in compilers. If your bug report is good, we will do our best to help you get a corrected version of the library; if the bug report is poor, we won't do anything about it (except maybe ask you to send a better report). Send your report to: @email{gmp-bugs@@gmplib.org}. If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address. @node Integer Functions, Rational Number Functions, Reporting Bugs, Top @comment node-name, next, previous, up @chapter Integer Functions @cindex Integer functions This chapter describes the GMP functions for performing integer arithmetic. These functions start with the prefix @code{mpz_}. GMP integers are stored in objects of type @code{mpz_t}. @menu * Initializing Integers:: * Assigning Integers:: * Simultaneous Integer Init & Assign:: * Converting Integers:: * Integer Arithmetic:: * Integer Division:: * Integer Exponentiation:: * Integer Roots:: * Number Theoretic Functions:: * Integer Comparisons:: * Integer Logic and Bit Fiddling:: * I/O of Integers:: * Integer Random Numbers:: * Integer Import and Export:: * Miscellaneous Integer Functions:: * Integer Special Functions:: @end menu @node Initializing Integers, Assigning Integers, Integer Functions, Integer Functions @comment node-name, next, previous, up @section Initialization Functions @cindex Integer initialization functions @cindex Initialization functions The functions for integer arithmetic assume that all integer objects are initialized. You do that by calling the function @code{mpz_init}. For example, @example @{ mpz_t integ; mpz_init (integ); @dots{} mpz_add (integ, @dots{}); @dots{} mpz_sub (integ, @dots{}); /* Unless the program is about to exit, do ... */ mpz_clear (integ); @} @end example As you can see, you can store new values any number of times, once an object is initialized. @deftypefun void mpz_init (mpz_t @var{x}) Initialize @var{x}, and set its value to 0. @end deftypefun @deftypefun void mpz_inits (mpz_t @var{x}, ...) Initialize a NULL-terminated list of @code{mpz_t} variables, and set their values to 0. @end deftypefun @deftypefun void mpz_init2 (mpz_t @var{x}, mp_bitcnt_t @var{n}) Initialize @var{x}, with space for @var{n}-bit numbers, and set its value to 0. Calling this function instead of @code{mpz_init} or @code{mpz_inits} is never necessary; reallocation is handled automatically by GMP when needed. While @var{n} defines the initial space, @var{x} will grow automatically in the normal way, if necessary, for subsequent values stored. @code{mpz_init2} makes it possible to avoid such reallocations if a maximum size is known in advance. In preparation for an operation, GMP often allocates one limb more than ultimately needed. To make sure GMP will not perform reallocation for @var{x}, you need to add the number of bits in @code{mp_limb_t} to @var{n}. @end deftypefun @deftypefun void mpz_clear (mpz_t @var{x}) Free the space occupied by @var{x}. Call this function for all @code{mpz_t} variables when you are done with them. @end deftypefun @deftypefun void mpz_clears (mpz_t @var{x}, ...) Free the space occupied by a NULL-terminated list of @code{mpz_t} variables. @end deftypefun @deftypefun void mpz_realloc2 (mpz_t @var{x}, mp_bitcnt_t @var{n}) Change the space allocated for @var{x} to @var{n} bits. The value in @var{x} is preserved if it fits, or is set to 0 if not. Calling this function is never necessary; reallocation is handled automatically by GMP when needed. But this function can be used to increase the space for a variable in order to avoid repeated automatic reallocations, or to decrease it to give memory back to the heap. @end deftypefun @node Assigning Integers, Simultaneous Integer Init & Assign, Initializing Integers, Integer Functions @comment node-name, next, previous, up @section Assignment Functions @cindex Integer assignment functions @cindex Assignment functions These functions assign new values to already initialized integers (@pxref{Initializing Integers}). @deftypefun void mpz_set (mpz_t @var{rop}, mpz_t @var{op}) @deftypefunx void mpz_set_ui (mpz_t @var{rop}, unsigned long int @var{op}) @deftypefunx void mpz_set_si (mpz_t @var{rop}, signed long int @var{op}) @deftypefunx void mpz_set_d (mpz_t @var{rop}, double @var{op}) @deftypefunx void mpz_set_q (mpz_t @var{rop}, mpq_t @var{op}) @deftypefunx void mpz_set_f (mpz_t @var{rop}, mpf_t @var{op}) Set the value of @var{rop} from @var{op}. @code{mpz_set_d}, @code{mpz_set_q} and @code{mpz_set_f} truncate @var{op} to make it an integer. @end deftypefun @deftypefun int mpz_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base}) Set the value of @var{rop} from @var{str}, a null-terminated C string in base @var{base}. White space is allowed in the string, and is simply ignored. The @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and @code{0B} for binary, @code{0} for octal, or decimal otherwise. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value. For bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. This function returns 0 if the entire string is a valid number in base @var{base}. Otherwise it returns @minus{}1. @c @c It turns out that it is not entirely true that this function ignores @c white-space. It does ignore it between digits, but not after a minus sign @c or within or after ``0x''. Some thought was given to disallowing all @c whitespace, but that would be an incompatible change, whitespace has been @c documented as ignored ever since GMP 1. @c @end deftypefun @deftypefun void mpz_swap (mpz_t @var{rop1}, mpz_t @var{rop2}) Swap the values @var{rop1} and @var{rop2} efficiently. @end deftypefun @node Simultaneous Integer Init & Assign, Converting Integers, Assigning Integers, Integer Functions @comment node-name, next, previous, up @section Combined Initialization and Assignment Functions @cindex Integer assignment functions @cindex Assignment functions @cindex Integer initialization functions @cindex Initialization functions For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions' names have the form @code{mpz_init_set@dots{}} Here is an example of using one: @example @{ mpz_t pie; mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10); @dots{} mpz_sub (pie, @dots{}); @dots{} mpz_clear (pie); @} @end example @noindent Once the integer has been initialized by any of the @code{mpz_init_set@dots{}} functions, it can be used as the source or destination operand for the ordinary integer functions. Don't use an initialize-and-set function on a variable already initialized! @deftypefun void mpz_init_set (mpz_t @var{rop}, mpz_t @var{op}) @deftypefunx void mpz_init_set_ui (mpz_t @var{rop}, unsigned long int @var{op}) @deftypefunx void mpz_init_set_si (mpz_t @var{rop}, signed long int @var{op}) @deftypefunx void mpz_init_set_d (mpz_t @var{rop}, double @var{op}) Initialize @var{rop} with limb space and set the initial numeric value from @var{op}. @end deftypefun @deftypefun int mpz_init_set_str (mpz_t @var{rop}, char *@var{str}, int @var{base}) Initialize @var{rop} and set its value like @code{mpz_set_str} (see its documentation above for details). If the string is a correct base @var{base} number, the function returns 0; if an error occurs it returns @minus{}1. @var{rop} is initialized even if an error occurs. (I.e., you have to call @code{mpz_clear} for it.) @end deftypefun @node Converting Integers, Integer Arithmetic, Simultaneous Integer Init & Assign, Integer Functions @comment node-name, next, previous, up @section Conversion Functions @cindex Integer conversion functions @cindex Conversion functions This section describes functions for converting GMP integers to standard C types. Functions for converting @emph{to} GMP integers are described in @ref{Assigning Integers} and @ref{I/O of Integers}. @deftypefun {unsigned long int} mpz_get_ui (mpz_t @var{op}) Return the value of @var{op} as an @code{unsigned long}. If @var{op} is too big to fit an @code{unsigned long} then just the least significant bits that do fit are returned. The sign of @var{op} is ignored, only the absolute value is used. @end deftypefun @deftypefun {signed long int} mpz_get_si (mpz_t @var{op}) If @var{op} fits into a @code{signed long int} return the value of @var{op}. Otherwise return the least significant part of @var{op}, with the same sign as @var{op}. If @var{op} is too big to fit in a @code{signed long int}, the returned result is probably not very useful. To find out if the value will fit, use the function @code{mpz_fits_slong_p}. @end deftypefun @deftypefun double mpz_get_d (mpz_t @var{op}) Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding towards zero). If the exponent from the conversion is too big, the result is system dependent. An infinity is returned where available. A hardware overflow trap may or may not occur. @end deftypefun @deftypefun double mpz_get_d_2exp (signed long int *@var{exp}, mpz_t @var{op}) Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding towards zero), and returning the exponent separately. The return value is in the range @math{0.5@le{}@GMPabs{@var{d}}<1} and the exponent is stored to @code{*@var{exp}}. @m{@var{d} * 2^{exp}, @var{d} * 2^@var{exp}} is the (truncated) @var{op} value. If @var{op} is zero, the return is @math{0.0} and 0 is stored to @code{*@var{exp}}. @cindex @code{frexp} This is similar to the standard C @code{frexp} function (@pxref{Normalization Functions,,, libc, The GNU C Library Reference Manual}). @end deftypefun @deftypefun {char *} mpz_get_str (char *@var{str}, int @var{base}, mpz_t @var{op}) Convert @var{op} to a string of digits in base @var{base}. The base argument may vary from 2 to 62 or from @minus{}2 to @minus{}36. For @var{base} in the range 2..36, digits and lower-case letters are used; for @minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. If @var{str} is @code{NULL}, the result string is allocated using the current allocation function (@pxref{Custom Allocation}). The block will be @code{strlen(str)+1} bytes, that being exactly enough for the string and null-terminator. If @var{str} is not @code{NULL}, it should point to a block of storage large enough for the result, that being @code{mpz_sizeinbase (@var{op}, @var{base}) + 2}. The two extra bytes are for a possible minus sign, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given @var{str}. @end deftypefun @need 2000 @node Integer Arithmetic, Integer Division, Converting Integers, Integer Functions @comment node-name, next, previous, up @section Arithmetic Functions @cindex Integer arithmetic functions @cindex Arithmetic functions @deftypefun void mpz_add (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @deftypefunx void mpz_add_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2}) Set @var{rop} to @math{@var{op1} + @var{op2}}. @end deftypefun @deftypefun void mpz_sub (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @deftypefunx void mpz_sub_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2}) @deftypefunx void mpz_ui_sub (mpz_t @var{rop}, unsigned long int @var{op1}, mpz_t @var{op2}) Set @var{rop} to @var{op1} @minus{} @var{op2}. @end deftypefun @deftypefun void mpz_mul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @deftypefunx void mpz_mul_si (mpz_t @var{rop}, mpz_t @var{op1}, long int @var{op2}) @deftypefunx void mpz_mul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2}) Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}. @end deftypefun @deftypefun void mpz_addmul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @deftypefunx void mpz_addmul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2}) Set @var{rop} to @math{@var{rop} + @var{op1} @GMPtimes{} @var{op2}}. @end deftypefun @deftypefun void mpz_submul (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @deftypefunx void mpz_submul_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2}) Set @var{rop} to @math{@var{rop} - @var{op1} @GMPtimes{} @var{op2}}. @end deftypefun @deftypefun void mpz_mul_2exp (mpz_t @var{rop}, mpz_t @var{op1}, mp_bitcnt_t @var{op2}) @cindex Bit shift left Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to @var{op2}}. This operation can also be defined as a left shift by @var{op2} bits. @end deftypefun @deftypefun void mpz_neg (mpz_t @var{rop}, mpz_t @var{op}) Set @var{rop} to @minus{}@var{op}. @end deftypefun @deftypefun void mpz_abs (mpz_t @var{rop}, mpz_t @var{op}) Set @var{rop} to the absolute value of @var{op}. @end deftypefun @need 2000 @node Integer Division, Integer Exponentiation, Integer Arithmetic, Integer Functions @section Division Functions @cindex Integer division functions @cindex Division functions Division is undefined if the divisor is zero. Passing a zero divisor to the division or modulo functions (including the modular powering functions @code{mpz_powm} and @code{mpz_powm_ui}), will cause an intentional division by zero. This lets a program handle arithmetic exceptions in these functions the same way as for normal C @code{int} arithmetic. @c Separate deftypefun groups for cdiv, fdiv and tdiv produce a blank line @c between each, and seem to let tex do a better job of page breaks than an @c @sp 1 in the middle of one big set. @deftypefun void mpz_cdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx void mpz_cdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx void mpz_cdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d}) @maybepagebreak @deftypefunx {unsigned long int} mpz_cdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_cdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_cdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_cdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}}) @maybepagebreak @deftypefunx void mpz_cdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}}) @deftypefunx void mpz_cdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}}) @end deftypefun @deftypefun void mpz_fdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx void mpz_fdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx void mpz_fdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d}) @maybepagebreak @deftypefunx {unsigned long int} mpz_fdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_fdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_fdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_fdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}}) @maybepagebreak @deftypefunx void mpz_fdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}}) @deftypefunx void mpz_fdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}}) @end deftypefun @deftypefun void mpz_tdiv_q (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx void mpz_tdiv_r (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx void mpz_tdiv_qr (mpz_t @var{q}, mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d}) @maybepagebreak @deftypefunx {unsigned long int} mpz_tdiv_q_ui (mpz_t @var{q}, mpz_t @var{n}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_tdiv_r_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_tdiv_qr_ui (mpz_t @var{q}, mpz_t @var{r}, @w{mpz_t @var{n}}, @w{unsigned long int @var{d}}) @deftypefunx {unsigned long int} mpz_tdiv_ui (mpz_t @var{n}, @w{unsigned long int @var{d}}) @maybepagebreak @deftypefunx void mpz_tdiv_q_2exp (mpz_t @var{q}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}}) @deftypefunx void mpz_tdiv_r_2exp (mpz_t @var{r}, mpz_t @var{n}, @w{mp_bitcnt_t @var{b}}) @cindex Bit shift right @sp 1 Divide @var{n} by @var{d}, forming a quotient @var{q} and/or remainder @var{r}. For the @code{2exp} functions, @m{@var{d}=2^b, @var{d}=2^@var{b}}. The rounding is in three styles, each suiting different applications. @itemize @bullet @item @code{cdiv} rounds @var{q} up towards @m{+\infty, +infinity}, and @var{r} will have the opposite sign to @var{d}. The @code{c} stands for ``ceil''. @item @code{fdiv} rounds @var{q} down towards @m{-\infty, @minus{}infinity}, and @var{r} will have the same sign as @var{d}. The @code{f} stands for ``floor''. @item @code{tdiv} rounds @var{q} towards zero, and @var{r} will have the same sign as @var{n}. The @code{t} stands for ``truncate''. @end itemize In all cases @var{q} and @var{r} will satisfy @m{@var{n}=@var{q}@var{d}+@var{r}, @var{n}=@var{q}*@var{d}+@var{r}}, and @var{r} will satisfy @math{0@le{}@GMPabs{@var{r}}<@GMPabs{@var{d}}}. The @code{q} functions calculate only the quotient, the @code{r} functions only the remainder, and the @code{qr} functions calculate both. Note that for @code{qr} the same variable cannot be passed for both @var{q} and @var{r}, or results will be unpredictable. For the @code{ui} variants the return value is the remainder, and in fact returning the remainder is all the @code{div_ui} functions do. For @code{tdiv} and @code{cdiv} the remainder can be negative, so for those the return value is the absolute value of the remainder. For the @code{2exp} variants the divisor is @m{2^b,2^@var{b}}. These functions are implemented as right shifts and bit masks, but of course they round the same as the other functions. For positive @var{n} both @code{mpz_fdiv_q_2exp} and @code{mpz_tdiv_q_2exp} are simple bitwise right shifts. For negative @var{n}, @code{mpz_fdiv_q_2exp} is effectively an arithmetic right shift treating @var{n} as twos complement the same as the bitwise logical functions do, whereas @code{mpz_tdiv_q_2exp} effectively treats @var{n} as sign and magnitude. @end deftypefun @deftypefun void mpz_mod (mpz_t @var{r}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx {unsigned long int} mpz_mod_ui (mpz_t @var{r}, mpz_t @var{n}, @w{unsigned long int @var{d}}) Set @var{r} to @var{n} @code{mod} @var{d}. The sign of the divisor is ignored; the result is always non-negative. @code{mpz_mod_ui} is identical to @code{mpz_fdiv_r_ui} above, returning the remainder as well as setting @var{r}. See @code{mpz_fdiv_ui} above if only the return value is wanted. @end deftypefun @deftypefun void mpz_divexact (mpz_t @var{q}, mpz_t @var{n}, mpz_t @var{d}) @deftypefunx void mpz_divexact_ui (mpz_t @var{q}, mpz_t @var{n}, unsigned long @var{d}) @cindex Exact division functions Set @var{q} to @var{n}/@var{d}. These functions produce correct results only when it is known in advance that @var{d} divides @var{n}. These routines are much faster than the other division functions, and are the best choice when exact division is known to occur, for example reducing a rational to lowest terms. @end deftypefun @deftypefun int mpz_divisible_p (mpz_t @var{n}, mpz_t @var{d}) @deftypefunx int mpz_divisible_ui_p (mpz_t @var{n}, unsigned long int @var{d}) @deftypefunx int mpz_divisible_2exp_p (mpz_t @var{n}, mp_bitcnt_t @var{b}) @cindex Divisibility functions Return non-zero if @var{n} is exactly divisible by @var{d}, or in the case of @code{mpz_divisible_2exp_p} by @m{2^b,2^@var{b}}. @var{n} is divisible by @var{d} if there exists an integer @var{q} satisfying @math{@var{n} = @var{q}@GMPmultiply{}@var{d}}. Unlike the other division functions, @math{@var{d}=0} is accepted and following the rule it can be seen that only 0 is considered divisible by 0. @end deftypefun @deftypefun int mpz_congruent_p (mpz_t @var{n}, mpz_t @var{c}, mpz_t @var{d}) @deftypefunx int mpz_congruent_ui_p (mpz_t @var{n}, unsigned long int @var{c}, unsigned long int @var{d}) @deftypefunx int mpz_congruent_2exp_p (mpz_t @var{n}, mpz_t @var{c}, mp_bitcnt_t @var{b}) @cindex Divisibility functions @cindex Congruence functions Return non-zero if @var{n} is congruent to @var{c} modulo @var{d}, or in the case of @code{mpz_congruent_2exp_p} modulo @m{2^b,2^@var{b}}. @var{n} is congruent to @var{c} mod @var{d} if there exists an integer @var{q} satisfying @math{@var{n} = @var{c} + @var{q}@GMPmultiply{}@var{d}}. Unlike the other division functions, @math{@var{d}=0} is accepted and following the rule it can be seen that @var{n} and @var{c} are considered congruent mod 0 only when exactly equal. @end deftypefun @need 2000 @node Integer Exponentiation, Integer Roots, Integer Division, Integer Functions @section Exponentiation Functions @cindex Integer exponentiation functions @cindex Exponentiation functions @cindex Powering functions @deftypefun void mpz_powm (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod}) @deftypefunx void mpz_powm_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp}, mpz_t @var{mod}) Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp}) modulo @var{mod}}. Negative @var{exp} is supported if an inverse @math{@var{base}^@W{-1} @bmod @var{mod}} exists (see @code{mpz_invert} in @ref{Number Theoretic Functions}). If an inverse doesn't exist then a divide by zero is raised. @end deftypefun @deftypefun void mpz_powm_sec (mpz_t @var{rop}, mpz_t @var{base}, mpz_t @var{exp}, mpz_t @var{mod}) Set @var{rop} to @m{base^{exp} \bmod mod, (@var{base} raised to @var{exp}) modulo @var{mod}}. It is required that @math{@var{exp} > 0} and that @var{mod} is odd. This function is designed to take the same time and have the same cache access patterns for any two same-size arguments, assuming that function arguments are placed at the same position and that the machine state is identical upon function entry. This function is intended for cryptographic purposes, where resilience to side-channel attacks is desired. @end deftypefun @deftypefun void mpz_pow_ui (mpz_t @var{rop}, mpz_t @var{base}, unsigned long int @var{exp}) @deftypefunx void mpz_ui_pow_ui (mpz_t @var{rop}, unsigned long int @var{base}, unsigned long int @var{exp}) Set @var{rop} to @m{base^{exp}, @var{base} raised to @var{exp}}. The case @math{0^0} yields 1. @end deftypefun @need 2000 @node Integer Roots, Number Theoretic Functions, Integer Exponentiation, Integer Functions @section Root Extraction Functions @cindex Integer root functions @cindex Root extraction functions @deftypefun int mpz_root (mpz_t @var{rop}, mpz_t @var{op}, unsigned long int @var{n}) Set @var{rop} to @m{\lfloor\root n \of {op}\rfloor@C{},} the truncated integer part of the @var{n}th root of @var{op}. Return non-zero if the computation was exact, i.e., if @var{op} is @var{rop} to the @var{n}th power. @end deftypefun @deftypefun void mpz_rootrem (mpz_t @var{root}, mpz_t @var{rem}, mpz_t @var{u}, unsigned long int @var{n}) Set @var{root} to @m{\lfloor\root n \of {u}\rfloor@C{},} the truncated integer part of the @var{n}th root of @var{u}. Set @var{rem} to the remainder, @m{(@var{u} - @var{root}^n), @var{u}@minus{}@var{root}**@var{n}}. @end deftypefun @deftypefun void mpz_sqrt (mpz_t @var{rop}, mpz_t @var{op}) Set @var{rop} to @m{\lfloor\sqrt{@var{op}}\rfloor@C{},} the truncated integer part of the square root of @var{op}. @end deftypefun @deftypefun void mpz_sqrtrem (mpz_t @var{rop1}, mpz_t @var{rop2}, mpz_t @var{op}) Set @var{rop1} to @m{\lfloor\sqrt{@var{op}}\rfloor, the truncated integer part of the square root of @var{op}}, like @code{mpz_sqrt}. Set @var{rop2} to the remainder @m{(@var{op} - @var{rop1}^2), @var{op}@minus{}@var{rop1}*@var{rop1}}, which will be zero if @var{op} is a perfect square. If @var{rop1} and @var{rop2} are the same variable, the results are undefined. @end deftypefun @deftypefun int mpz_perfect_power_p (mpz_t @var{op}) @cindex Perfect power functions @cindex Root testing functions Return non-zero if @var{op} is a perfect power, i.e., if there exist integers @m{a,@var{a}} and @m{b,@var{b}}, with @m{b>1, @var{b}>1}, such that @m{@var{op}=a^b, @var{op} equals @var{a} raised to the power @var{b}}. Under this definition both 0 and 1 are considered to be perfect powers. Negative values of @var{op} are accepted, but of course can only be odd perfect powers. @end deftypefun @deftypefun int mpz_perfect_square_p (mpz_t @var{op}) @cindex Perfect square functions @cindex Root testing functions Return non-zero if @var{op} is a perfect square, i.e., if the square root of @var{op} is an integer. Under this definition both 0 and 1 are considered to be perfect squares. @end deftypefun @need 2000 @node Number Theoretic Functions, Integer Comparisons, Integer Roots, Integer Functions @section Number Theoretic Functions @cindex Number theoretic functions @deftypefun int mpz_probab_prime_p (mpz_t @var{n}, int @var{reps}) @cindex Prime testing functions @cindex Probable prime testing functions Determine whether @var{n} is prime. Return 2 if @var{n} is definitely prime, return 1 if @var{n} is probably prime (without being certain), or return 0 if @var{n} is definitely composite. This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. The argument @var{reps} controls how many such tests are done; a higher value will reduce the chances of a composite being returned as ``probably prime''. 25 is a reasonable number; a composite number will then be identified as a prime with a probability of less than @m{2^{-50},2^(-50)}. Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime. @end deftypefun @deftypefun void mpz_nextprime (mpz_t @var{rop}, mpz_t @var{op}) @cindex Next prime function Set @var{rop} to the next prime greater than @var{op}. This function uses a probabilistic algorithm to identify primes. For practical purposes it's adequate, the chance of a composite passing will be extremely small. @end deftypefun @c mpz_prime_p not implemented as of gmp 3.0. @c @deftypefun int mpz_prime_p (mpz_t @var{n}) @c Return non-zero if @var{n} is prime and zero if @var{n} is a non-prime. @c This function is far slower than @code{mpz_probab_prime_p}, but then it @c never returns non-zero for composite numbers. @c (For practical purposes, using @code{mpz_probab_prime_p} is adequate. @c The likelihood of a programming error or hardware malfunction is orders @c of magnitudes greater than the likelihood for a composite to pass as a @c prime, if the @var{reps} argument is in the suggested range.) @c @end deftypefun @deftypefun void mpz_gcd (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @cindex Greatest common divisor functions @cindex GCD functions Set @var{rop} to the greatest common divisor of @var{op1} and @var{op2}. The result is always positive even if one or both input operands are negative. Except if both inputs are zero; then this function defines @math{gcd(0,0) = 0}. @end deftypefun @deftypefun {unsigned long int} mpz_gcd_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long int @var{op2}) Compute the greatest common divisor of @var{op1} and @var{op2}. If @var{rop} is not @code{NULL}, store the result there. If the result is small enough to fit in an @code{unsigned long int}, it is returned. If the result does not fit, 0 is returned, and the result is equal to the argument @var{op1}. Note that the result will always fit if @var{op2} is non-zero. @end deftypefun @deftypefun void mpz_gcdext (mpz_t @var{g}, mpz_t @var{s}, mpz_t @var{t}, mpz_t @var{a}, mpz_t @var{b}) @cindex Extended GCD @cindex GCD extended Set @var{g} to the greatest common divisor of @var{a} and @var{b}, and in addition set @var{s} and @var{t} to coefficients satisfying @math{@var{a}@GMPmultiply{}@var{s} + @var{b}@GMPmultiply{}@var{t} = @var{g}}. The value in @var{g} is always positive, even if one or both of @var{a} and @var{b} are negative (or zero if both inputs are zero). The values in @var{s} and @var{t} are chosen such that normally, @math{@GMPabs{@var{s}} < @GMPabs{@var{b}} / (2 @var{g})} and @math{@GMPabs{@var{t}} < @GMPabs{@var{a}} / (2 @var{g})}, and these relations define @var{s} and @var{t} uniquely. There are a few exceptional cases: If @math{@GMPabs{@var{a}} = @GMPabs{@var{b}}}, then @math{@var{s} = 0}, @math{@var{t} = sgn(@var{b})}. Otherwise, @math{@var{s} = sgn(@var{a})} if @math{@var{b} = 0} or @math{@GMPabs{@var{b}} = 2 @var{g}}, and @math{@var{t} = sgn(@var{b})} if @math{@var{a} = 0} or @math{@GMPabs{@var{a}} = 2 @var{g}}. In all cases, @math{@var{s} = 0} if and only if @math{@var{g} = @GMPabs{@var{b}}}, i.e., if @var{b} divides @var{a} or @math{@var{a} = @var{b} = 0}. If @var{t} is @code{NULL} then that value is not computed. @end deftypefun @deftypefun void mpz_lcm (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @deftypefunx void mpz_lcm_ui (mpz_t @var{rop}, mpz_t @var{op1}, unsigned long @var{op2}) @cindex Least common multiple functions @cindex LCM functions Set @var{rop} to the least common multiple of @var{op1} and @var{op2}. @var{rop} is always positive, irrespective of the signs of @var{op1} and @var{op2}. @var{rop} will be zero if either @var{op1} or @var{op2} is zero. @end deftypefun @deftypefun int mpz_invert (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) @cindex Modular inverse functions @cindex Inverse modulo functions Compute the inverse of @var{op1} modulo @var{op2} and put the result in @var{rop}. If the inverse exists, the return value is non-zero and @var{rop} will satisfy @math{0 < @var{rop} < @GMPabs{@var{op2}}}. If an inverse doesn't exist the return value is zero and @var{rop} is undefined. The behaviour of this function is undefined when @var{op2} is zero. @end deftypefun @deftypefun int mpz_jacobi (mpz_t @var{a}, mpz_t @var{b}) @cindex Jacobi symbol functions Calculate the Jacobi symbol @m{\left(a \over b\right), (@var{a}/@var{b})}. This is defined only for @var{b} odd. @end deftypefun @deftypefun int mpz_legendre (mpz_t @var{a}, mpz_t @var{p}) @cindex Legendre symbol functions Calculate the Legendre symbol @m{\left(a \over p\right), (@var{a}/@var{p})}. This is defined only for @var{p} an odd positive prime, and for such @var{p} it's identical to the Jacobi symbol. @end deftypefun @deftypefun int mpz_kronecker (mpz_t @var{a}, mpz_t @var{b}) @deftypefunx int mpz_kronecker_si (mpz_t @var{a}, long @var{b}) @deftypefunx int mpz_kronecker_ui (mpz_t @var{a}, unsigned long @var{b}) @deftypefunx int mpz_si_kronecker (long @var{a}, mpz_t @var{b}) @deftypefunx int mpz_ui_kronecker (unsigned long @var{a}, mpz_t @var{b}) @cindex Kronecker symbol functions Calculate the Jacobi symbol @m{\left(a \over b\right), (@var{a}/@var{b})} with the Kronecker extension @m{\left(a \over 2\right) = \left(2 \over a\right), (a/2)=(2/a)} when @math{a} odd, or @m{\left(a \over 2\right) = 0, (a/2)=0} when @math{a} even. When @var{b} is odd the Jacobi symbol and Kronecker symbol are identical, so @code{mpz_kronecker_ui} etc can be used for mixed precision Jacobi symbols too. For more information see Henri Cohen section 1.4.2 (@pxref{References}), or any number theory textbook. See also the example program @file{demos/qcn.c} which uses @code{mpz_kronecker_ui}. @end deftypefun @deftypefun {mp_bitcnt_t} mpz_remove (mpz_t @var{rop}, mpz_t @var{op}, mpz_t @var{f}) @cindex Remove factor functions @cindex Factor removal functions Remove all occurrences of the factor @var{f} from @var{op} and store the result in @var{rop}. The return value is how many such occurrences were removed. @end deftypefun @deftypefun void mpz_fac_ui (mpz_t @var{rop}, unsigned long int @var{n}) @deftypefunx void mpz_2fac_ui (mpz_t @var{rop}, unsigned long int @var{n}) @deftypefunx void mpz_mfac_uiui (mpz_t @var{rop}, unsigned long int @var{n}, unsigned long int @var{m}) @cindex Factorial functions Set @var{rop} to the factorial of @var{n}: @code{mpz_fac_ui} computes the plain factorial @var{n}!, @code{mpz_2fac_ui} computes the double-factorial @var{n}!!, and @code{mpz_mfac_uiui} the @var{m}-multi-factorial @m{n!^{(m)}, @var{n}!^(@var{m})}. @end deftypefun @deftypefun void mpz_primorial_ui (mpz_t @var{rop}, unsigned long int @var{n}) @cindex Primorial functions Set @var{rop} to the primorial of @var{n}, i.e. the product of all positive prime numbers @math{@le{}@var{n}}. @end deftypefun @deftypefun void mpz_bin_ui (mpz_t @var{rop}, mpz_t @var{n}, unsigned long int @var{k}) @deftypefunx void mpz_bin_uiui (mpz_t @var{rop}, unsigned long int @var{n}, @w{unsigned long int @var{k}}) @cindex Binomial coefficient functions Compute the binomial coefficient @m{\left({n}\atop{k}\right), @var{n} over @var{k}} and store the result in @var{rop}. Negative values of @var{n} are supported by @code{mpz_bin_ui}, using the identity @m{\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right), bin(-n@C{}k) = (-1)^k * bin(n+k-1@C{}k)}, see Knuth volume 1 section 1.2.6 part G. @end deftypefun @deftypefun void mpz_fib_ui (mpz_t @var{fn}, unsigned long int @var{n}) @deftypefunx void mpz_fib2_ui (mpz_t @var{fn}, mpz_t @var{fnsub1}, unsigned long int @var{n}) @cindex Fibonacci sequence functions @code{mpz_fib_ui} sets @var{fn} to to @m{F_n,F[n]}, the @var{n}'th Fibonacci number. @code{mpz_fib2_ui} sets @var{fn} to @m{F_n,F[n]}, and @var{fnsub1} to @m{F_{n-1},F[n-1]}. These functions are designed for calculating isolated Fibonacci numbers. When a sequence of values is wanted it's best to start with @code{mpz_fib2_ui} and iterate the defining @m{F_{n+1} = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or similar. @end deftypefun @deftypefun void mpz_lucnum_ui (mpz_t @var{ln}, unsigned long int @var{n}) @deftypefunx void mpz_lucnum2_ui (mpz_t @var{ln}, mpz_t @var{lnsub1}, unsigned long int @var{n}) @cindex Lucas number functions @code{mpz_lucnum_ui} sets @var{ln} to to @m{L_n,L[n]}, the @var{n}'th Lucas number. @code{mpz_lucnum2_ui} sets @var{ln} to @m{L_n,L[n]}, and @var{lnsub1} to @m{L_{n-1},L[n-1]}. These functions are designed for calculating isolated Lucas numbers. When a sequence of values is wanted it's best to start with @code{mpz_lucnum2_ui} and iterate the defining @m{L_{n+1} = L_n + L_{n-1}, L[n+1]=L[n]+L[n-1]} or similar. The Fibonacci numbers and Lucas numbers are related sequences, so it's never necessary to call both @code{mpz_fib2_ui} and @code{mpz_lucnum2_ui}. The formulas for going from Fibonacci to Lucas can be found in @ref{Lucas Numbers Algorithm}, the reverse is straightforward too. @end deftypefun @node Integer Comparisons, Integer Logic and Bit Fiddling, Number Theoretic Functions, Integer Functions @comment node-name, next, previous, up @section Comparison Functions @cindex Integer comparison functions @cindex Comparison functions @deftypefn Function int mpz_cmp (mpz_t @var{op1}, mpz_t @var{op2}) @deftypefnx Function int mpz_cmp_d (mpz_t @var{op1}, double @var{op2}) @deftypefnx Macro int mpz_cmp_si (mpz_t @var{op1}, signed long int @var{op2}) @deftypefnx Macro int mpz_cmp_ui (mpz_t @var{op1}, unsigned long int @var{op2}) Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} > @var{op2}}, zero if @math{@var{op1} = @var{op2}}, or a negative value if @math{@var{op1} < @var{op2}}. @code{mpz_cmp_ui} and @code{mpz_cmp_si} are macros and will evaluate their arguments more than once. @code{mpz_cmp_d} can be called with an infinity, but results are undefined for a NaN. @end deftypefn @deftypefn Function int mpz_cmpabs (mpz_t @var{op1}, mpz_t @var{op2}) @deftypefnx Function int mpz_cmpabs_d (mpz_t @var{op1}, double @var{op2}) @deftypefnx Function int mpz_cmpabs_ui (mpz_t @var{op1}, unsigned long int @var{op2}) Compare the absolute values of @var{op1} and @var{op2}. Return a positive value if @math{@GMPabs{@var{op1}} > @GMPabs{@var{op2}}}, zero if @math{@GMPabs{@var{op1}} = @GMPabs{@var{op2}}}, or a negative value if @math{@GMPabs{@var{op1}} < @GMPabs{@var{op2}}}. @code{mpz_cmpabs_d} can be called with an infinity, but results are undefined for a NaN. @end deftypefn @deftypefn Macro int mpz_sgn (mpz_t @var{op}) @cindex Sign tests @cindex Integer sign tests Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and @math{-1} if @math{@var{op} < 0}. This function is actually implemented as a macro. It evaluates its argument multiple times. @end deftypefn @node Integer Logic and Bit Fiddling, I/O of Integers, Integer Comparisons, Integer Functions @comment node-name, next, previous, up @section Logical and Bit Manipulation Functions @cindex Logical functions @cindex Bit manipulation functions @cindex Integer logical functions @cindex Integer bit manipulation functions These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0. @deftypefun void mpz_and (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) Set @var{rop} to @var{op1} bitwise-and @var{op2}. @end deftypefun @deftypefun void mpz_ior (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) Set @var{rop} to @var{op1} bitwise inclusive-or @var{op2}. @end deftypefun @deftypefun void mpz_xor (mpz_t @var{rop}, mpz_t @var{op1}, mpz_t @var{op2}) Set @var{rop} to @var{op1} bitwise exclusive-or @var{op2}. @end deftypefun @deftypefun void mpz_com (mpz_t @var{rop}, mpz_t @var{op}) Set @var{rop} to the one's complement of @var{op}. @end deftypefun @deftypefun {mp_bitcnt_t} mpz_popcount (mpz_t @var{op}) If @math{@var{op}@ge{}0}, return the population count of @var{op}, which is the number of 1 bits in the binary representation. If @math{@var{op}<0}, the number of 1s is infinite, and the return value is the largest possible @code{mp_bitcnt_t}. @end deftypefun @deftypefun {mp_bitcnt_t} mpz_hamdist (mpz_t @var{op1}, mpz_t @var{op2}) If @var{op1} and @var{op2} are both @math{@ge{}0} or both @math{<0}, return the hamming distance between the two operands, which is the number of bit positions where @var{op1} and @var{op2} have different bit values. If one operand is @math{@ge{}0} and the other @math{<0} then the number of bits different is infinite, and the return value is the largest possible @code{mp_bitcnt_t}. @end deftypefun @deftypefun {mp_bitcnt_t} mpz_scan0 (mpz_t @var{op}, mp_bitcnt_t @var{starting_bit}) @deftypefunx {mp_bitcnt_t} mpz_scan1 (mpz_t @var{op}, mp_bitcnt_t @var{starting_bit}) @cindex Bit scanning functions @cindex Scan bit functions Scan @var{op}, starting from bit @var{starting_bit}, towards more significant bits, until the first 0 or 1 bit (respectively) is found. Return the index of the found bit. If the bit at @var{starting_bit} is already what's sought, then @var{starting_bit} is returned. If there's no bit found, then the largest possible @code{mp_bitcnt_t} is returned. This will happen in @code{mpz_scan0} past the end of a negative number, or @code{mpz_scan1} past the end of a nonnegative number. @end deftypefun @deftypefun void mpz_setbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index}) Set bit @var{bit_index} in @var{rop}. @end deftypefun @deftypefun void mpz_clrbit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index}) Clear bit @var{bit_index} in @var{rop}. @end deftypefun @deftypefun void mpz_combit (mpz_t @var{rop}, mp_bitcnt_t @var{bit_index}) Complement bit @var{bit_index} in @var{rop}. @end deftypefun @deftypefun int mpz_tstbit (mpz_t @var{op}, mp_bitcnt_t @var{bit_index}) Test bit @var{bit_index} in @var{op} and return 0 or 1 accordingly. @end deftypefun @node I/O of Integers, Integer Random Numbers, Integer Logic and Bit Fiddling, Integer Functions @comment node-name, next, previous, up @section Input and Output Functions @cindex Integer input and output functions @cindex Input functions @cindex Output functions @cindex I/O functions Functions that perform input from a stdio stream, and functions that output to a stdio stream, of @code{mpz} numbers. Passing a @code{NULL} pointer for a @var{stream} argument to any of these functions will make them read from @code{stdin} and write to @code{stdout}, respectively. When using any of these functions, it is a good idea to include @file{stdio.h} before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes for these functions. See also @ref{Formatted Output} and @ref{Formatted Input}. @deftypefun size_t mpz_out_str (FILE *@var{stream}, int @var{base}, mpz_t @var{op}) Output @var{op} on stdio stream @var{stream}, as a string of digits in base @var{base}. The base argument may vary from 2 to 62 or from @minus{}2 to @minus{}36. For @var{base} in the range 2..36, digits and lower-case letters are used; for @minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. Return the number of bytes written, or if an error occurred, return 0. @end deftypefun @deftypefun size_t mpz_inp_str (mpz_t @var{rop}, FILE *@var{stream}, int @var{base}) Input a possibly white-space preceded string in base @var{base} from stdio stream @var{stream}, and put the read integer in @var{rop}. The @var{base} may vary from 2 to 62, or if @var{base} is 0, then the leading characters are used: @code{0x} and @code{0X} for hexadecimal, @code{0b} and @code{0B} for binary, @code{0} for octal, or decimal otherwise. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value. For bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. Return the number of bytes read, or if an error occurred, return 0. @end deftypefun @deftypefun size_t mpz_out_raw (FILE *@var{stream}, mpz_t @var{op}) Output @var{op} on stdio stream @var{stream}, in raw binary format. The integer is written in a portable format, with 4 bytes of size information, and that many bytes of limbs. Both the size and the limbs are written in decreasing significance order (i.e., in big-endian). The output can be read with @code{mpz_inp_raw}. Return the number of bytes written, or if an error occurred, return 0. The output of this can not be read by @code{mpz_inp_raw} from GMP 1, because of changes necessary for compatibility between 32-bit and 64-bit machines. @end deftypefun @deftypefun size_t mpz_inp_raw (mpz_t @var{rop}, FILE *@var{stream}) Input from stdio stream @var{stream} in the format written by @code{mpz_out_raw}, and put the result in @var{rop}. Return the number of bytes read, or if an error occurred, return 0. This routine can read the output from @code{mpz_out_raw} also from GMP 1, in spite of changes necessary for compatibility between 32-bit and 64-bit machines. @end deftypefun @need 2000 @node Integer Random Numbers, Integer Import and Export, I/O of Integers, Integer Functions @comment node-name, next, previous, up @section Random Number Functions @cindex Integer random number functions @cindex Random number functions The random number functions of GMP come in two groups; older function that rely on a global state, and newer functions that accept a state parameter that is read and modified. Please see the @ref{Random Number Functions} for more information on how to use and not to use random number functions. @deftypefun void mpz_urandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{n}) Generate a uniformly distributed random integer in the range 0 to @m{2^n-1, 2^@var{n}@minus{}1}, inclusive. The variable @var{state} must be initialized by calling one of the @code{gmp_randinit} functions (@ref{Random State Initialization}) before invoking this function. @end deftypefun @deftypefun void mpz_urandomm (mpz_t @var{rop}, gmp_randstate_t @var{state}, mpz_t @var{n}) Generate a uniform random integer in the range 0 to @math{@var{n}-1}, inclusive. The variable @var{state} must be initialized by calling one of the @code{gmp_randinit} functions (@ref{Random State Initialization}) before invoking this function. @end deftypefun @deftypefun void mpz_rrandomb (mpz_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{n}) Generate a random integer with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. The random number will be in the range 0 to @m{2^n-1, 2^@var{n}@minus{}1}, inclusive. The variable @var{state} must be initialized by calling one of the @code{gmp_randinit} functions (@ref{Random State Initialization}) before invoking this function. @end deftypefun @deftypefun void mpz_random (mpz_t @var{rop}, mp_size_t @var{max_size}) Generate a random integer of at most @var{max_size} limbs. The generated random number doesn't satisfy any particular requirements of randomness. Negative random numbers are generated when @var{max_size} is negative. This function is obsolete. Use @code{mpz_urandomb} or @code{mpz_urandomm} instead. @end deftypefun @deftypefun void mpz_random2 (mpz_t @var{rop}, mp_size_t @var{max_size}) Generate a random integer of at most @var{max_size} limbs, with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when @var{max_size} is negative. This function is obsolete. Use @code{mpz_rrandomb} instead. @end deftypefun @node Integer Import and Export, Miscellaneous Integer Functions, Integer Random Numbers, Integer Functions @section Integer Import and Export @code{mpz_t} variables can be converted to and from arbitrary words of binary data with the following functions. @deftypefun void mpz_import (mpz_t @var{rop}, size_t @var{count}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, const void *@var{op}) @cindex Integer import @cindex Import Set @var{rop} from an array of word data at @var{op}. The parameters specify the format of the data. @var{count} many words are read, each @var{size} bytes. @var{order} can be 1 for most significant word first or -1 for least significant first. Within each word @var{endian} can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU@. The most significant @var{nails} bits of each word are skipped, this can be 0 to use the full words. There is no sign taken from the data, @var{rop} will simply be a positive integer. An application can handle any sign itself, and apply it for instance with @code{mpz_neg}. There are no data alignment restrictions on @var{op}, any address is allowed. Here's an example converting an array of @code{unsigned long} data, most significant element first, and host byte order within each value. @example unsigned long a[20]; /* Initialize @var{z} and @var{a} */ mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a); @end example This example assumes the full @code{sizeof} bytes are used for data in the given type, which is usually true, and certainly true for @code{unsigned long} everywhere we know of. However on Cray vector systems it may be noted that @code{short} and @code{int} are always stored in 8 bytes (and with @code{sizeof} indicating that) but use only 32 or 46 bits. The @var{nails} feature can account for this, by passing for instance @code{8*sizeof(int)-INT_BIT}. @end deftypefun @deftypefun {void *} mpz_export (void *@var{rop}, size_t *@var{countp}, int @var{order}, size_t @var{size}, int @var{endian}, size_t @var{nails}, mpz_t @var{op}) @cindex Integer export @cindex Export Fill @var{rop} with word data from @var{op}. The parameters specify the format of the data produced. Each word will be @var{size} bytes and @var{order} can be 1 for most significant word first or -1 for least significant first. Within each word @var{endian} can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU@. The most significant @var{nails} bits of each word are unused and set to zero, this can be 0 to produce full words. The number of words produced is written to @code{*@var{countp}}, or @var{countp} can be @code{NULL} to discard the count. @var{rop} must have enough space for the data, or if @var{rop} is @code{NULL} then a result array of the necessary size is allocated using the current GMP allocation function (@pxref{Custom Allocation}). In either case the return value is the destination used, either @var{rop} or the allocated block. If @var{op} is non-zero then the most significant word produced will be non-zero. If @var{op} is zero then the count returned will be zero and nothing written to @var{rop}. If @var{rop} is @code{NULL} in this case, no block is allocated, just @code{NULL} is returned. The sign of @var{op} is ignored, just the absolute value is exported. An application can use @code{mpz_sgn} to get the sign and handle it as desired. (@pxref{Integer Comparisons}) There are no data alignment restrictions on @var{rop}, any address is allowed. When an application is allocating space itself the required size can be determined with a calculation like the following. Since @code{mpz_sizeinbase} always returns at least 1, @code{count} here will be at least one, which avoids any portability problems with @code{malloc(0)}, though if @code{z} is zero no space at all is actually needed (or written). @example numb = 8*size - nail; count = (mpz_sizeinbase (z, 2) + numb-1) / numb; p = malloc (count * size); @end example @end deftypefun @need 2000 @node Miscellaneous Integer Functions, Integer Special Functions, Integer Import and Export, Integer Functions @comment node-name, next, previous, up @section Miscellaneous Functions @cindex Miscellaneous integer functions @cindex Integer miscellaneous functions @deftypefun int mpz_fits_ulong_p (mpz_t @var{op}) @deftypefunx int mpz_fits_slong_p (mpz_t @var{op}) @deftypefunx int mpz_fits_uint_p (mpz_t @var{op}) @deftypefunx int mpz_fits_sint_p (mpz_t @var{op}) @deftypefunx int mpz_fits_ushort_p (mpz_t @var{op}) @deftypefunx int mpz_fits_sshort_p (mpz_t @var{op}) Return non-zero iff the value of @var{op} fits in an @code{unsigned long int}, @code{signed long int}, @code{unsigned int}, @code{signed int}, @code{unsigned short int}, or @code{signed short int}, respectively. Otherwise, return zero. @end deftypefun @deftypefn Macro int mpz_odd_p (mpz_t @var{op}) @deftypefnx Macro int mpz_even_p (mpz_t @var{op}) Determine whether @var{op} is odd or even, respectively. Return non-zero if yes, zero if no. These macros evaluate their argument more than once. @end deftypefn @deftypefun size_t mpz_sizeinbase (mpz_t @var{op}, int @var{base}) @cindex Size in digits @cindex Digits in an integer Return the size of @var{op} measured in number of digits in the given @var{base}. @var{base} can vary from 2 to 62. The sign of @var{op} is ignored, just the absolute value is used. The result will be either exact or 1 too big. If @var{base} is a power of 2, the result is always exact. If @var{op} is zero the return value is always 1. This function can be used to determine the space required when converting @var{op} to a string. The right amount of allocation is normally two more than the value returned by @code{mpz_sizeinbase}, one extra for a minus sign and one for the null-terminator. @cindex Most significant bit It will be noted that @code{mpz_sizeinbase(@var{op},2)} can be used to locate the most significant 1 bit in @var{op}, counting from 1. (Unlike the bitwise functions which start from 0, @xref{Integer Logic and Bit Fiddling,, Logical and Bit Manipulation Functions}.) @end deftypefun @node Integer Special Functions, , Miscellaneous Integer Functions, Integer Functions @section Special Functions @cindex Special integer functions @cindex Integer special functions The functions in this section are for various special purposes. Most applications will not need them. @deftypefun void mpz_array_init (mpz_t @var{integer_array}, mp_size_t @var{array_size}, @w{mp_size_t @var{fixed_num_bits}}) This is a special type of initialization. @strong{Fixed} space of @var{fixed_num_bits} is allocated to each of the @var{array_size} integers in @var{integer_array}. There is no way to free the storage allocated by this function. Don't call @code{mpz_clear}! The @var{integer_array} parameter is the first @code{mpz_t} in the array. For example, @example mpz_t arr[20000]; mpz_array_init (arr[0], 20000, 512); @end example @c In case anyone's wondering, yes this parameter style is a bit anomalous, @c it'd probably be nicer if it was "arr" instead of "arr[0]". Obviously the @c two differ only in the declaration, not the pointer value, but changing is @c not possible since it'd provoke warnings or errors in existing sources. This function is only intended for programs that create a large number of integers and need to reduce memory usage by avoiding the overheads of allocating and reallocating lots of small blocks. In normal programs this function is not recommended. The space allocated to each integer by this function will not be automatically increased, unlike the normal @code{mpz_init}, so an application must ensure it is sufficient for any value stored. The following space requirements apply to various routines, @itemize @bullet @item @code{mpz_abs}, @code{mpz_neg}, @code{mpz_set}, @code{mpz_set_si} and @code{mpz_set_ui} need room for the value they store. @item @code{mpz_add}, @code{mpz_add_ui}, @code{mpz_sub} and @code{mpz_sub_ui} need room for the larger of the two operands, plus an extra @code{mp_bits_per_limb}. @item @code{mpz_mul}, @code{mpz_mul_ui} and @code{mpz_mul_si} need room for the sum of the number of bits in their operands, but each rounded up to a multiple of @code{mp_bits_per_limb}. @item @code{mpz_swap} can be used between two array variables, but not between an array and a normal variable. @end itemize For other functions, or if in doubt, the suggestion is to calculate in a regular @code{mpz_init} variable and copy the result to an array variable with @code{mpz_set}. @end deftypefun @deftypefun {void *} _mpz_realloc (mpz_t @var{integer}, mp_size_t @var{new_alloc}) Change the space for @var{integer} to @var{new_alloc} limbs. The value in @var{integer} is preserved if it fits, or is set to 0 if not. The return value is not useful to applications and should be ignored. @code{mpz_realloc2} is the preferred way to accomplish allocation changes like this. @code{mpz_realloc2} and @code{_mpz_realloc} are the same except that @code{_mpz_realloc} takes its size in limbs. @end deftypefun @deftypefun mp_limb_t mpz_getlimbn (mpz_t @var{op}, mp_size_t @var{n}) Return limb number @var{n} from @var{op}. The sign of @var{op} is ignored, just the absolute value is used. The least significant limb is number 0. @code{mpz_size} can be used to find how many limbs make up @var{op}. @code{mpz_getlimbn} returns zero if @var{n} is outside the range 0 to @code{mpz_size(@var{op})-1}. @end deftypefun @deftypefun size_t mpz_size (mpz_t @var{op}) Return the size of @var{op} measured in number of limbs. If @var{op} is zero, the returned value will be zero. @c (@xref{Nomenclature}, for an explanation of the concept @dfn{limb}.) @end deftypefun @node Rational Number Functions, Floating-point Functions, Integer Functions, Top @comment node-name, next, previous, up @chapter Rational Number Functions @cindex Rational number functions This chapter describes the GMP functions for performing arithmetic on rational numbers. These functions start with the prefix @code{mpq_}. Rational numbers are stored in objects of type @code{mpq_t}. All rational arithmetic functions assume operands have a canonical form, and canonicalize their result. The canonical from means that the denominator and the numerator have no common factors, and that the denominator is positive. Zero has the unique representation 0/1. Pure assignment functions do not canonicalize the assigned variable. It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable. @deftypefun void mpq_canonicalize (mpq_t @var{op}) Remove any factors that are common to the numerator and denominator of @var{op}, and make the denominator positive. @end deftypefun @menu * Initializing Rationals:: * Rational Conversions:: * Rational Arithmetic:: * Comparing Rationals:: * Applying Integer Functions:: * I/O of Rationals:: @end menu @node Initializing Rationals, Rational Conversions, Rational Number Functions, Rational Number Functions @comment node-name, next, previous, up @section Initialization and Assignment Functions @cindex Rational assignment functions @cindex Assignment functions @cindex Rational initialization functions @cindex Initialization functions @deftypefun void mpq_init (mpq_t @var{x}) Initialize @var{x} and set it to 0/1. Each variable should normally only be initialized once, or at least cleared out (using the function @code{mpq_clear}) between each initialization. @end deftypefun @deftypefun void mpq_inits (mpq_t @var{x}, ...) Initialize a NULL-terminated list of @code{mpq_t} variables, and set their values to 0/1. @end deftypefun @deftypefun void mpq_clear (mpq_t @var{x}) Free the space occupied by @var{x}. Make sure to call this function for all @code{mpq_t} variables when you are done with them. @end deftypefun @deftypefun void mpq_clears (mpq_t @var{x}, ...) Free the space occupied by a NULL-terminated list of @code{mpq_t} variables. @end deftypefun @deftypefun void mpq_set (mpq_t @var{rop}, mpq_t @var{op}) @deftypefunx void mpq_set_z (mpq_t @var{rop}, mpz_t @var{op}) Assign @var{rop} from @var{op}. @end deftypefun @deftypefun void mpq_set_ui (mpq_t @var{rop}, unsigned long int @var{op1}, unsigned long int @var{op2}) @deftypefunx void mpq_set_si (mpq_t @var{rop}, signed long int @var{op1}, unsigned long int @var{op2}) Set the value of @var{rop} to @var{op1}/@var{op2}. Note that if @var{op1} and @var{op2} have common factors, @var{rop} has to be passed to @code{mpq_canonicalize} before any operations are performed on @var{rop}. @end deftypefun @deftypefun int mpq_set_str (mpq_t @var{rop}, char *@var{str}, int @var{base}) Set @var{rop} from a null-terminated string @var{str} in the given @var{base}. The string can be an integer like ``41'' or a fraction like ``41/152''. The fraction must be in canonical form (@pxref{Rational Number Functions}), or if not then @code{mpq_canonicalize} must be called. The numerator and optional denominator are parsed the same as in @code{mpz_set_str} (@pxref{Assigning Integers}). White space is allowed in the string, and is simply ignored. The @var{base} can vary from 2 to 62, or if @var{base} is 0 then the leading characters are used: @code{0x} or @code{0X} for hex, @code{0b} or @code{0B} for binary, @code{0} for octal, or decimal otherwise. Note that this is done separately for the numerator and denominator, so for instance @code{0xEF/100} is 239/100, whereas @code{0xEF/0x100} is 239/256. The return value is 0 if the entire string is a valid number, or @minus{}1 if not. @end deftypefun @deftypefun void mpq_swap (mpq_t @var{rop1}, mpq_t @var{rop2}) Swap the values @var{rop1} and @var{rop2} efficiently. @end deftypefun @need 2000 @node Rational Conversions, Rational Arithmetic, Initializing Rationals, Rational Number Functions @comment node-name, next, previous, up @section Conversion Functions @cindex Rational conversion functions @cindex Conversion functions @deftypefun double mpq_get_d (mpq_t @var{op}) Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding towards zero). If the exponent from the conversion is too big or too small to fit a @code{double} then the result is system dependent. For too big an infinity is returned when available. For too small @math{0.0} is normally returned. Hardware overflow, underflow and denorm traps may or may not occur. @end deftypefun @deftypefun void mpq_set_d (mpq_t @var{rop}, double @var{op}) @deftypefunx void mpq_set_f (mpq_t @var{rop}, mpf_t @var{op}) Set @var{rop} to the value of @var{op}. There is no rounding, this conversion is exact. @end deftypefun @deftypefun {char *} mpq_get_str (char *@var{str}, int @var{base}, mpq_t @var{op}) Convert @var{op} to a string of digits in base @var{base}. The base may vary from 2 to 36. The string will be of the form @samp{num/den}, or if the denominator is 1 then just @samp{num}. If @var{str} is @code{NULL}, the result string is allocated using the current allocation function (@pxref{Custom Allocation}). The block will be @code{strlen(str)+1} bytes, that being exactly enough for the string and null-terminator. If @var{str} is not @code{NULL}, it should point to a block of storage large enough for the result, that being @example mpz_sizeinbase (mpq_numref(@var{op}), @var{base}) + mpz_sizeinbase (mpq_denref(@var{op}), @var{base}) + 3 @end example The three extra bytes are for a possible minus sign, possible slash, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given @var{str}. @end deftypefun @node Rational Arithmetic, Comparing Rationals, Rational Conversions, Rational Number Functions @comment node-name, next, previous, up @section Arithmetic Functions @cindex Rational arithmetic functions @cindex Arithmetic functions @deftypefun void mpq_add (mpq_t @var{sum}, mpq_t @var{addend1}, mpq_t @var{addend2}) Set @var{sum} to @var{addend1} + @var{addend2}. @end deftypefun @deftypefun void mpq_sub (mpq_t @var{difference}, mpq_t @var{minuend}, mpq_t @var{subtrahend}) Set @var{difference} to @var{minuend} @minus{} @var{subtrahend}. @end deftypefun @deftypefun void mpq_mul (mpq_t @var{product}, mpq_t @var{multiplier}, mpq_t @var{multiplicand}) Set @var{product} to @math{@var{multiplier} @GMPtimes{} @var{multiplicand}}. @end deftypefun @deftypefun void mpq_mul_2exp (mpq_t @var{rop}, mpq_t @var{op1}, mp_bitcnt_t @var{op2}) Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to @var{op2}}. @end deftypefun @deftypefun void mpq_div (mpq_t @var{quotient}, mpq_t @var{dividend}, mpq_t @var{divisor}) @cindex Division functions Set @var{quotient} to @var{dividend}/@var{divisor}. @end deftypefun @deftypefun void mpq_div_2exp (mpq_t @var{rop}, mpq_t @var{op1}, mp_bitcnt_t @var{op2}) Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to @var{op2}}. @end deftypefun @deftypefun void mpq_neg (mpq_t @var{negated_operand}, mpq_t @var{operand}) Set @var{negated_operand} to @minus{}@var{operand}. @end deftypefun @deftypefun void mpq_abs (mpq_t @var{rop}, mpq_t @var{op}) Set @var{rop} to the absolute value of @var{op}. @end deftypefun @deftypefun void mpq_inv (mpq_t @var{inverted_number}, mpq_t @var{number}) Set @var{inverted_number} to 1/@var{number}. If the new denominator is zero, this routine will divide by zero. @end deftypefun @node Comparing Rationals, Applying Integer Functions, Rational Arithmetic, Rational Number Functions @comment node-name, next, previous, up @section Comparison Functions @cindex Rational comparison functions @cindex Comparison functions @deftypefun int mpq_cmp (mpq_t @var{op1}, mpq_t @var{op2}) Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} > @var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if @math{@var{op1} < @var{op2}}. To determine if two rationals are equal, @code{mpq_equal} is faster than @code{mpq_cmp}. @end deftypefun @deftypefn Macro int mpq_cmp_ui (mpq_t @var{op1}, unsigned long int @var{num2}, unsigned long int @var{den2}) @deftypefnx Macro int mpq_cmp_si (mpq_t @var{op1}, long int @var{num2}, unsigned long int @var{den2}) Compare @var{op1} and @var{num2}/@var{den2}. Return a positive value if @math{@var{op1} > @var{num2}/@var{den2}}, zero if @math{@var{op1} = @var{num2}/@var{den2}}, and a negative value if @math{@var{op1} < @var{num2}/@var{den2}}. @var{num2} and @var{den2} are allowed to have common factors. These functions are implemented as a macros and evaluate their arguments multiple times. @end deftypefn @deftypefn Macro int mpq_sgn (mpq_t @var{op}) @cindex Sign tests @cindex Rational sign tests Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and @math{-1} if @math{@var{op} < 0}. This function is actually implemented as a macro. It evaluates its argument multiple times. @end deftypefn @deftypefun int mpq_equal (mpq_t @var{op1}, mpq_t @var{op2}) Return non-zero if @var{op1} and @var{op2} are equal, zero if they are non-equal. Although @code{mpq_cmp} can be used for the same purpose, this function is much faster. @end deftypefun @node Applying Integer Functions, I/O of Rationals, Comparing Rationals, Rational Number Functions @comment node-name, next, previous, up @section Applying Integer Functions to Rationals @cindex Rational numerator and denominator @cindex Numerator and denominator The set of @code{mpq} functions is quite small. In particular, there are few functions for either input or output. The following functions give direct access to the numerator and denominator of an @code{mpq_t}. Note that if an assignment to the numerator and/or denominator could take an @code{mpq_t} out of the canonical form described at the start of this chapter (@pxref{Rational Number Functions}) then @code{mpq_canonicalize} must be called before any other @code{mpq} functions are applied to that @code{mpq_t}. @deftypefn Macro mpz_t mpq_numref (mpq_t @var{op}) @deftypefnx Macro mpz_t mpq_denref (mpq_t @var{op}) Return a reference to the numerator and denominator of @var{op}, respectively. The @code{mpz} functions can be used on the result of these macros. @end deftypefn @deftypefun void mpq_get_num (mpz_t @var{numerator}, mpq_t @var{rational}) @deftypefunx void mpq_get_den (mpz_t @var{denominator}, mpq_t @var{rational}) @deftypefunx void mpq_set_num (mpq_t @var{rational}, mpz_t @var{numerator}) @deftypefunx void mpq_set_den (mpq_t @var{rational}, mpz_t @var{denominator}) Get or set the numerator or denominator of a rational. These functions are equivalent to calling @code{mpz_set} with an appropriate @code{mpq_numref} or @code{mpq_denref}. Direct use of @code{mpq_numref} or @code{mpq_denref} is recommended instead of these functions. @end deftypefun @need 2000 @node I/O of Rationals, , Applying Integer Functions, Rational Number Functions @comment node-name, next, previous, up @section Input and Output Functions @cindex Rational input and output functions @cindex Input functions @cindex Output functions @cindex I/O functions Functions that perform input from a stdio stream, and functions that output to a stdio stream, of @code{mpq} numbers. Passing a @code{NULL} pointer for a @var{stream} argument to any of these functions will make them read from @code{stdin} and write to @code{stdout}, respectively. When using any of these functions, it is a good idea to include @file{stdio.h} before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes for these functions. See also @ref{Formatted Output} and @ref{Formatted Input}. @deftypefun size_t mpq_out_str (FILE *@var{stream}, int @var{base}, mpq_t @var{op}) Output @var{op} on stdio stream @var{stream}, as a string of digits in base @var{base}. The base may vary from 2 to 36. Output is in the form @samp{num/den} or if the denominator is 1 then just @samp{num}. Return the number of bytes written, or if an error occurred, return 0. @end deftypefun @deftypefun size_t mpq_inp_str (mpq_t @var{rop}, FILE *@var{stream}, int @var{base}) Read a string of digits from @var{stream} and convert them to a rational in @var{rop}. Any initial white-space characters are read and discarded. Return the number of characters read (including white space), or 0 if a rational could not be read. The input can be a fraction like @samp{17/63} or just an integer like @samp{123}. Reading stops at the first character not in this form, and white space is not permitted within the string. If the input might not be in canonical form, then @code{mpq_canonicalize} must be called (@pxref{Rational Number Functions}). The @var{base} can be between 2 and 36, or can be 0 in which case the leading characters of the string determine the base, @samp{0x} or @samp{0X} for hexadecimal, @samp{0} for octal, or decimal otherwise. The leading characters are examined separately for the numerator and denominator of a fraction, so for instance @samp{0x10/11} is @math{16/11}, whereas @samp{0x10/0x11} is @math{16/17}. @end deftypefun @node Floating-point Functions, Low-level Functions, Rational Number Functions, Top @comment node-name, next, previous, up @chapter Floating-point Functions @cindex Floating-point functions @cindex Float functions @cindex User-defined precision @cindex Precision of floats GMP floating point numbers are stored in objects of type @code{mpf_t} and functions operating on them have an @code{mpf_} prefix. The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time. The exponent of each float is a fixed precision, one machine word on most systems. In the current implementation the exponent is a count of limbs, so for example on a 32-bit system this means a range of roughly @math{2^@W{-68719476768}} to @math{2^@W{68719476736}}, or on a 64-bit system this will be greater. Note however @code{mpf_get_str} can only return an exponent which fits an @code{mp_exp_t} and currently @code{mpf_set_str} doesn't accept exponents bigger than a @code{long}. Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high. All calculations are performed to the precision of the destination variable. Each function is defined to calculate with ``infinite precision'' followed by a truncation to the destination precision, but of course the work done is only what's needed to determine a result under that definition. The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn't be concerned by such details. The mantissa in stored in binary, as might be imagined from the fact precisions are expressed in bits. One consequence of this is that decimal fractions like @math{0.1} cannot be represented exactly. The same is true of plain IEEE @code{double} floats. This makes both highly unsuitable for calculations involving money or other values that should be exact decimal fractions. (Suitably scaled integers, or perhaps rationals, are better choices.) @code{mpf} functions and variables have no special notion of infinity or not-a-number, and applications must take care not to overflow the exponent or results will be unpredictable. This might change in a future release. Note that the @code{mpf} functions are @emph{not} intended as a smooth extension to IEEE P754 arithmetic. In particular results obtained on one computer often differ from the results on a computer with a different word size. @menu * Initializing Floats:: * Assigning Floats:: * Simultaneous Float Init & Assign:: * Converting Floats:: * Float Arithmetic:: * Float Comparison:: * I/O of Floats:: * Miscellaneous Float Functions:: @end menu @node Initializing Floats, Assigning Floats, Floating-point Functions, Floating-point Functions @comment node-name, next, previous, up @section Initialization Functions @cindex Float initialization functions @cindex Initialization functions @deftypefun void mpf_set_default_prec (mp_bitcnt_t @var{prec}) Set the default precision to be @strong{at least} @var{prec} bits. All subsequent calls to @code{mpf_init} will use this precision, but previously initialized variables are unaffected. @end deftypefun @deftypefun {mp_bitcnt_t} mpf_get_default_prec (void) Return the default precision actually used. @end deftypefun An @code{mpf_t} object must be initialized before storing the first value in it. The functions @code{mpf_init} and @code{mpf_init2} are used for that purpose. @deftypefun void mpf_init (mpf_t @var{x}) Initialize @var{x} to 0. Normally, a variable should be initialized once only or at least be cleared, using @code{mpf_clear}, between initializations. The precision of @var{x} is undefined unless a default precision has already been established by a call to @code{mpf_set_default_prec}. @end deftypefun @deftypefun void mpf_init2 (mpf_t @var{x}, mp_bitcnt_t @var{prec}) Initialize @var{x} to 0 and set its precision to be @strong{at least} @var{prec} bits. Normally, a variable should be initialized once only or at least be cleared, using @code{mpf_clear}, between initializations. @end deftypefun @deftypefun void mpf_inits (mpf_t @var{x}, ...) Initialize a NULL-terminated list of @code{mpf_t} variables, and set their values to 0. The precision of the initialized variables is undefined unless a default precision has already been established by a call to @code{mpf_set_default_prec}. @end deftypefun @deftypefun void mpf_clear (mpf_t @var{x}) Free the space occupied by @var{x}. Make sure to call this function for all @code{mpf_t} variables when you are done with them. @end deftypefun @deftypefun void mpf_clears (mpf_t @var{x}, ...) Free the space occupied by a NULL-terminated list of @code{mpf_t} variables. @end deftypefun @need 2000 Here is an example on how to initialize floating-point variables: @example @{ mpf_t x, y; mpf_init (x); /* use default precision */ mpf_init2 (y, 256); /* precision @emph{at least} 256 bits */ @dots{} /* Unless the program is about to exit, do ... */ mpf_clear (x); mpf_clear (y); @} @end example The following three functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers. @deftypefun {mp_bitcnt_t} mpf_get_prec (mpf_t @var{op}) Return the current precision of @var{op}, in bits. @end deftypefun @deftypefun void mpf_set_prec (mpf_t @var{rop}, mp_bitcnt_t @var{prec}) Set the precision of @var{rop} to be @strong{at least} @var{prec} bits. The value in @var{rop} will be truncated to the new precision. This function requires a call to @code{realloc}, and so should not be used in a tight loop. @end deftypefun @deftypefun void mpf_set_prec_raw (mpf_t @var{rop}, mp_bitcnt_t @var{prec}) Set the precision of @var{rop} to be @strong{at least} @var{prec} bits, without changing the memory allocated. @var{prec} must be no more than the allocated precision for @var{rop}, that being the precision when @var{rop} was initialized, or in the most recent @code{mpf_set_prec}. The value in @var{rop} is unchanged, and in particular if it had a higher precision than @var{prec} it will retain that higher precision. New values written to @var{rop} will use the new @var{prec}. Before calling @code{mpf_clear} or the full @code{mpf_set_prec}, another @code{mpf_set_prec_raw} call must be made to restore @var{rop} to its original allocated precision. Failing to do so will have unpredictable results. @code{mpf_get_prec} can be used before @code{mpf_set_prec_raw} to get the original allocated precision. After @code{mpf_set_prec_raw} it reflects the @var{prec} value set. @code{mpf_set_prec_raw} is an efficient way to use an @code{mpf_t} variable at different precisions during a calculation, perhaps to gradually increase precision in an iteration, or just to use various different precisions for different purposes during a calculation. @end deftypefun @need 2000 @node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions @comment node-name, next, previous, up @section Assignment Functions @cindex Float assignment functions @cindex Assignment functions These functions assign new values to already initialized floats (@pxref{Initializing Floats}). @deftypefun void mpf_set (mpf_t @var{rop}, mpf_t @var{op}) @deftypefunx void mpf_set_ui (mpf_t @var{rop}, unsigned long int @var{op}) @deftypefunx void mpf_set_si (mpf_t @var{rop}, signed long int @var{op}) @deftypefunx void mpf_set_d (mpf_t @var{rop}, double @var{op}) @deftypefunx void mpf_set_z (mpf_t @var{rop}, mpz_t @var{op}) @deftypefunx void mpf_set_q (mpf_t @var{rop}, mpq_t @var{op}) Set the value of @var{rop} from @var{op}. @end deftypefun @deftypefun int mpf_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base}) Set the value of @var{rop} from the string in @var{str}. The string is of the form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}. @samp{M} is the mantissa and @samp{N} is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if @var{base} is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing @code{localeconv}. The argument @var{base} may be in the ranges 2 to 62, or @minus{}62 to @minus{}2. Negative values are used to specify that the exponent is in decimal. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value; for bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. Unlike the corresponding @code{mpz} function, the base will not be determined from the leading characters of the string if @var{base} is 0. This is so that numbers like @samp{0.23} are not interpreted as octal. White space is allowed in the string, and is simply ignored. [This is not really true; white-space is ignored in the beginning of the string and within the mantissa, but not in other places, such as after a minus sign or in the exponent. We are considering changing the definition of this function, making it fail when there is any white-space in the input, since that makes a lot of sense. Please tell us your opinion about this change. Do you really want it to accept @nicode{"3 14"} as meaning 314 as it does now?] This function returns 0 if the entire string is a valid number in base @var{base}. Otherwise it returns @minus{}1. @end deftypefun @deftypefun void mpf_swap (mpf_t @var{rop1}, mpf_t @var{rop2}) Swap @var{rop1} and @var{rop2} efficiently. Both the values and the precisions of the two variables are swapped. @end deftypefun @node Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions @comment node-name, next, previous, up @section Combined Initialization and Assignment Functions @cindex Float assignment functions @cindex Assignment functions @cindex Float initialization functions @cindex Initialization functions For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions' names have the form @code{mpf_init_set@dots{}} Once the float has been initialized by any of the @code{mpf_init_set@dots{}} functions, it can be used as the source or destination operand for the ordinary float functions. Don't use an initialize-and-set function on a variable already initialized! @deftypefun void mpf_init_set (mpf_t @var{rop}, mpf_t @var{op}) @deftypefunx void mpf_init_set_ui (mpf_t @var{rop}, unsigned long int @var{op}) @deftypefunx void mpf_init_set_si (mpf_t @var{rop}, signed long int @var{op}) @deftypefunx void mpf_init_set_d (mpf_t @var{rop}, double @var{op}) Initialize @var{rop} and set its value from @var{op}. The precision of @var{rop} will be taken from the active default precision, as set by @code{mpf_set_default_prec}. @end deftypefun @deftypefun int mpf_init_set_str (mpf_t @var{rop}, char *@var{str}, int @var{base}) Initialize @var{rop} and set its value from the string in @var{str}. See @code{mpf_set_str} above for details on the assignment operation. Note that @var{rop} is initialized even if an error occurs. (I.e., you have to call @code{mpf_clear} for it.) The precision of @var{rop} will be taken from the active default precision, as set by @code{mpf_set_default_prec}. @end deftypefun @node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions @comment node-name, next, previous, up @section Conversion Functions @cindex Float conversion functions @cindex Conversion functions @deftypefun double mpf_get_d (mpf_t @var{op}) Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding towards zero). If the exponent in @var{op} is too big or too small to fit a @code{double} then the result is system dependent. For too big an infinity is returned when available. For too small @math{0.0} is normally returned. Hardware overflow, underflow and denorm traps may or may not occur. @end deftypefun @deftypefun double mpf_get_d_2exp (signed long int *@var{exp}, mpf_t @var{op}) Convert @var{op} to a @code{double}, truncating if necessary (i.e.@: rounding towards zero), and with an exponent returned separately. The return value is in the range @math{0.5@le{}@GMPabs{@var{d}}<1} and the exponent is stored to @code{*@var{exp}}. @m{@var{d} * 2^{exp}, @var{d} * 2^@var{exp}} is the (truncated) @var{op} value. If @var{op} is zero, the return is @math{0.0} and 0 is stored to @code{*@var{exp}}. @cindex @code{frexp} This is similar to the standard C @code{frexp} function (@pxref{Normalization Functions,,, libc, The GNU C Library Reference Manual}). @end deftypefun @deftypefun long mpf_get_si (mpf_t @var{op}) @deftypefunx {unsigned long} mpf_get_ui (mpf_t @var{op}) Convert @var{op} to a @code{long} or @code{unsigned long}, truncating any fraction part. If @var{op} is too big for the return type, the result is undefined. See also @code{mpf_fits_slong_p} and @code{mpf_fits_ulong_p} (@pxref{Miscellaneous Float Functions}). @end deftypefun @deftypefun {char *} mpf_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op}) Convert @var{op} to a string of digits in base @var{base}. The base argument may vary from 2 to 62 or from @minus{}2 to @minus{}36. Up to @var{n_digits} digits will be generated. Trailing zeros are not returned. No more digits than can be accurately represented by @var{op} are ever generated. If @var{n_digits} is 0 then that accurate maximum number of digits are generated. For @var{base} in the range 2..36, digits and lower-case letters are used; for @minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. If @var{str} is @code{NULL}, the result string is allocated using the current allocation function (@pxref{Custom Allocation}). The block will be @code{strlen(str)+1} bytes, that being exactly enough for the string and null-terminator. If @var{str} is not @code{NULL}, it should point to a block of @math{@var{n_digits} + 2} bytes, that being enough for the mantissa, a possible minus sign, and a null-terminator. When @var{n_digits} is 0 to get all significant digits, an application won't be able to know the space required, and @var{str} should be @code{NULL} in that case. The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. The applicable exponent is written through the @var{expptr} pointer. For example, the number 3.1416 would be returned as string @nicode{"31416"} and exponent 1. When @var{op} is zero, an empty string is produced and the exponent returned is 0. A pointer to the result string is returned, being either the allocated block or the given @var{str}. @end deftypefun @node Float Arithmetic, Float Comparison, Converting Floats, Floating-point Functions @comment node-name, next, previous, up @section Arithmetic Functions @cindex Float arithmetic functions @cindex Arithmetic functions @deftypefun void mpf_add (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2}) @deftypefunx void mpf_add_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2}) Set @var{rop} to @math{@var{op1} + @var{op2}}. @end deftypefun @deftypefun void mpf_sub (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2}) @deftypefunx void mpf_ui_sub (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2}) @deftypefunx void mpf_sub_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2}) Set @var{rop} to @var{op1} @minus{} @var{op2}. @end deftypefun @deftypefun void mpf_mul (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2}) @deftypefunx void mpf_mul_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2}) Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}}. @end deftypefun Division is undefined if the divisor is zero, and passing a zero divisor to the divide functions will make these functions intentionally divide by zero. This lets the user handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions. @deftypefun void mpf_div (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2}) @deftypefunx void mpf_ui_div (mpf_t @var{rop}, unsigned long int @var{op1}, mpf_t @var{op2}) @deftypefunx void mpf_div_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2}) @cindex Division functions Set @var{rop} to @var{op1}/@var{op2}. @end deftypefun @deftypefun void mpf_sqrt (mpf_t @var{rop}, mpf_t @var{op}) @deftypefunx void mpf_sqrt_ui (mpf_t @var{rop}, unsigned long int @var{op}) @cindex Root extraction functions Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}. @end deftypefun @deftypefun void mpf_pow_ui (mpf_t @var{rop}, mpf_t @var{op1}, unsigned long int @var{op2}) @cindex Exponentiation functions @cindex Powering functions Set @var{rop} to @m{@var{op1}^{op2}, @var{op1} raised to the power @var{op2}}. @end deftypefun @deftypefun void mpf_neg (mpf_t @var{rop}, mpf_t @var{op}) Set @var{rop} to @minus{}@var{op}. @end deftypefun @deftypefun void mpf_abs (mpf_t @var{rop}, mpf_t @var{op}) Set @var{rop} to the absolute value of @var{op}. @end deftypefun @deftypefun void mpf_mul_2exp (mpf_t @var{rop}, mpf_t @var{op1}, mp_bitcnt_t @var{op2}) Set @var{rop} to @m{@var{op1} \times 2^{op2}, @var{op1} times 2 raised to @var{op2}}. @end deftypefun @deftypefun void mpf_div_2exp (mpf_t @var{rop}, mpf_t @var{op1}, mp_bitcnt_t @var{op2}) Set @var{rop} to @m{@var{op1}/2^{op2}, @var{op1} divided by 2 raised to @var{op2}}. @end deftypefun @node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions @comment node-name, next, previous, up @section Comparison Functions @cindex Float comparison functions @cindex Comparison functions @deftypefun int mpf_cmp (mpf_t @var{op1}, mpf_t @var{op2}) @deftypefunx int mpf_cmp_d (mpf_t @var{op1}, double @var{op2}) @deftypefunx int mpf_cmp_ui (mpf_t @var{op1}, unsigned long int @var{op2}) @deftypefunx int mpf_cmp_si (mpf_t @var{op1}, signed long int @var{op2}) Compare @var{op1} and @var{op2}. Return a positive value if @math{@var{op1} > @var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if @math{@var{op1} < @var{op2}}. @code{mpf_cmp_d} can be called with an infinity, but results are undefined for a NaN. @end deftypefun @deftypefun int mpf_eq (mpf_t @var{op1}, mpf_t @var{op2}, mp_bitcnt_t op3) Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are equal, zero otherwise. I.e., test if @var{op1} and @var{op2} are approximately equal. Caution 1: All version of GMP up to version 4.2.4 compared just whole limbs, meaning sometimes more than @var{op3} bits, sometimes fewer. Caution 2: This function will consider XXX11...111 and XX100...000 different, even if ... is replaced by a semi-infinite number of bits. Such numbers are really just one ulp off, and should be considered equal. @end deftypefun @deftypefun void mpf_reldiff (mpf_t @var{rop}, mpf_t @var{op1}, mpf_t @var{op2}) Compute the relative difference between @var{op1} and @var{op2} and store the result in @var{rop}. This is @math{@GMPabs{@var{op1}-@var{op2}}/@var{op1}}. @end deftypefun @deftypefn Macro int mpf_sgn (mpf_t @var{op}) @cindex Sign tests @cindex Float sign tests Return @math{+1} if @math{@var{op} > 0}, 0 if @math{@var{op} = 0}, and @math{-1} if @math{@var{op} < 0}. This function is actually implemented as a macro. It evaluates its argument multiple times. @end deftypefn @node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions @comment node-name, next, previous, up @section Input and Output Functions @cindex Float input and output functions @cindex Input functions @cindex Output functions @cindex I/O functions Functions that perform input from a stdio stream, and functions that output to a stdio stream, of @code{mpf} numbers. Passing a @code{NULL} pointer for a @var{stream} argument to any of these functions will make them read from @code{stdin} and write to @code{stdout}, respectively. When using any of these functions, it is a good idea to include @file{stdio.h} before @file{gmp.h}, since that will allow @file{gmp.h} to define prototypes for these functions. See also @ref{Formatted Output} and @ref{Formatted Input}. @deftypefun size_t mpf_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n_digits}, mpf_t @var{op}) Print @var{op} to @var{stream}, as a string of digits. Return the number of bytes written, or if an error occurred, return 0. The mantissa is prefixed with an @samp{0.} and is in the given @var{base}, which may vary from 2 to 62 or from @minus{}2 to @minus{}36. An exponent is then printed, separated by an @samp{e}, or if the base is greater than 10 then by an @samp{@@}. The exponent is always in decimal. The decimal point follows the current locale, on systems providing @code{localeconv}. For @var{base} in the range 2..36, digits and lower-case letters are used; for @minus{}2..@minus{}36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. Up to @var{n_digits} will be printed from the mantissa, except that no more digits than are accurately representable by @var{op} will be printed. @var{n_digits} can be 0 to select that accurate maximum. @end deftypefun @deftypefun size_t mpf_inp_str (mpf_t @var{rop}, FILE *@var{stream}, int @var{base}) Read a string in base @var{base} from @var{stream}, and put the read float in @var{rop}. The string is of the form @samp{M@@N} or, if the base is 10 or less, alternatively @samp{MeN}. @samp{M} is the mantissa and @samp{N} is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if @var{base} is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing @code{localeconv}. The argument @var{base} may be in the ranges 2 to 36, or @minus{}36 to @minus{}2. Negative values are used to specify that the exponent is in decimal. Unlike the corresponding @code{mpz} function, the base will not be determined from the leading characters of the string if @var{base} is 0. This is so that numbers like @samp{0.23} are not interpreted as octal. Return the number of bytes read, or if an error occurred, return 0. @end deftypefun @c @deftypefun void mpf_out_raw (FILE *@var{stream}, mpf_t @var{float}) @c Output @var{float} on stdio stream @var{stream}, in raw binary @c format. The float is written in a portable format, with 4 bytes of @c size information, and that many bytes of limbs. Both the size and the @c limbs are written in decreasing significance order. @c @end deftypefun @c @deftypefun void mpf_inp_raw (mpf_t @var{float}, FILE *@var{stream}) @c Input from stdio stream @var{stream} in the format written by @c @code{mpf_out_raw}, and put the result in @var{float}. @c @end deftypefun @node Miscellaneous Float Functions, , I/O of Floats, Floating-point Functions @comment node-name, next, previous, up @section Miscellaneous Functions @cindex Miscellaneous float functions @cindex Float miscellaneous functions @deftypefun void mpf_ceil (mpf_t @var{rop}, mpf_t @var{op}) @deftypefunx void mpf_floor (mpf_t @var{rop}, mpf_t @var{op}) @deftypefunx void mpf_trunc (mpf_t @var{rop}, mpf_t @var{op}) @cindex Rounding functions @cindex Float rounding functions Set @var{rop} to @var{op} rounded to an integer. @code{mpf_ceil} rounds to the next higher integer, @code{mpf_floor} to the next lower, and @code{mpf_trunc} to the integer towards zero. @end deftypefun @deftypefun int mpf_integer_p (mpf_t @var{op}) Return non-zero if @var{op} is an integer. @end deftypefun @deftypefun int mpf_fits_ulong_p (mpf_t @var{op}) @deftypefunx int mpf_fits_slong_p (mpf_t @var{op}) @deftypefunx int mpf_fits_uint_p (mpf_t @var{op}) @deftypefunx int mpf_fits_sint_p (mpf_t @var{op}) @deftypefunx int mpf_fits_ushort_p (mpf_t @var{op}) @deftypefunx int mpf_fits_sshort_p (mpf_t @var{op}) Return non-zero if @var{op} would fit in the respective C data type, when truncated to an integer. @end deftypefun @deftypefun void mpf_urandomb (mpf_t @var{rop}, gmp_randstate_t @var{state}, mp_bitcnt_t @var{nbits}) @cindex Random number functions @cindex Float random number functions Generate a uniformly distributed random float in @var{rop}, such that @math{0 @le{} @var{rop} < 1}, with @var{nbits} significant bits in the mantissa or less if the precision of @var{rop} is smaller. The variable @var{state} must be initialized by calling one of the @code{gmp_randinit} functions (@ref{Random State Initialization}) before invoking this function. @end deftypefun @deftypefun void mpf_random2 (mpf_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{exp}) Generate a random float of at most @var{max_size} limbs, with long strings of zeros and ones in the binary representation. The exponent of the number is in the interval @minus{}@var{exp} to @var{exp} (in limbs). This function is useful for testing functions and algorithms, since these kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated when @var{max_size} is negative. @end deftypefun @c @deftypefun size_t mpf_size (mpf_t @var{op}) @c Return the size of @var{op} measured in number of limbs. If @var{op} is @c zero, the returned value will be zero. (@xref{Nomenclature}, for an @c explanation of the concept @dfn{limb}.) @c @c @strong{This function is obsolete. It will disappear from future GMP @c releases.} @c @end deftypefun @node Low-level Functions, Random Number Functions, Floating-point Functions, Top @comment node-name, next, previous, up @chapter Low-level Functions @cindex Low-level functions This chapter describes low-level GMP functions, used to implement the high-level GMP functions, but also intended for time-critical user code. These functions start with the prefix @code{mpn_}. @c 1. Some of these function clobber input operands. @c The @code{mpn} functions are designed to be as fast as possible, @strong{not} to provide a coherent calling interface. The different functions have somewhat similar interfaces, but there are variations that make them hard to use. These functions do as little as possible apart from the real multiple precision computation, so that no time is spent on things that not all callers need. A source operand is specified by a pointer to the least significant limb and a limb count. A destination operand is specified by just a pointer. It is the responsibility of the caller to ensure that the destination has enough space for storing the result. With this way of specifying operands, it is possible to perform computations on subranges of an argument, and store the result into a subrange of a destination. A common requirement for all functions is that each source area needs at least one limb. No size argument may be zero. Unless otherwise stated, in-place operations are allowed where source and destination are the same, but not where they only partly overlap. The @code{mpn} functions are the base for the implementation of the @code{mpz_}, @code{mpf_}, and @code{mpq_} functions. This example adds the number beginning at @var{s1p} and the number beginning at @var{s2p} and writes the sum at @var{destp}. All areas have @var{n} limbs. @example cy = mpn_add_n (destp, s1p, s2p, n) @end example It should be noted that the @code{mpn} functions make no attempt to identify high or low zero limbs on their operands, or other special forms. On random data such cases will be unlikely and it'd be wasteful for every function to check every time. An application knowing something about its data can take steps to trim or perhaps split its calculations. @c @c For reference, within gmp mpz_t operands never have high zero limbs, and @c we rate low zero limbs as unlikely too (or something an application should @c handle). This is a prime motivation for not stripping zero limbs in say @c mpn_mul_n etc. @c @c Other applications doing variable-length calculations will quite likely do @c something similar to mpz. And even if not then it's highly likely zero @c limb stripping can be done at just a few judicious points, which will be @c more efficient than having lots of mpn functions checking every time. @sp 1 @noindent In the notation used below, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, @{@var{s1p}, @var{s1n}@}. @deftypefun mp_limb_t mpn_add_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Add @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the @var{n} least significant limbs of the result to @var{rp}. Return carry, either 0 or 1. This is the lowest-level function for addition. It is the preferred function for addition, since it is written in assembly for most CPUs. For addition of a variable to itself (i.e., @var{s1p} equals @var{s2p}) use @code{mpn_lshift} with a count of 1 for optimal speed. @end deftypefun @deftypefun mp_limb_t mpn_add_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb}) Add @{@var{s1p}, @var{n}@} and @var{s2limb}, and write the @var{n} least significant limbs of the result to @var{rp}. Return carry, either 0 or 1. @end deftypefun @deftypefun mp_limb_t mpn_add (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n}) Add @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the @var{s1n} least significant limbs of the result to @var{rp}. Return carry, either 0 or 1. This function requires that @var{s1n} is greater than or equal to @var{s2n}. @end deftypefun @deftypefun mp_limb_t mpn_sub_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Subtract @{@var{s2p}, @var{n}@} from @{@var{s1p}, @var{n}@}, and write the @var{n} least significant limbs of the result to @var{rp}. Return borrow, either 0 or 1. This is the lowest-level function for subtraction. It is the preferred function for subtraction, since it is written in assembly for most CPUs. @end deftypefun @deftypefun mp_limb_t mpn_sub_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb}) Subtract @var{s2limb} from @{@var{s1p}, @var{n}@}, and write the @var{n} least significant limbs of the result to @var{rp}. Return borrow, either 0 or 1. @end deftypefun @deftypefun mp_limb_t mpn_sub (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n}) Subtract @{@var{s2p}, @var{s2n}@} from @{@var{s1p}, @var{s1n}@}, and write the @var{s1n} least significant limbs of the result to @var{rp}. Return borrow, either 0 or 1. This function requires that @var{s1n} is greater than or equal to @var{s2n}. @end deftypefun @deftypefun mp_limb_t mpn_neg (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}) Perform the negation of @{@var{sp}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}. Return carry-out. @end deftypefun @deftypefun void mpn_mul_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Multiply @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the 2*@var{n}-limb result to @var{rp}. The destination has to have space for 2*@var{n} limbs, even if the product's most significant limb is zero. No overlap is permitted between the destination and either source. If the two input operands are the same, use @code{mpn_sqr}. @end deftypefun @deftypefun mp_limb_t mpn_mul (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, const mp_limb_t *@var{s2p}, mp_size_t @var{s2n}) Multiply @{@var{s1p}, @var{s1n}@} and @{@var{s2p}, @var{s2n}@}, and write the (@var{s1n}+@var{s2n})-limb result to @var{rp}. Return the most significant limb of the result. The destination has to have space for @var{s1n} + @var{s2n} limbs, even if the product's most significant limb is zero. No overlap is permitted between the destination and either source. This function requires that @var{s1n} is greater than or equal to @var{s2n}. @end deftypefun @deftypefun void mpn_sqr (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}) Compute the square of @{@var{s1p}, @var{n}@} and write the 2*@var{n}-limb result to @var{rp}. The destination has to have space for 2*@var{n} limbs, even if the result's most significant limb is zero. No overlap is permitted between the destination and the source. @end deftypefun @deftypefun mp_limb_t mpn_mul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb}) Multiply @{@var{s1p}, @var{n}@} by @var{s2limb}, and write the @var{n} least significant limbs of the product to @var{rp}. Return the most significant limb of the product. @{@var{s1p}, @var{n}@} and @{@var{rp}, @var{n}@} are allowed to overlap provided @math{@var{rp} @le{} @var{s1p}}. This is a low-level function that is a building block for general multiplication as well as other operations in GMP@. It is written in assembly for most CPUs. Don't call this function if @var{s2limb} is a power of 2; use @code{mpn_lshift} with a count equal to the logarithm of @var{s2limb} instead, for optimal speed. @end deftypefun @deftypefun mp_limb_t mpn_addmul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb}) Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and add the @var{n} least significant limbs of the product to @{@var{rp}, @var{n}@} and write the result to @var{rp}. Return the most significant limb of the product, plus carry-out from the addition. This is a low-level function that is a building block for general multiplication as well as other operations in GMP@. It is written in assembly for most CPUs. @end deftypefun @deftypefun mp_limb_t mpn_submul_1 (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}, mp_limb_t @var{s2limb}) Multiply @{@var{s1p}, @var{n}@} and @var{s2limb}, and subtract the @var{n} least significant limbs of the product from @{@var{rp}, @var{n}@} and write the result to @var{rp}. Return the most significant limb of the product, plus borrow-out from the subtraction. This is a low-level function that is a building block for general multiplication and division as well as other operations in GMP@. It is written in assembly for most CPUs. @end deftypefun @deftypefun void mpn_tdiv_qr (mp_limb_t *@var{qp}, mp_limb_t *@var{rp}, mp_size_t @var{qxn}, const mp_limb_t *@var{np}, mp_size_t @var{nn}, const mp_limb_t *@var{dp}, mp_size_t @var{dn}) Divide @{@var{np}, @var{nn}@} by @{@var{dp}, @var{dn}@} and put the quotient at @{@var{qp}, @var{nn}@minus{}@var{dn}+1@} and the remainder at @{@var{rp}, @var{dn}@}. The quotient is rounded towards 0. No overlap is permitted between arguments, except that @var{np} might equal @var{rp}. The dividend size @var{nn} must be greater than or equal to divisor size @var{dn}. The most significant limb of the divisor must be non-zero. The @var{qxn} operand must be zero. @end deftypefun @deftypefun mp_limb_t mpn_divrem (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n}) [This function is obsolete. Please call @code{mpn_tdiv_qr} instead for best performance.] Divide @{@var{rs2p}, @var{rs2n}@} by @{@var{s3p}, @var{s3n}@}, and write the quotient at @var{r1p}, with the exception of the most significant limb, which is returned. The remainder replaces the dividend at @var{rs2p}; it will be @var{s3n} limbs long (i.e., as many limbs as the divisor). In addition to an integer quotient, @var{qxn} fraction limbs are developed, and stored after the integral limbs. For most usages, @var{qxn} will be zero. It is required that @var{rs2n} is greater than or equal to @var{s3n}. It is required that the most significant bit of the divisor is set. If the quotient is not needed, pass @var{rs2p} + @var{s3n} as @var{r1p}. Aside from that special case, no overlap between arguments is permitted. Return the most significant limb of the quotient, either 0 or 1. The area at @var{r1p} needs to be @var{rs2n} @minus{} @var{s3n} + @var{qxn} limbs large. @end deftypefun @deftypefn Function mp_limb_t mpn_divrem_1 (mp_limb_t *@var{r1p}, mp_size_t @var{qxn}, @w{mp_limb_t *@var{s2p}}, mp_size_t @var{s2n}, mp_limb_t @var{s3limb}) @deftypefnx Macro mp_limb_t mpn_divmod_1 (mp_limb_t *@var{r1p}, mp_limb_t *@var{s2p}, @w{mp_size_t @var{s2n}}, @w{mp_limb_t @var{s3limb}}) Divide @{@var{s2p}, @var{s2n}@} by @var{s3limb}, and write the quotient at @var{r1p}. Return the remainder. The integer quotient is written to @{@var{r1p}+@var{qxn}, @var{s2n}@} and in addition @var{qxn} fraction limbs are developed and written to @{@var{r1p}, @var{qxn}@}. Either or both @var{s2n} and @var{qxn} can be zero. For most usages, @var{qxn} will be zero. @code{mpn_divmod_1} exists for upward source compatibility and is simply a macro calling @code{mpn_divrem_1} with a @var{qxn} of 0. The areas at @var{r1p} and @var{s2p} have to be identical or completely separate, not partially overlapping. @end deftypefn @deftypefun mp_limb_t mpn_divmod (mp_limb_t *@var{r1p}, mp_limb_t *@var{rs2p}, mp_size_t @var{rs2n}, const mp_limb_t *@var{s3p}, mp_size_t @var{s3n}) [This function is obsolete. Please call @code{mpn_tdiv_qr} instead for best performance.] @end deftypefun @deftypefn Macro mp_limb_t mpn_divexact_by3 (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}}) @deftypefnx Function mp_limb_t mpn_divexact_by3c (mp_limb_t *@var{rp}, mp_limb_t *@var{sp}, @w{mp_size_t @var{n}}, mp_limb_t @var{carry}) Divide @{@var{sp}, @var{n}@} by 3, expecting it to divide exactly, and writing the result to @{@var{rp}, @var{n}@}. If 3 divides exactly, the return value is zero and the result is the quotient. If not, the return value is non-zero and the result won't be anything useful. @code{mpn_divexact_by3c} takes an initial carry parameter, which can be the return value from a previous call, so a large calculation can be done piece by piece from low to high. @code{mpn_divexact_by3} is simply a macro calling @code{mpn_divexact_by3c} with a 0 carry parameter. These routines use a multiply-by-inverse and will be faster than @code{mpn_divrem_1} on CPUs with fast multiplication but slow division. The source @math{a}, result @math{q}, size @math{n}, initial carry @math{i}, and return value @math{c} satisfy @m{cb^n+a-i=3q, c*b^n + a-i = 3*q}, where @m{b=2\GMPraise{@code{GMP\_NUMB\_BITS}}, b=2^GMP_NUMB_BITS}. The return @math{c} is always 0, 1 or 2, and the initial carry @math{i} must also be 0, 1 or 2 (these are both borrows really). When @math{c=0} clearly @math{q=(a-i)/3}. When @m{c \neq 0, c!=0}, the remainder @math{(a-i) @bmod{} 3} is given by @math{3-c}, because @math{b @equiv{} 1 @bmod{} 3} (when @code{mp_bits_per_limb} is even, which is always so currently). @end deftypefn @deftypefun mp_limb_t mpn_mod_1 (const mp_limb_t *@var{s1p}, mp_size_t @var{s1n}, mp_limb_t @var{s2limb}) Divide @{@var{s1p}, @var{s1n}@} by @var{s2limb}, and return the remainder. @var{s1n} can be zero. @end deftypefun @deftypefun mp_limb_t mpn_lshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count}) Shift @{@var{sp}, @var{n}@} left by @var{count} bits, and write the result to @{@var{rp}, @var{n}@}. The bits shifted out at the left are returned in the least significant @var{count} bits of the return value (the rest of the return value is zero). @var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1. The regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided @math{@var{rp} @ge{} @var{sp}}. This function is written in assembly for most CPUs. @end deftypefun @deftypefun mp_limb_t mpn_rshift (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}, unsigned int @var{count}) Shift @{@var{sp}, @var{n}@} right by @var{count} bits, and write the result to @{@var{rp}, @var{n}@}. The bits shifted out at the right are returned in the most significant @var{count} bits of the return value (the rest of the return value is zero). @var{count} must be in the range 1 to @nicode{mp_bits_per_limb}@minus{}1. The regions @{@var{sp}, @var{n}@} and @{@var{rp}, @var{n}@} may overlap, provided @math{@var{rp} @le{} @var{sp}}. This function is written in assembly for most CPUs. @end deftypefun @deftypefun int mpn_cmp (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Compare @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@} and return a positive value if @math{@var{s1} > @var{s2}}, 0 if they are equal, or a negative value if @math{@var{s1} < @var{s2}}. @end deftypefun @deftypefun mp_size_t mpn_gcd (mp_limb_t *@var{rp}, mp_limb_t *@var{xp}, mp_size_t @var{xn}, mp_limb_t *@var{yp}, mp_size_t @var{yn}) Set @{@var{rp}, @var{retval}@} to the greatest common divisor of @{@var{xp}, @var{xn}@} and @{@var{yp}, @var{yn}@}. The result can be up to @var{yn} limbs, the return value is the actual number produced. Both source operands are destroyed. It is required that @math{@var{xn} @ge @var{yn} > 0}, and the most significant limb of @{@var{yp}, @var{yn}@} must be non-zero. No overlap is permitted between @{@var{xp}, @var{xn}@} and @{@var{yp}, @var{yn}@}. @end deftypefun @deftypefun mp_limb_t mpn_gcd_1 (const mp_limb_t *@var{xp}, mp_size_t @var{xn}, mp_limb_t @var{ylimb}) Return the greatest common divisor of @{@var{xp}, @var{xn}@} and @var{ylimb}. Both operands must be non-zero. @end deftypefun @deftypefun mp_size_t mpn_gcdext (mp_limb_t *@var{gp}, mp_limb_t *@var{sp}, mp_size_t *@var{sn}, mp_limb_t *@var{up}, mp_size_t @var{un}, mp_limb_t *@var{vp}, mp_size_t @var{vn}) Let @m{U,@var{U}} be defined by @{@var{up}, @var{un}@} and let @m{V,@var{V}} be defined by @{@var{vp}, @var{vn}@}. Compute the greatest common divisor @math{G} of @math{U} and @math{V}. Compute a cofactor @math{S} such that @math{G = US + VT}. The second cofactor @var{T} is not computed but can easily be obtained from @m{(G - US) / V, (@var{G} - @var{U}*@var{S}) / @var{V}} (the division will be exact). It is required that @math{@var{un} @ge @var{vn} > 0}, and the most significant limb of @{@var{vp}, @var{vn}@} must be non-zero. @math{S} satisfies @math{S = 1} or @math{@GMPabs{S} < V / (2 G)}. @math{S = 0} if and only if @math{V} divides @math{U} (i.e., @math{G = V}). Store @math{G} at @var{gp} and let the return value define its limb count. Store @math{S} at @var{sp} and let |*@var{sn}| define its limb count. @math{S} can be negative; when this happens *@var{sn} will be negative. The area at @var{gp} should have room for @var{vn} limbs and the area at @var{sp} should have room for @math{@var{vn}+1} limbs. Both source operands are destroyed. Compatibility notes: GMP 4.3.0 and 4.3.1 defined @math{S} less strictly. Earlier as well as later GMP releases define @math{S} as described here. GMP releases before GMP 4.3.0 required additional space for both input and output areas. More precisely, the areas @{@var{up}, @math{@var{un}+1}@} and @{@var{vp}, @math{@var{vn}+1}@} were destroyed (i.e.@: the operands plus an extra limb past the end of each), and the areas pointed to by @var{gp} and @var{sp} should each have room for @math{@var{un}+1} limbs. @end deftypefun @deftypefun mp_size_t mpn_sqrtrem (mp_limb_t *@var{r1p}, mp_limb_t *@var{r2p}, const mp_limb_t *@var{sp}, mp_size_t @var{n}) Compute the square root of @{@var{sp}, @var{n}@} and put the result at @{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and the remainder at @{@var{r2p}, @var{retval}@}. @var{r2p} needs space for @var{n} limbs, but the return value indicates how many are produced. The most significant limb of @{@var{sp}, @var{n}@} must be non-zero. The areas @{@var{r1p}, @math{@GMPceil{@var{n}/2}}@} and @{@var{sp}, @var{n}@} must be completely separate. The areas @{@var{r2p}, @var{n}@} and @{@var{sp}, @var{n}@} must be either identical or completely separate. If the remainder is not wanted then @var{r2p} can be @code{NULL}, and in this case the return value is zero or non-zero according to whether the remainder would have been zero or non-zero. A return value of zero indicates a perfect square. See also @code{mpn_perfect_square_p}. @end deftypefun @deftypefun mp_size_t mpn_get_str (unsigned char *@var{str}, int @var{base}, mp_limb_t *@var{s1p}, mp_size_t @var{s1n}) Convert @{@var{s1p}, @var{s1n}@} to a raw unsigned char array at @var{str} in base @var{base}, and return the number of characters produced. There may be leading zeros in the string. The string is not in ASCII; to convert it to printable format, add the ASCII codes for @samp{0} or @samp{A}, depending on the base and range. @var{base} can vary from 2 to 256. The most significant limb of the input @{@var{s1p}, @var{s1n}@} must be non-zero. The input @{@var{s1p}, @var{s1n}@} is clobbered, except when @var{base} is a power of 2, in which case it's unchanged. The area at @var{str} has to have space for the largest possible number represented by a @var{s1n} long limb array, plus one extra character. @end deftypefun @deftypefun mp_size_t mpn_set_str (mp_limb_t *@var{rp}, const unsigned char *@var{str}, size_t @var{strsize}, int @var{base}) Convert bytes @{@var{str},@var{strsize}@} in the given @var{base} to limbs at @var{rp}. @math{@var{str}[0]} is the most significant byte and @math{@var{str}[@var{strsize}-1]} is the least significant. Each byte should be a value in the range 0 to @math{@var{base}-1}, not an ASCII character. @var{base} can vary from 2 to 256. The return value is the number of limbs written to @var{rp}. If the most significant input byte is non-zero then the high limb at @var{rp} will be non-zero, and only that exact number of limbs will be required there. If the most significant input byte is zero then there may be high zero limbs written to @var{rp} and included in the return value. @var{strsize} must be at least 1, and no overlap is permitted between @{@var{str},@var{strsize}@} and the result at @var{rp}. @end deftypefun @deftypefun {mp_bitcnt_t} mpn_scan0 (const mp_limb_t *@var{s1p}, mp_bitcnt_t @var{bit}) Scan @var{s1p} from bit position @var{bit} for the next clear bit. It is required that there be a clear bit within the area at @var{s1p} at or beyond bit position @var{bit}, so that the function has something to return. @end deftypefun @deftypefun {mp_bitcnt_t} mpn_scan1 (const mp_limb_t *@var{s1p}, mp_bitcnt_t @var{bit}) Scan @var{s1p} from bit position @var{bit} for the next set bit. It is required that there be a set bit within the area at @var{s1p} at or beyond bit position @var{bit}, so that the function has something to return. @end deftypefun @deftypefun void mpn_random (mp_limb_t *@var{r1p}, mp_size_t @var{r1n}) @deftypefunx void mpn_random2 (mp_limb_t *@var{r1p}, mp_size_t @var{r1n}) Generate a random number of length @var{r1n} and store it at @var{r1p}. The most significant limb is always non-zero. @code{mpn_random} generates uniformly distributed limb data, @code{mpn_random2} generates long strings of zeros and ones in the binary representation. @code{mpn_random2} is intended for testing the correctness of the @code{mpn} routines. @end deftypefun @deftypefun {mp_bitcnt_t} mpn_popcount (const mp_limb_t *@var{s1p}, mp_size_t @var{n}) Count the number of set bits in @{@var{s1p}, @var{n}@}. @end deftypefun @deftypefun {mp_bitcnt_t} mpn_hamdist (const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Compute the hamming distance between @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, which is the number of bit positions where the two operands have different bit values. @end deftypefun @deftypefun int mpn_perfect_square_p (const mp_limb_t *@var{s1p}, mp_size_t @var{n}) Return non-zero iff @{@var{s1p}, @var{n}@} is a perfect square. The most significant limb of the input @{@var{s1p}, @var{n}@} must be non-zero. @end deftypefun @deftypefun void mpn_and_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_ior_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_xor_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical exclusive or of @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_andn_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and the bitwise complement of @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_iorn_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and the bitwise complement of @{@var{s2p}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_nand_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical and of @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the bitwise complement of the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_nior_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical inclusive or of @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the bitwise complement of the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_xnor_n (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, const mp_limb_t *@var{s2p}, mp_size_t @var{n}) Perform the bitwise logical exclusive or of @{@var{s1p}, @var{n}@} and @{@var{s2p}, @var{n}@}, and write the bitwise complement of the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_com (mp_limb_t *@var{rp}, const mp_limb_t *@var{sp}, mp_size_t @var{n}) Perform the bitwise complement of @{@var{sp}, @var{n}@}, and write the result to @{@var{rp}, @var{n}@}. @end deftypefun @deftypefun void mpn_copyi (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}) Copy from @{@var{s1p}, @var{n}@} to @{@var{rp}, @var{n}@}, increasingly. @end deftypefun @deftypefun void mpn_copyd (mp_limb_t *@var{rp}, const mp_limb_t *@var{s1p}, mp_size_t @var{n}) Copy from @{@var{s1p}, @var{n}@} to @{@var{rp}, @var{n}@}, decreasingly. @end deftypefun @deftypefun void mpn_zero (mp_limb_t *@var{rp}, mp_size_t @var{n}) Zero @{@var{rp}, @var{n}@}. @end deftypefun @sp 1 @section Nails @cindex Nails @strong{Everything in this section is highly experimental and may disappear or be subject to incompatible changes in a future version of GMP.} Nails are an experimental feature whereby a few bits are left unused at the top of each @code{mp_limb_t}. This can significantly improve carry handling on some processors. All the @code{mpn} functions accepting limb data will expect the nail bits to be zero on entry, and will return data with the nails similarly all zero. This applies both to limb vectors and to single limb arguments. Nails can be enabled by configuring with @samp{--enable-nails}. By default the number of bits will be chosen according to what suits the host processor, but a particular number can be selected with @samp{--enable-nails=N}. At the mpn level, a nail build is neither source nor binary compatible with a non-nail build, strictly speaking. But programs acting on limbs only through the mpn functions are likely to work equally well with either build, and judicious use of the definitions below should make any program compatible with either build, at the source level. For the higher level routines, meaning @code{mpz} etc, a nail build should be fully source and binary compatible with a non-nail build. @defmac GMP_NAIL_BITS @defmacx GMP_NUMB_BITS @defmacx GMP_LIMB_BITS @code{GMP_NAIL_BITS} is the number of nail bits, or 0 when nails are not in use. @code{GMP_NUMB_BITS} is the number of data bits in a limb. @code{GMP_LIMB_BITS} is the total number of bits in an @code{mp_limb_t}. In all cases @example GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS @end example @end defmac @defmac GMP_NAIL_MASK @defmacx GMP_NUMB_MASK Bit masks for the nail and number parts of a limb. @code{GMP_NAIL_MASK} is 0 when nails are not in use. @code{GMP_NAIL_MASK} is not often needed, since the nail part can be obtained with @code{x >> GMP_NUMB_BITS}, and that means one less large constant, which can help various RISC chips. @end defmac @defmac GMP_NUMB_MAX The maximum value that can be stored in the number part of a limb. This is the same as @code{GMP_NUMB_MASK}, but can be used for clarity when doing comparisons rather than bit-wise operations. @end defmac The term ``nails'' comes from finger or toe nails, which are at the ends of a limb (arm or leg). ``numb'' is short for number, but is also how the developers felt after trying for a long time to come up with sensible names for these things. In the future (the distant future most likely) a non-zero nail might be permitted, giving non-unique representations for numbers in a limb vector. This would help vector processors since carries would only ever need to propagate one or two limbs. @node Random Number Functions, Formatted Output, Low-level Functions, Top @chapter Random Number Functions @cindex Random number functions Sequences of pseudo-random numbers in GMP are generated using a variable of type @code{gmp_randstate_t}, which holds an algorithm selection and a current state. Such a variable must be initialized by a call to one of the @code{gmp_randinit} functions, and can be seeded with one of the @code{gmp_randseed} functions. The functions actually generating random numbers are described in @ref{Integer Random Numbers}, and @ref{Miscellaneous Float Functions}. The older style random number functions don't accept a @code{gmp_randstate_t} parameter but instead share a global variable of that type. They use a default algorithm and are currently not seeded (though perhaps that will change in the future). The new functions accepting a @code{gmp_randstate_t} are recommended for applications that care about randomness. @menu * Random State Initialization:: * Random State Seeding:: * Random State Miscellaneous:: @end menu @node Random State Initialization, Random State Seeding, Random Number Functions, Random Number Functions @section Random State Initialization @cindex Random number state @cindex Initialization functions @deftypefun void gmp_randinit_default (gmp_randstate_t @var{state}) Initialize @var{state} with a default algorithm. This will be a compromise between speed and randomness, and is recommended for applications with no special requirements. Currently this is @code{gmp_randinit_mt}. @end deftypefun @deftypefun void gmp_randinit_mt (gmp_randstate_t @var{state}) @cindex Mersenne twister random numbers Initialize @var{state} for a Mersenne Twister algorithm. This algorithm is fast and has good randomness properties. @end deftypefun @deftypefun void gmp_randinit_lc_2exp (gmp_randstate_t @var{state}, mpz_t @var{a}, @w{unsigned long @var{c}}, @w{mp_bitcnt_t @var{m2exp}}) @cindex Linear congruential random numbers Initialize @var{state} with a linear congruential algorithm @m{X = (@var{a}X + @var{c}) @bmod 2^{m2exp}, X = (@var{a}*X + @var{c}) mod 2^@var{m2exp}}. The low bits of @math{X} in this algorithm are not very random. The least significant bit will have a period no more than 2, and the second bit no more than 4, etc. For this reason only the high half of each @math{X} is actually used. When a random number of more than @math{@var{m2exp}/2} bits is to be generated, multiple iterations of the recurrence are used and the results concatenated. @end deftypefun @deftypefun int gmp_randinit_lc_2exp_size (gmp_randstate_t @var{state}, mp_bitcnt_t @var{size}) @cindex Linear congruential random numbers Initialize @var{state} for a linear congruential algorithm as per @code{gmp_randinit_lc_2exp}. @var{a}, @var{c} and @var{m2exp} are selected from a table, chosen so that @var{size} bits (or more) of each @math{X} will be used, i.e.@: @math{@var{m2exp}/2 @ge{} @var{size}}. If successful the return value is non-zero. If @var{size} is bigger than the table data provides then the return value is zero. The maximum @var{size} currently supported is 128. @end deftypefun @deftypefun void gmp_randinit_set (gmp_randstate_t @var{rop}, gmp_randstate_t @var{op}) Initialize @var{rop} with a copy of the algorithm and state from @var{op}. @end deftypefun @c Although gmp_randinit, gmp_errno and related constants are obsolete, we @c still put @findex entries for them, since they're still documented and @c someone might be looking them up when perusing old application code. @deftypefun void gmp_randinit (gmp_randstate_t @var{state}, @w{gmp_randalg_t @var{alg}}, @dots{}) @strong{This function is obsolete.} @findex GMP_RAND_ALG_LC @findex GMP_RAND_ALG_DEFAULT Initialize @var{state} with an algorithm selected by @var{alg}. The only choice is @code{GMP_RAND_ALG_LC}, which is @code{gmp_randinit_lc_2exp_size} described above. A third parameter of type @code{unsigned long} is required, this is the @var{size} for that function. @code{GMP_RAND_ALG_DEFAULT} or 0 are the same as @code{GMP_RAND_ALG_LC}. @c For reference, this is the only place gmp_errno has been documented, and @c due to being non thread safe we won't be adding to it's uses. @findex gmp_errno @findex GMP_ERROR_UNSUPPORTED_ARGUMENT @findex GMP_ERROR_INVALID_ARGUMENT @code{gmp_randinit} sets bits in the global variable @code{gmp_errno} to indicate an error. @code{GMP_ERROR_UNSUPPORTED_ARGUMENT} if @var{alg} is unsupported, or @code{GMP_ERROR_INVALID_ARGUMENT} if the @var{size} parameter is too big. It may be noted this error reporting is not thread safe (a good reason to use @code{gmp_randinit_lc_2exp_size} instead). @end deftypefun @deftypefun void gmp_randclear (gmp_randstate_t @var{state}) Free all memory occupied by @var{state}. @end deftypefun @node Random State Seeding, Random State Miscellaneous, Random State Initialization, Random Number Functions @section Random State Seeding @cindex Random number seeding @cindex Seeding random numbers @deftypefun void gmp_randseed (gmp_randstate_t @var{state}, mpz_t @var{seed}) @deftypefunx void gmp_randseed_ui (gmp_randstate_t @var{state}, @w{unsigned long int @var{seed}}) Set an initial seed value into @var{state}. The size of a seed determines how many different sequences of random numbers that it's possible to generate. The ``quality'' of the seed is the randomness of a given seed compared to the previous seed used, and this affects the randomness of separate number sequences. The method for choosing a seed is critical if the generated numbers are to be used for important applications, such as generating cryptographic keys. Traditionally the system time has been used to seed, but care needs to be taken with this. If an application seeds often and the resolution of the system clock is low, then the same sequence of numbers might be repeated. Also, the system time is quite easy to guess, so if unpredictability is required then it should definitely not be the only source for the seed value. On some systems there's a special device @file{/dev/random} which provides random data better suited for use as a seed. @end deftypefun @node Random State Miscellaneous, , Random State Seeding, Random Number Functions @section Random State Miscellaneous @deftypefun {unsigned long} gmp_urandomb_ui (gmp_randstate_t @var{state}, unsigned long @var{n}) Return a uniformly distributed random number of @var{n} bits, i.e.@: in the range 0 to @m{2^n-1,2^@var{n}-1} inclusive. @var{n} must be less than or equal to the number of bits in an @code{unsigned long}. @end deftypefun @deftypefun {unsigned long} gmp_urandomm_ui (gmp_randstate_t @var{state}, unsigned long @var{n}) Return a uniformly distributed random number in the range 0 to @math{@var{n}-1}, inclusive. @end deftypefun @node Formatted Output, Formatted Input, Random Number Functions, Top @chapter Formatted Output @cindex Formatted output @cindex @code{printf} formatted output @menu * Formatted Output Strings:: * Formatted Output Functions:: * C++ Formatted Output:: @end menu @node Formatted Output Strings, Formatted Output Functions, Formatted Output, Formatted Output @section Format Strings @code{gmp_printf} and friends accept format strings similar to the standard C @code{printf} (@pxref{Formatted Output,, Formatted Output, libc, The GNU C Library Reference Manual}). A format specification is of the form @example % [flags] [width] [.[precision]] [type] conv @end example GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t} and @code{mpf_t} respectively, @samp{M} for @code{mp_limb_t}, and @samp{N} for an @code{mp_limb_t} array. @samp{Z}, @samp{Q}, @samp{M} and @samp{N} behave like integers. @samp{Q} will print a @samp{/} and a denominator, if needed. @samp{F} behaves like a float. For example, @example mpz_t z; gmp_printf ("%s is an mpz %Zd\n", "here", z); mpq_t q; gmp_printf ("a hex rational: %#40Qx\n", q); mpf_t f; int n; gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n); mp_limb_t l; gmp_printf ("limb %Mu\n", l); const mp_limb_t *ptr; mp_size_t size; gmp_printf ("limb array %Nx\n", ptr, size); @end example For @samp{N} the limbs are expected least significant first, as per the @code{mpn} functions (@pxref{Low-level Functions}). A negative size can be given to print the value as a negative. All the standard C @code{printf} types behave the same as the C library @code{printf}, and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to @code{printf} and only the GMP extensions handled directly. The flags accepted are as follows. GLIBC style @nisamp{'} is only for the standard C types (not the GMP types), and only if the C library supports it. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{0} @tab pad with zeros (rather than spaces) @item @nicode{#} @tab show the base with @samp{0x}, @samp{0X} or @samp{0} @item @nicode{+} @tab always show a sign @item (space) @tab show a space or a @samp{-} sign @item @nicode{'} @tab group digits, GLIBC style (not GMP types) @end multitable @end quotation The optional width and precision can be given as a number within the format string, or as a @samp{*} to take an extra parameter of type @code{int}, the same as the standard @code{printf}. The standard types accepted are as follows. @samp{h} and @samp{l} are portable, the rest will depend on the compiler (or include files) for the type and the C library for the output. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{h} @tab @nicode{short} @item @nicode{hh} @tab @nicode{char} @item @nicode{j} @tab @nicode{intmax_t} or @nicode{uintmax_t} @item @nicode{l} @tab @nicode{long} or @nicode{wchar_t} @item @nicode{ll} @tab @nicode{long long} @item @nicode{L} @tab @nicode{long double} @item @nicode{q} @tab @nicode{quad_t} or @nicode{u_quad_t} @item @nicode{t} @tab @nicode{ptrdiff_t} @item @nicode{z} @tab @nicode{size_t} @end multitable @end quotation @noindent The GMP types are @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{F} @tab @nicode{mpf_t}, float conversions @item @nicode{Q} @tab @nicode{mpq_t}, integer conversions @item @nicode{M} @tab @nicode{mp_limb_t}, integer conversions @item @nicode{N} @tab @nicode{mp_limb_t} array, integer conversions @item @nicode{Z} @tab @nicode{mpz_t}, integer conversions @end multitable @end quotation The conversions accepted are as follows. @samp{a} and @samp{A} are always supported for @code{mpf_t} but depend on the C library for standard C float types. @samp{m} and @samp{p} depend on the C library. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{a} @nicode{A} @tab hex floats, C99 style @item @nicode{c} @tab character @item @nicode{d} @tab decimal integer @item @nicode{e} @nicode{E} @tab scientific format float @item @nicode{f} @tab fixed point float @item @nicode{i} @tab same as @nicode{d} @item @nicode{g} @nicode{G} @tab fixed or scientific float @item @nicode{m} @tab @code{strerror} string, GLIBC style @item @nicode{n} @tab store characters written so far @item @nicode{o} @tab octal integer @item @nicode{p} @tab pointer @item @nicode{s} @tab string @item @nicode{u} @tab unsigned integer @item @nicode{x} @nicode{X} @tab hex integer @end multitable @end quotation @samp{o}, @samp{x} and @samp{X} are unsigned for the standard C types, but for types @samp{Z}, @samp{Q} and @samp{N} they are signed. @samp{u} is not meaningful for @samp{Z}, @samp{Q} and @samp{N}. @samp{M} is a proxy for the C library @samp{l} or @samp{L}, according to the size of @code{mp_limb_t}. Unsigned conversions will be usual, but a signed conversion can be used and will interpret the value as a twos complement negative. @samp{n} can be used with any type, even the GMP types. Other types or conversions that might be accepted by the C library @code{printf} cannot be used through @code{gmp_printf}, this includes for instance extensions registered with GLIBC @code{register_printf_function}. Also currently there's no support for POSIX @samp{$} style numbered arguments (perhaps this will be added in the future). The precision field has its usual meaning for integer @samp{Z} and float @samp{F} types, but is currently undefined for @samp{Q} and should not be used with that. @code{mpf_t} conversions only ever generate as many digits as can be accurately represented by the operand, the same as @code{mpf_get_str} does. Zeros will be used if necessary to pad to the requested precision. This happens even for an @samp{f} conversion of an @code{mpf_t} which is an integer, for instance @math{2^@W{1024}} in an @code{mpf_t} of 128 bits precision will only produce about 40 digits, then pad with zeros to the decimal point. An empty precision field like @samp{%.Fe} or @samp{%.Ff} can be used to specifically request just the significant digits. Without any dot and thus no precision field, a precision value of 6 will be used. Note that these rules mean that @samp{%Ff}, @samp{%.Ff}, and @samp{%.0Ff} will all be different. The decimal point character (or string) is taken from the current locale settings on systems which provide @code{localeconv} (@pxref{Locales,, Locales and Internationalization, libc, The GNU C Library Reference Manual}). The C library will normally do the same for standard float output. The format string is only interpreted as plain @code{char}s, multibyte characters are not recognised. Perhaps this will change in the future. @node Formatted Output Functions, C++ Formatted Output, Formatted Output Strings, Formatted Output @section Functions @cindex Output functions Each of the following functions is similar to the corresponding C library function. The basic @code{printf} forms take a variable argument list. The @code{vprintf} forms take an argument pointer, see @ref{Variadic Functions,, Variadic Functions, libc, The GNU C Library Reference Manual}, or @samp{man 3 va_start}. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. The file based functions @code{gmp_printf} and @code{gmp_fprintf} will return @math{-1} to indicate a write error. Output is not ``atomic'', so partial output may be produced if a write error occurs. All the functions can return @math{-1} if the C library @code{printf} variant in use returns @math{-1}, but this shouldn't normally occur. @deftypefun int gmp_printf (const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vprintf (const char *@var{fmt}, va_list @var{ap}) Print to the standard output @code{stdout}. Return the number of characters written, or @math{-1} if an error occurred. @end deftypefun @deftypefun int gmp_fprintf (FILE *@var{fp}, const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vfprintf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap}) Print to the stream @var{fp}. Return the number of characters written, or @math{-1} if an error occurred. @end deftypefun @deftypefun int gmp_sprintf (char *@var{buf}, const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vsprintf (char *@var{buf}, const char *@var{fmt}, va_list @var{ap}) Form a null-terminated string in @var{buf}. Return the number of characters written, excluding the terminating null. No overlap is permitted between the space at @var{buf} and the string @var{fmt}. These functions are not recommended, since there's no protection against exceeding the space available at @var{buf}. @end deftypefun @deftypefun int gmp_snprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vsnprintf (char *@var{buf}, size_t @var{size}, const char *@var{fmt}, va_list @var{ap}) Form a null-terminated string in @var{buf}. No more than @var{size} bytes will be written. To get the full output, @var{size} must be enough for the string and null-terminator. The return value is the total number of characters which ought to have been produced, excluding the terminating null. If @math{@var{retval} @ge{} @var{size}} then the actual output has been truncated to the first @math{@var{size}-1} characters, and a null appended. No overlap is permitted between the region @{@var{buf},@var{size}@} and the @var{fmt} string. Notice the return value is in ISO C99 @code{snprintf} style. This is so even if the C library @code{vsnprintf} is the older GLIBC 2.0.x style. @end deftypefun @deftypefun int gmp_asprintf (char **@var{pp}, const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vasprintf (char **@var{pp}, const char *@var{fmt}, va_list @var{ap}) Form a null-terminated string in a block of memory obtained from the current memory allocation function (@pxref{Custom Allocation}). The block will be the size of the string and null-terminator. The address of the block in stored to *@var{pp}. The return value is the number of characters produced, excluding the null-terminator. Unlike the C library @code{asprintf}, @code{gmp_asprintf} doesn't return @math{-1} if there's no more memory available, it lets the current allocation function handle that. @end deftypefun @deftypefun int gmp_obstack_printf (struct obstack *@var{ob}, const char *@var{fmt}, @dots{}) @deftypefunx int gmp_obstack_vprintf (struct obstack *@var{ob}, const char *@var{fmt}, va_list @var{ap}) @cindex @code{obstack} output Append to the current object in @var{ob}. The return value is the number of characters written. A null-terminator is not written. @var{fmt} cannot be within the current object in @var{ob}, since that object might move as it grows. These functions are available only when the C library provides the obstack feature, which probably means only on GNU systems, see @ref{Obstacks,, Obstacks, libc, The GNU C Library Reference Manual}. @end deftypefun @node C++ Formatted Output, , Formatted Output Functions, Formatted Output @section C++ Formatted Output @cindex C++ @code{ostream} output @cindex @code{ostream} output The following functions are provided in @file{libgmpxx} (@pxref{Headers and Libraries}), which is built if C++ support is enabled (@pxref{Build Options}). Prototypes are available from @code{}. @deftypefun ostream& operator<< (ostream& @var{stream}, mpz_t @var{op}) Print @var{op} to @var{stream}, using its @code{ios} formatting settings. @code{ios::width} is reset to 0 after output, the same as the standard @code{ostream operator<<} routines do. In hex or octal, @var{op} is printed as a signed number, the same as for decimal. This is unlike the standard @code{operator<<} routines on @code{int} etc, which instead give twos complement. @end deftypefun @deftypefun ostream& operator<< (ostream& @var{stream}, mpq_t @var{op}) Print @var{op} to @var{stream}, using its @code{ios} formatting settings. @code{ios::width} is reset to 0 after output, the same as the standard @code{ostream operator<<} routines do. Output will be a fraction like @samp{5/9}, or if the denominator is 1 then just a plain integer like @samp{123}. In hex or octal, @var{op} is printed as a signed value, the same as for decimal. If @code{ios::showbase} is set then a base indicator is shown on both the numerator and denominator (if the denominator is required). @end deftypefun @deftypefun ostream& operator<< (ostream& @var{stream}, mpf_t @var{op}) Print @var{op} to @var{stream}, using its @code{ios} formatting settings. @code{ios::width} is reset to 0 after output, the same as the standard @code{ostream operator<<} routines do. The decimal point follows the standard library float @code{operator<<}, which on recent systems means the @code{std::locale} imbued on @var{stream}. Hex and octal are supported, unlike the standard @code{operator<<} on @code{double}. The mantissa will be in hex or octal, the exponent will be in decimal. For hex the exponent delimiter is an @samp{@@}. This is as per @code{mpf_out_str}. @code{ios::showbase} is supported, and will put a base on the mantissa, for example hex @samp{0x1.8} or @samp{0x0.8}, or octal @samp{01.4} or @samp{00.4}. This last form is slightly strange, but at least differentiates itself from decimal. @end deftypefun These operators mean that GMP types can be printed in the usual C++ way, for example, @example mpz_t z; int n; ... cout << "iteration " << n << " value " << z << "\n"; @end example But note that @code{ostream} output (and @code{istream} input, @pxref{C++ Formatted Input}) is the only overloading available for the GMP types and that for instance using @code{+} with an @code{mpz_t} will have unpredictable results. For classes with overloading, see @ref{C++ Class Interface}. @node Formatted Input, C++ Class Interface, Formatted Output, Top @chapter Formatted Input @cindex Formatted input @cindex @code{scanf} formatted input @menu * Formatted Input Strings:: * Formatted Input Functions:: * C++ Formatted Input:: @end menu @node Formatted Input Strings, Formatted Input Functions, Formatted Input, Formatted Input @section Formatted Input Strings @code{gmp_scanf} and friends accept format strings similar to the standard C @code{scanf} (@pxref{Formatted Input,, Formatted Input, libc, The GNU C Library Reference Manual}). A format specification is of the form @example % [flags] [width] [type] conv @end example GMP adds types @samp{Z}, @samp{Q} and @samp{F} for @code{mpz_t}, @code{mpq_t} and @code{mpf_t} respectively. @samp{Z} and @samp{Q} behave like integers. @samp{Q} will read a @samp{/} and a denominator, if present. @samp{F} behaves like a float. GMP variables don't require an @code{&} when passed to @code{gmp_scanf}, since they're already ``call-by-reference''. For example, @example /* to read say "a(5) = 1234" */ int n; mpz_t z; gmp_scanf ("a(%d) = %Zd\n", &n, z); mpq_t q1, q2; gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2); /* to read say "topleft (1.55,-2.66)" */ mpf_t x, y; char buf[32]; gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y); @end example All the standard C @code{scanf} types behave the same as in the C library @code{scanf}, and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to @code{scanf} and only the GMP extensions handled directly. The flags accepted are as follows. @samp{a} and @samp{'} will depend on support from the C library, and @samp{'} cannot be used with GMP types. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{*} @tab read but don't store @item @nicode{a} @tab allocate a buffer (string conversions) @item @nicode{'} @tab grouped digits, GLIBC style (not GMP types) @end multitable @end quotation The standard types accepted are as follows. @samp{h} and @samp{l} are portable, the rest will depend on the compiler (or include files) for the type and the C library for the input. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{h} @tab @nicode{short} @item @nicode{hh} @tab @nicode{char} @item @nicode{j} @tab @nicode{intmax_t} or @nicode{uintmax_t} @item @nicode{l} @tab @nicode{long int}, @nicode{double} or @nicode{wchar_t} @item @nicode{ll} @tab @nicode{long long} @item @nicode{L} @tab @nicode{long double} @item @nicode{q} @tab @nicode{quad_t} or @nicode{u_quad_t} @item @nicode{t} @tab @nicode{ptrdiff_t} @item @nicode{z} @tab @nicode{size_t} @end multitable @end quotation @noindent The GMP types are @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{F} @tab @nicode{mpf_t}, float conversions @item @nicode{Q} @tab @nicode{mpq_t}, integer conversions @item @nicode{Z} @tab @nicode{mpz_t}, integer conversions @end multitable @end quotation The conversions accepted are as follows. @samp{p} and @samp{[} will depend on support from the C library, the rest are standard. @quotation @multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item @nicode{c} @tab character or characters @item @nicode{d} @tab decimal integer @item @nicode{e} @nicode{E} @nicode{f} @nicode{g} @nicode{G} @tab float @item @nicode{i} @tab integer with base indicator @item @nicode{n} @tab characters read so far @item @nicode{o} @tab octal integer @item @nicode{p} @tab pointer @item @nicode{s} @tab string of non-whitespace characters @item @nicode{u} @tab decimal integer @item @nicode{x} @nicode{X} @tab hex integer @item @nicode{[} @tab string of characters in a set @end multitable @end quotation @samp{e}, @samp{E}, @samp{f}, @samp{g} and @samp{G} are identical, they all read either fixed point or scientific format, and either upper or lower case @samp{e} for the exponent in scientific format. C99 style hex float format (@code{printf %a}, @pxref{Formatted Output Strings}) is always accepted for @code{mpf_t}, but for the standard float types it will depend on the C library. @samp{x} and @samp{X} are identical, both accept both upper and lower case hexadecimal. @samp{o}, @samp{u}, @samp{x} and @samp{X} all read positive or negative values. For the standard C types these are described as ``unsigned'' conversions, but that merely affects certain overflow handling, negatives are still allowed (per @code{strtoul}, @pxref{Parsing of Integers,, Parsing of Integers, libc, The GNU C Library Reference Manual}). For GMP types there are no overflows, so @samp{d} and @samp{u} are identical. @samp{Q} type reads the numerator and (optional) denominator as given. If the value might not be in canonical form then @code{mpq_canonicalize} must be called before using it in any calculations (@pxref{Rational Number Functions}). @samp{Qi} will read a base specification separately for the numerator and denominator. For example @samp{0x10/11} would be 16/11, whereas @samp{0x10/0x11} would be 16/17. @samp{n} can be used with any of the types above, even the GMP types. @samp{*} to suppress assignment is allowed, though in that case it would do nothing at all. Other conversions or types that might be accepted by the C library @code{scanf} cannot be used through @code{gmp_scanf}. Whitespace is read and discarded before a field, except for @samp{c} and @samp{[} conversions. For float conversions, the decimal point character (or string) expected is taken from the current locale settings on systems which provide @code{localeconv} (@pxref{Locales,, Locales and Internationalization, libc, The GNU C Library Reference Manual}). The C library will normally do the same for standard float input. The format string is only interpreted as plain @code{char}s, multibyte characters are not recognised. Perhaps this will change in the future. @node Formatted Input Functions, C++ Formatted Input, Formatted Input Strings, Formatted Input @section Formatted Input Functions @cindex Input functions Each of the following functions is similar to the corresponding C library function. The plain @code{scanf} forms take a variable argument list. The @code{vscanf} forms take an argument pointer, see @ref{Variadic Functions,, Variadic Functions, libc, The GNU C Library Reference Manual}, or @samp{man 3 va_start}. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. No overlap is permitted between the @var{fmt} string and any of the results produced. @deftypefun int gmp_scanf (const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vscanf (const char *@var{fmt}, va_list @var{ap}) Read from the standard input @code{stdin}. @end deftypefun @deftypefun int gmp_fscanf (FILE *@var{fp}, const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vfscanf (FILE *@var{fp}, const char *@var{fmt}, va_list @var{ap}) Read from the stream @var{fp}. @end deftypefun @deftypefun int gmp_sscanf (const char *@var{s}, const char *@var{fmt}, @dots{}) @deftypefunx int gmp_vsscanf (const char *@var{s}, const char *@var{fmt}, va_list @var{ap}) Read from a null-terminated string @var{s}. @end deftypefun The return value from each of these functions is the same as the standard C99 @code{scanf}, namely the number of fields successfully parsed and stored. @samp{%n} fields and fields read but suppressed by @samp{*} don't count towards the return value. If end of input (or a file error) is reached before a character for a field or a literal, and if no previous non-suppressed fields have matched, then the return value is @code{EOF} instead of 0. A whitespace character in the format string is only an optional match and doesn't induce an @code{EOF} in this fashion. Leading whitespace read and discarded for a field don't count as characters for that field. For the GMP types, input parsing follows C99 rules, namely one character of lookahead is used and characters are read while they continue to meet the format requirements. If this doesn't provide a complete number then the function terminates, with that field not stored nor counted towards the return value. For instance with @code{mpf_t} an input @samp{1.23e-XYZ} would be read up to the @samp{X} and that character pushed back since it's not a digit. The string @samp{1.23e-} would then be considered invalid since an @samp{e} must be followed by at least one digit. For the standard C types, in the current implementation GMP calls the C library @code{scanf} functions, which might have looser rules about what constitutes a valid input. Note that @code{gmp_sscanf} is the same as @code{gmp_fscanf} and only does one character of lookahead when parsing. Although clearly it could look at its entire input, it is deliberately made identical to @code{gmp_fscanf}, the same way C99 @code{sscanf} is the same as @code{fscanf}. @node C++ Formatted Input, , Formatted Input Functions, Formatted Input @section C++ Formatted Input @cindex C++ @code{istream} input @cindex @code{istream} input The following functions are provided in @file{libgmpxx} (@pxref{Headers and Libraries}), which is built only if C++ support is enabled (@pxref{Build Options}). Prototypes are available from @code{}. @deftypefun istream& operator>> (istream& @var{stream}, mpz_t @var{rop}) Read @var{rop} from @var{stream}, using its @code{ios} formatting settings. @end deftypefun @deftypefun istream& operator>> (istream& @var{stream}, mpq_t @var{rop}) An integer like @samp{123} will be read, or a fraction like @samp{5/9}. No whitespace is allowed around the @samp{/}. If the fraction is not in canonical form then @code{mpq_canonicalize} must be called (@pxref{Rational Number Functions}) before operating on it. As per integer input, an @samp{0} or @samp{0x} base indicator is read when none of @code{ios::dec}, @code{ios::oct} or @code{ios::hex} are set. This is done separately for numerator and denominator, so that for instance @samp{0x10/11} is @math{16/11} and @samp{0x10/0x11} is @math{16/17}. @end deftypefun @deftypefun istream& operator>> (istream& @var{stream}, mpf_t @var{rop}) Read @var{rop} from @var{stream}, using its @code{ios} formatting settings. Hex or octal floats are not supported, but might be in the future, or perhaps it's best to accept only what the standard float @code{operator>>} does. @end deftypefun Note that digit grouping specified by the @code{istream} locale is currently not accepted. Perhaps this will change in the future. @sp 1 These operators mean that GMP types can be read in the usual C++ way, for example, @example mpz_t z; ... cin >> z; @end example But note that @code{istream} input (and @code{ostream} output, @pxref{C++ Formatted Output}) is the only overloading available for the GMP types and that for instance using @code{+} with an @code{mpz_t} will have unpredictable results. For classes with overloading, see @ref{C++ Class Interface}. @node C++ Class Interface, Custom Allocation, Formatted Input, Top @chapter C++ Class Interface @cindex C++ interface This chapter describes the C++ class based interface to GMP. All GMP C language types and functions can be used in C++ programs, since @file{gmp.h} has @code{extern "C"} qualifiers, but the class interface offers overloaded functions and operators which may be more convenient. Due to the implementation of this interface, a reasonably recent C++ compiler is required, one supporting namespaces, partial specialization of templates and member templates. For GCC this means version 2.91 or later. @strong{Everything described in this chapter is to be considered preliminary and might be subject to incompatible changes if some unforeseen difficulty reveals itself.} @menu * C++ Interface General:: * C++ Interface Integers:: * C++ Interface Rationals:: * C++ Interface Floats:: * C++ Interface Random Numbers:: * C++ Interface Limitations:: @end menu @node C++ Interface General, C++ Interface Integers, C++ Class Interface, C++ Class Interface @section C++ Interface General @noindent All the C++ classes and functions are available with @cindex @code{gmpxx.h} @example #include @end example Programs should be linked with the @file{libgmpxx} and @file{libgmp} libraries. For example, @example g++ mycxxprog.cc -lgmpxx -lgmp @end example @noindent The classes defined are @deftp Class mpz_class @deftpx Class mpq_class @deftpx Class mpf_class @end deftp The standard operators and various standard functions are overloaded to allow arithmetic with these classes. For example, @example int main (void) @{ mpz_class a, b, c; a = 1234; b = "-5678"; c = a+b; cout << "sum is " << c << "\n"; cout << "absolute value is " << abs(c) << "\n"; return 0; @} @end example An important feature of the implementation is that an expression like @code{a=b+c} results in a single call to the corresponding @code{mpz_add}, without using a temporary for the @code{b+c} part. Expressions which by their nature imply intermediate values, like @code{a=b*c+d*e}, still use temporaries though. The classes can be freely intermixed in expressions, as can the classes and the standard types @code{long}, @code{unsigned long} and @code{double}. Smaller types like @code{int} or @code{float} can also be intermixed, since C++ will promote them. Note that @code{bool} is not accepted directly, but must be explicitly cast to an @code{int} first. This is because C++ will automatically convert any pointer to a @code{bool}, so if GMP accepted @code{bool} it would make all sorts of invalid class and pointer combinations compile but almost certainly not do anything sensible. Conversions back from the classes to standard C++ types aren't done automatically, instead member functions like @code{get_si} are provided (see the following sections for details). Also there are no automatic conversions from the classes to the corresponding GMP C types, instead a reference to the underlying C object can be obtained with the following functions, @deftypefun mpz_t mpz_class::get_mpz_t () @deftypefunx mpq_t mpq_class::get_mpq_t () @deftypefunx mpf_t mpf_class::get_mpf_t () @end deftypefun These can be used to call a C function which doesn't have a C++ class interface. For example to set @code{a} to the GCD of @code{b} and @code{c}, @example mpz_class a, b, c; ... mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t()); @end example In the other direction, a class can be initialized from the corresponding GMP C type, or assigned to if an explicit constructor is used. In both cases this makes a copy of the value, it doesn't create any sort of association. For example, @example mpz_t z; // ... init and calculate z ... mpz_class x(z); mpz_class y; y = mpz_class (z); @end example There are no namespace setups in @file{gmpxx.h}, all types and functions are simply put into the global namespace. This is what @file{gmp.h} has done in the past, and continues to do for compatibility. The extras provided by @file{gmpxx.h} follow GMP naming conventions and are unlikely to clash with anything. @node C++ Interface Integers, C++ Interface Rationals, C++ Interface General, C++ Class Interface @section C++ Interface Integers @deftypefun {} mpz_class::mpz_class (type @var{n}) Construct an @code{mpz_class}. All the standard C++ types may be used, except @code{long long} and @code{long double}, and all the GMP C++ classes can be used, although conversions from @code{mpq_class} and @code{mpf_class} are @code{explicit}. Any necessary conversion follows the corresponding C function, for example @code{double} follows @code{mpz_set_d} (@pxref{Assigning Integers}). @end deftypefun @deftypefun explicit mpz_class::mpz_class (mpz_t @var{z}) Construct an @code{mpz_class} from an @code{mpz_t}. The value in @var{z} is copied into the new @code{mpz_class}, there won't be any permanent association between it and @var{z}. @end deftypefun @deftypefun explicit mpz_class::mpz_class (const char *@var{s}, int @var{base} = 0) @deftypefunx explicit mpz_class::mpz_class (const string& @var{s}, int @var{base} = 0) Construct an @code{mpz_class} converted from a string using @code{mpz_set_str} (@pxref{Assigning Integers}). If the string is not a valid integer, an @code{std::invalid_argument} exception is thrown. The same applies to @code{operator=}. @end deftypefun @deftypefun mpz_class operator"" _mpz (const char *@var{str}) With C++11 compilers, integers can be constructed with the syntax @code{123_mpz} which is equivalent to @code{mpz_class("123")}. @end deftypefun @deftypefun mpz_class operator/ (mpz_class @var{a}, mpz_class @var{d}) @deftypefunx mpz_class operator% (mpz_class @var{a}, mpz_class @var{d}) Divisions involving @code{mpz_class} round towards zero, as per the @code{mpz_tdiv_q} and @code{mpz_tdiv_r} functions (@pxref{Integer Division}). This is the same as the C99 @code{/} and @code{%} operators. The @code{mpz_fdiv@dots{}} or @code{mpz_cdiv@dots{}} functions can always be called directly if desired. For example, @example mpz_class q, a, d; ... mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t()); @end example @end deftypefun @deftypefun mpz_class abs (mpz_class @var{op}) @deftypefunx int cmp (mpz_class @var{op1}, type @var{op2}) @deftypefunx int cmp (type @var{op1}, mpz_class @var{op2}) @maybepagebreak @deftypefunx bool mpz_class::fits_sint_p (void) @deftypefunx bool mpz_class::fits_slong_p (void) @deftypefunx bool mpz_class::fits_sshort_p (void) @maybepagebreak @deftypefunx bool mpz_class::fits_uint_p (void) @deftypefunx bool mpz_class::fits_ulong_p (void) @deftypefunx bool mpz_class::fits_ushort_p (void) @maybepagebreak @deftypefunx double mpz_class::get_d (void) @deftypefunx long mpz_class::get_si (void) @deftypefunx string mpz_class::get_str (int @var{base} = 10) @deftypefunx {unsigned long} mpz_class::get_ui (void) @maybepagebreak @deftypefunx int mpz_class::set_str (const char *@var{str}, int @var{base}) @deftypefunx int mpz_class::set_str (const string& @var{str}, int @var{base}) @deftypefunx int sgn (mpz_class @var{op}) @deftypefunx mpz_class sqrt (mpz_class @var{op}) @maybepagebreak @deftypefunx void mpz_class::swap (mpz_class& @var{op}) @deftypefunx void swap (mpz_class& @var{op1}, mpz_class& @var{op2}) These functions provide a C++ class interface to the corresponding GMP C routines. @code{cmp} can be used with any of the classes or the standard C++ types, except @code{long long} and @code{long double}. @end deftypefun @sp 1 Overloaded operators for combinations of @code{mpz_class} and @code{double} are provided for completeness, but it should be noted that if the given @code{double} is not an integer then the way any rounding is done is currently unspecified. The rounding might take place at the start, in the middle, or at the end of the operation, and it might change in the future. Conversions between @code{mpz_class} and @code{double}, however, are defined to follow the corresponding C functions @code{mpz_get_d} and @code{mpz_set_d}. And comparisons are always made exactly, as per @code{mpz_cmp_d}. @node C++ Interface Rationals, C++ Interface Floats, C++ Interface Integers, C++ Class Interface @section C++ Interface Rationals In all the following constructors, if a fraction is given then it should be in canonical form, or if not then @code{mpq_class::canonicalize} called. @deftypefun {} mpq_class::mpq_class (type @var{op}) @deftypefunx {} mpq_class::mpq_class (integer @var{num}, integer @var{den}) Construct an @code{mpq_class}. The initial value can be a single value of any type (conversion from @code{mpf_class} is @code{explicit}), or a pair of integers (@code{mpz_class} or standard C++ integer types) representing a fraction, except that @code{long long} and @code{long double} are not supported. For example, @example mpq_class q (99); mpq_class q (1.75); mpq_class q (1, 3); @end example @end deftypefun @deftypefun explicit mpq_class::mpq_class (mpq_t @var{q}) Construct an @code{mpq_class} from an @code{mpq_t}. The value in @var{q} is copied into the new @code{mpq_class}, there won't be any permanent association between it and @var{q}. @end deftypefun @deftypefun explicit mpq_class::mpq_class (const char *@var{s}, int @var{base} = 0) @deftypefunx explicit mpq_class::mpq_class (const string& @var{s}, int @var{base} = 0) Construct an @code{mpq_class} converted from a string using @code{mpq_set_str} (@pxref{Initializing Rationals}). If the string is not a valid rational, an @code{std::invalid_argument} exception is thrown. The same applies to @code{operator=}. @end deftypefun @deftypefun mpq_class operator"" _mpq (const char *@var{str}) With C++11 compilers, integral rationals can be constructed with the syntax @code{123_mpq} which is equivalent to @code{mpq_class(123_mpz)}. Other rationals can be built as @code{-1_mpq/2} or @code{0xb_mpq/123456_mpz}. @end deftypefun @deftypefun void mpq_class::canonicalize () Put an @code{mpq_class} into canonical form, as per @ref{Rational Number Functions}. All arithmetic operators require their operands in canonical form, and will return results in canonical form. @end deftypefun @deftypefun mpq_class abs (mpq_class @var{op}) @deftypefunx int cmp (mpq_class @var{op1}, type @var{op2}) @deftypefunx int cmp (type @var{op1}, mpq_class @var{op2}) @maybepagebreak @deftypefunx double mpq_class::get_d (void) @deftypefunx string mpq_class::get_str (int @var{base} = 10) @maybepagebreak @deftypefunx int mpq_class::set_str (const char *@var{str}, int @var{base}) @deftypefunx int mpq_class::set_str (const string& @var{str}, int @var{base}) @deftypefunx int sgn (mpq_class @var{op}) @maybepagebreak @deftypefunx void mpq_class::swap (mpq_class& @var{op}) @deftypefunx void swap (mpq_class& @var{op1}, mpq_class& @var{op2}) These functions provide a C++ class interface to the corresponding GMP C routines. @code{cmp} can be used with any of the classes or the standard C++ types, except @code{long long} and @code{long double}. @end deftypefun @deftypefun {mpz_class&} mpq_class::get_num () @deftypefunx {mpz_class&} mpq_class::get_den () Get a reference to an @code{mpz_class} which is the numerator or denominator of an @code{mpq_class}. This can be used both for read and write access. If the object returned is modified, it modifies the original @code{mpq_class}. If direct manipulation might produce a non-canonical value, then @code{mpq_class::canonicalize} must be called before further operations. @end deftypefun @deftypefun mpz_t mpq_class::get_num_mpz_t () @deftypefunx mpz_t mpq_class::get_den_mpz_t () Get a reference to the underlying @code{mpz_t} numerator or denominator of an @code{mpq_class}. This can be passed to C functions expecting an @code{mpz_t}. Any modifications made to the @code{mpz_t} will modify the original @code{mpq_class}. If direct manipulation might produce a non-canonical value, then @code{mpq_class::canonicalize} must be called before further operations. @end deftypefun @deftypefun istream& operator>> (istream& @var{stream}, mpq_class& @var{rop}); Read @var{rop} from @var{stream}, using its @code{ios} formatting settings, the same as @code{mpq_t operator>>} (@pxref{C++ Formatted Input}). If the @var{rop} read might not be in canonical form then @code{mpq_class::canonicalize} must be called. @end deftypefun @node C++ Interface Floats, C++ Interface Random Numbers, C++ Interface Rationals, C++ Class Interface @section C++ Interface Floats When an expression requires the use of temporary intermediate @code{mpf_class} values, like @code{f=g*h+x*y}, those temporaries will have the same precision as the destination @code{f}. Explicit constructors can be used if this doesn't suit. @deftypefun {} mpf_class::mpf_class (type @var{op}) @deftypefunx {} mpf_class::mpf_class (type @var{op}, mp_bitcnt_t @var{prec}) Construct an @code{mpf_class}. Any standard C++ type can be used, except @code{long long} and @code{long double}, and any of the GMP C++ classes can be used. If @var{prec} is given, the initial precision is that value, in bits. If @var{prec} is not given, then the initial precision is determined by the type of @var{op} given. An @code{mpz_class}, @code{mpq_class}, or C++ builtin type will give the default @code{mpf} precision (@pxref{Initializing Floats}). An @code{mpf_class} or expression will give the precision of that value. The precision of a binary expression is the higher of the two operands. @example mpf_class f(1.5); // default precision mpf_class f(1.5, 500); // 500 bits (at least) mpf_class f(x); // precision of x mpf_class f(abs(x)); // precision of x mpf_class f(-g, 1000); // 1000 bits (at least) mpf_class f(x+y); // greater of precisions of x and y @end example @end deftypefun @deftypefun explicit mpf_class::mpf_class (mpf_t @var{f}) @deftypefunx {} mpf_class::mpf_class (mpf_t @var{f}, mp_bitcnt_t @var{prec}) Construct an @code{mpf_class} from an @code{mpf_t}. The value in @var{f} is copied into the new @code{mpf_class}, there won't be any permanent association between it and @var{f}. If @var{prec} is given, the initial precision is that value, in bits. If @var{prec} is not given, then the initial precision is that of @var{f}. @end deftypefun @deftypefun explicit mpf_class::mpf_class (const char *@var{s}) @deftypefunx {} mpf_class::mpf_class (const char *@var{s}, mp_bitcnt_t @var{prec}, int @var{base} = 0) @deftypefunx explicit mpf_class::mpf_class (const string& @var{s}) @deftypefunx {} mpf_class::mpf_class (const string& @var{s}, mp_bitcnt_t @var{prec}, int @var{base} = 0) Construct an @code{mpf_class} converted from a string using @code{mpf_set_str} (@pxref{Assigning Floats}). If @var{prec} is given, the initial precision is that value, in bits. If not, the default @code{mpf} precision (@pxref{Initializing Floats}) is used. If the string is not a valid float, an @code{std::invalid_argument} exception is thrown. The same applies to @code{operator=}. @end deftypefun @deftypefun mpf_class operator"" _mpf (const char *@var{str}) With C++11 compilers, floats can be constructed with the syntax @code{1.23e-1_mpf} which is equivalent to @code{mpf_class("1.23e-1")}. @end deftypefun @deftypefun {mpf_class&} mpf_class::operator= (type @var{op}) Convert and store the given @var{op} value to an @code{mpf_class} object. The same types are accepted as for the constructors above. Note that @code{operator=} only stores a new value, it doesn't copy or change the precision of the destination, instead the value is truncated if necessary. This is the same as @code{mpf_set} etc. Note in particular this means for @code{mpf_class} a copy constructor is not the same as a default constructor plus assignment. @example mpf_class x (y); // x created with precision of y mpf_class x; // x created with default precision x = y; // value truncated to that precision @end example Applications using templated code may need to be careful about the assumptions the code makes in this area, when working with @code{mpf_class} values of various different or non-default precisions. For instance implementations of the standard @code{complex} template have been seen in both styles above, though of course @code{complex} is normally only actually specified for use with the builtin float types. @end deftypefun @deftypefun mpf_class abs (mpf_class @var{op}) @deftypefunx mpf_class ceil (mpf_class @var{op}) @deftypefunx int cmp (mpf_class @var{op1}, type @var{op2}) @deftypefunx int cmp (type @var{op1}, mpf_class @var{op2}) @maybepagebreak @deftypefunx bool mpf_class::fits_sint_p (void) @deftypefunx bool mpf_class::fits_slong_p (void) @deftypefunx bool mpf_class::fits_sshort_p (void) @maybepagebreak @deftypefunx bool mpf_class::fits_uint_p (void) @deftypefunx bool mpf_class::fits_ulong_p (void) @deftypefunx bool mpf_class::fits_ushort_p (void) @maybepagebreak @deftypefunx mpf_class floor (mpf_class @var{op}) @deftypefunx mpf_class hypot (mpf_class @var{op1}, mpf_class @var{op2}) @maybepagebreak @deftypefunx double mpf_class::get_d (void) @deftypefunx long mpf_class::get_si (void) @deftypefunx string mpf_class::get_str (mp_exp_t& @var{exp}, int @var{base} = 10, size_t @var{digits} = 0) @deftypefunx {unsigned long} mpf_class::get_ui (void) @maybepagebreak @deftypefunx int mpf_class::set_str (const char *@var{str}, int @var{base}) @deftypefunx int mpf_class::set_str (const string& @var{str}, int @var{base}) @deftypefunx int sgn (mpf_class @var{op}) @deftypefunx mpf_class sqrt (mpf_class @var{op}) @maybepagebreak @deftypefunx void mpf_class::swap (mpf_class& @var{op}) @deftypefunx void swap (mpf_class& @var{op1}, mpf_class& @var{op2}) @deftypefunx mpf_class trunc (mpf_class @var{op}) These functions provide a C++ class interface to the corresponding GMP C routines. @code{cmp} can be used with any of the classes or the standard C++ types, except @code{long long} and @code{long double}. The accuracy provided by @code{hypot} is not currently guaranteed. @end deftypefun @deftypefun {mp_bitcnt_t} mpf_class::get_prec () @deftypefunx void mpf_class::set_prec (mp_bitcnt_t @var{prec}) @deftypefunx void mpf_class::set_prec_raw (mp_bitcnt_t @var{prec}) Get or set the current precision of an @code{mpf_class}. The restrictions described for @code{mpf_set_prec_raw} (@pxref{Initializing Floats}) apply to @code{mpf_class::set_prec_raw}. Note in particular that the @code{mpf_class} must be restored to it's allocated precision before being destroyed. This must be done by application code, there's no automatic mechanism for it. @end deftypefun @node C++ Interface Random Numbers, C++ Interface Limitations, C++ Interface Floats, C++ Class Interface @section C++ Interface Random Numbers @deftp Class gmp_randclass The C++ class interface to the GMP random number functions uses @code{gmp_randclass} to hold an algorithm selection and current state, as per @code{gmp_randstate_t}. @end deftp @deftypefun {} gmp_randclass::gmp_randclass (void (*@var{randinit}) (gmp_randstate_t, @dots{}), @dots{}) Construct a @code{gmp_randclass}, using a call to the given @var{randinit} function (@pxref{Random State Initialization}). The arguments expected are the same as @var{randinit}, but with @code{mpz_class} instead of @code{mpz_t}. For example, @example gmp_randclass r1 (gmp_randinit_default); gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32); gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp); gmp_randclass r4 (gmp_randinit_mt); @end example @code{gmp_randinit_lc_2exp_size} will fail if the size requested is too big, an @code{std::length_error} exception is thrown in that case. @end deftypefun @deftypefun {} gmp_randclass::gmp_randclass (gmp_randalg_t @var{alg}, @dots{}) Construct a @code{gmp_randclass} using the same parameters as @code{gmp_randinit} (@pxref{Random State Initialization}). This function is obsolete and the above @var{randinit} style should be preferred. @end deftypefun @deftypefun void gmp_randclass::seed (unsigned long int @var{s}) @deftypefunx void gmp_randclass::seed (mpz_class @var{s}) Seed a random number generator. See @pxref{Random Number Functions}, for how to choose a good seed. @end deftypefun @deftypefun mpz_class gmp_randclass::get_z_bits (mp_bitcnt_t @var{bits}) @deftypefunx mpz_class gmp_randclass::get_z_bits (mpz_class @var{bits}) Generate a random integer with a specified number of bits. @end deftypefun @deftypefun mpz_class gmp_randclass::get_z_range (mpz_class @var{n}) Generate a random integer in the range 0 to @math{@var{n}-1} inclusive. @end deftypefun @deftypefun mpf_class gmp_randclass::get_f () @deftypefunx mpf_class gmp_randclass::get_f (mp_bitcnt_t @var{prec}) Generate a random float @var{f} in the range @math{0 <= @var{f} < 1}. @var{f} will be to @var{prec} bits precision, or if @var{prec} is not given then to the precision of the destination. For example, @example gmp_randclass r; ... mpf_class f (0, 512); // 512 bits precision f = r.get_f(); // random number, 512 bits @end example @end deftypefun @node C++ Interface Limitations, , C++ Interface Random Numbers, C++ Class Interface @section C++ Interface Limitations @table @asis @item @code{mpq_class} and Templated Reading A generic piece of template code probably won't know that @code{mpq_class} requires a @code{canonicalize} call if inputs read with @code{operator>>} might be non-canonical. This can lead to incorrect results. @code{operator>>} behaves as it does for reasons of efficiency. A canonicalize can be quite time consuming on large operands, and is best avoided if it's not necessary. But this potential difficulty reduces the usefulness of @code{mpq_class}. Perhaps a mechanism to tell @code{operator>>} what to do will be adopted in the future, maybe a preprocessor define, a global flag, or an @code{ios} flag pressed into service. Or maybe, at the risk of inconsistency, the @code{mpq_class} @code{operator>>} could canonicalize and leave @code{mpq_t} @code{operator>>} not doing so, for use on those occasions when that's acceptable. Send feedback or alternate ideas to @email{gmp-bugs@@gmplib.org}. @item Subclassing Subclassing the GMP C++ classes works, but is not currently recommended. Expressions involving subclasses resolve correctly (or seem to), but in normal C++ fashion the subclass doesn't inherit constructors and assignments. There's many of those in the GMP classes, and a good way to reestablish them in a subclass is not yet provided. @item Templated Expressions A subtle difficulty exists when using expressions together with application-defined template functions. Consider the following, with @code{T} intended to be some numeric type, @example template T fun (const T &, const T &); @end example @noindent When used with, say, plain @code{mpz_class} variables, it works fine: @code{T} is resolved as @code{mpz_class}. @example mpz_class f(1), g(2); fun (f, g); // Good @end example @noindent But when one of the arguments is an expression, it doesn't work. @example mpz_class f(1), g(2), h(3); fun (f, g+h); // Bad @end example This is because @code{g+h} ends up being a certain expression template type internal to @code{gmpxx.h}, which the C++ template resolution rules are unable to automatically convert to @code{mpz_class}. The workaround is simply to add an explicit cast. @example mpz_class f(1), g(2), h(3); fun (f, mpz_class(g+h)); // Good @end example Similarly, within @code{fun} it may be necessary to cast an expression to type @code{T} when calling a templated @code{fun2}. @example template void fun (T f, T g) @{ fun2 (f, f+g); // Bad @} template void fun (T f, T g) @{ fun2 (f, T(f+g)); // Good @} @end example @end table @node Custom Allocation, Language Bindings, C++ Class Interface, Top @comment node-name, next, previous, up @chapter Custom Allocation @cindex Custom allocation @cindex Memory allocation @cindex Allocation of memory By default GMP uses @code{malloc}, @code{realloc} and @code{free} for memory allocation, and if they fail GMP prints a message to the standard error output and terminates the program. Alternate functions can be specified, to allocate memory in a different way or to have a different error action on running out of memory. @deftypefun void mp_set_memory_functions (@* void *(*@var{alloc_func_ptr}) (size_t), @* void *(*@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (*@var{free_func_ptr}) (void *, size_t)) Replace the current allocation functions from the arguments. If an argument is @code{NULL}, the corresponding default function is used. These functions will be used for all memory allocation done by GMP, apart from temporary space from @code{alloca} if that function is available and GMP is configured to use it (@pxref{Build Options}). @strong{Be sure to call @code{mp_set_memory_functions} only when there are no active GMP objects allocated using the previous memory functions! Usually that means calling it before any other GMP function.} @end deftypefun The functions supplied should fit the following declarations: @deftypevr Function {void *} allocate_function (size_t @var{alloc_size}) Return a pointer to newly allocated space with at least @var{alloc_size} bytes. @end deftypevr @deftypevr Function {void *} reallocate_function (void *@var{ptr}, size_t @var{old_size}, size_t @var{new_size}) Resize a previously allocated block @var{ptr} of @var{old_size} bytes to be @var{new_size} bytes. The block may be moved if necessary or if desired, and in that case the smaller of @var{old_size} and @var{new_size} bytes must be copied to the new location. The return value is a pointer to the resized block, that being the new location if moved or just @var{ptr} if not. @var{ptr} is never @code{NULL}, it's always a previously allocated block. @var{new_size} may be bigger or smaller than @var{old_size}. @end deftypevr @deftypevr Function void free_function (void *@var{ptr}, size_t @var{size}) De-allocate the space pointed to by @var{ptr}. @var{ptr} is never @code{NULL}, it's always a previously allocated block of @var{size} bytes. @end deftypevr A @dfn{byte} here means the unit used by the @code{sizeof} operator. The @var{reallocate_function} parameter @var{old_size} and the @var{free_function} parameter @var{size} are passed for convenience, but of course they can be ignored if not needed by an implementation. The default functions using @code{malloc} and friends for instance don't use them. No error return is allowed from any of these functions, if they return then they must have performed the specified operation. In particular note that @var{allocate_function} or @var{reallocate_function} mustn't return @code{NULL}. Getting a different fatal error action is a good use for custom allocation functions, for example giving a graphical dialog rather than the default print to @code{stderr}. How much is possible when genuinely out of memory is another question though. There's currently no defined way for the allocation functions to recover from an error such as out of memory, they must terminate program execution. A @code{longjmp} or throwing a C++ exception will have undefined results. This may change in the future. GMP may use allocated blocks to hold pointers to other allocated blocks. This will limit the assumptions a conservative garbage collection scheme can make. Since the default GMP allocation uses @code{malloc} and friends, those functions will be linked in even if the first thing a program does is an @code{mp_set_memory_functions}. It's necessary to change the GMP sources if this is a problem. @sp 1 @deftypefun void mp_get_memory_functions (@* void *(**@var{alloc_func_ptr}) (size_t), @* void *(**@var{realloc_func_ptr}) (void *, size_t, size_t), @* void (**@var{free_func_ptr}) (void *, size_t)) Get the current allocation functions, storing function pointers to the locations given by the arguments. If an argument is @code{NULL}, that function pointer is not stored. @need 1000 For example, to get just the current free function, @example void (*freefunc) (void *, size_t); mp_get_memory_functions (NULL, NULL, &freefunc); @end example @end deftypefun @node Language Bindings, Algorithms, Custom Allocation, Top @chapter Language Bindings @cindex Language bindings @cindex Other languages The following packages and projects offer access to GMP from languages other than C, though perhaps with varying levels of functionality and efficiency. @c @spaceuref{U} is the same as @uref{U}, but with a couple of extra spaces @c in tex, just to separate the URL from the preceding text a bit. @iftex @macro spaceuref {U} @ @ @uref{\U\} @end macro @end iftex @ifnottex @macro spaceuref {U} @uref{\U\} @end macro @end ifnottex @sp 1 @table @asis @item C++ @itemize @bullet @item GMP C++ class interface, @pxref{C++ Class Interface} @* Straightforward interface, expression templates to eliminate temporaries. @item ALP @spaceuref{http://www-sop.inria.fr/saga/logiciels/ALP/} @* Linear algebra and polynomials using templates. @item Arithmos @spaceuref{http://cant.ua.ac.be/old/arithmos/} @* Rationals with infinities and square roots. @item CLN @spaceuref{http://www.ginac.de/CLN/} @* High level classes for arithmetic. @item LiDIA @spaceuref{http://www.cdc.informatik.tu-darmstadt.de/TI/LiDIA/} @* A C++ library for computational number theory. @item Linbox @spaceuref{http://www.linalg.org/} @* Sparse vectors and matrices. @item NTL @spaceuref{http://www.shoup.net/ntl/} @* A C++ number theory library. @end itemize @c @item D @c @itemize @bullet @c @item @c gmp-d @spaceuref{http://home.comcast.net/~benhinkle/gmp-d/} @c @end itemize @item Eiffel @itemize @bullet @item Eiffelroom @spaceuref{http://www.eiffelroom.org/node/442} @end itemize @item Fortran @itemize @bullet @item Omni F77 @spaceuref{http://phase.hpcc.jp/Omni/home.html} @* Arbitrary precision floats. @end itemize @item Haskell @itemize @bullet @item Glasgow Haskell Compiler @spaceuref{http://www.haskell.org/ghc/} @end itemize @item Java @itemize @bullet @item Kaffe @spaceuref{http://www.kaffe.org/} @item Kissme @spaceuref{http://kissme.sourceforge.net/} @end itemize @item Lisp @itemize @bullet @item GNU Common Lisp @spaceuref{http://www.gnu.org/software/gcl/gcl.html} @item Librep @spaceuref{http://librep.sourceforge.net/} @item @c FIXME: When there's a stable release with gmp support, just refer to it @c rather than bothering to talk about betas. XEmacs (21.5.18 beta and up) @spaceuref{http://www.xemacs.org} @* Optional big integers, rationals and floats using GMP. @end itemize @item M4 @itemize @bullet @item @c FIXME: When there's a stable release with gmp support, just refer to it @c rather than bothering to talk about betas. GNU m4 betas @spaceuref{http://www.seindal.dk/rene/gnu/} @* Optionally provides an arbitrary precision @code{mpeval}. @end itemize @item ML @itemize @bullet @item MLton compiler @spaceuref{http://mlton.org/} @end itemize @item Objective Caml @itemize @bullet @item MLGMP @spaceuref{http://www.di.ens.fr/~monniaux/programmes.html.en} @item Numerix @spaceuref{http://pauillac.inria.fr/~quercia/} @* Optionally using GMP. @end itemize @item Oz @itemize @bullet @item Mozart @spaceuref{http://www.mozart-oz.org/} @end itemize @item Pascal @itemize @bullet @item GNU Pascal Compiler @spaceuref{http://www.gnu-pascal.de/} @* GMP unit. @item Numerix @spaceuref{http://pauillac.inria.fr/~quercia/} @* For Free Pascal, optionally using GMP. @end itemize @item Perl @itemize @bullet @item GMP module, see @file{demos/perl} in the GMP sources (@pxref{Demonstration Programs}). @item Math::GMP @spaceuref{http://www.cpan.org/} @* Compatible with Math::BigInt, but not as many functions as the GMP module above. @item Math::BigInt::GMP @spaceuref{http://www.cpan.org/} @* Plug Math::GMP into normal Math::BigInt operations. @end itemize @need 1000 @item Pike @itemize @bullet @item mpz module in the standard distribution, @uref{http://pike.ida.liu.se/} @end itemize @need 500 @item Prolog @itemize @bullet @item SWI Prolog @spaceuref{http://www.swi-prolog.org/} @* Arbitrary precision floats. @end itemize @item Python @itemize @bullet @item GMPY @uref{http://code.google.com/p/gmpy/} @end itemize @item Ruby @itemize @bullet @item http://rubygems.org/gems/gmp @end itemize @item Scheme @itemize @bullet @item GNU Guile (upcoming 1.8) @spaceuref{http://www.gnu.org/software/guile/guile.html} @item RScheme @spaceuref{http://www.rscheme.org/} @item STklos @spaceuref{http://www.stklos.org/} @c @c For reference, MzScheme uses some of gmp, but (as of version 205) it only @c has copies of some of the generic C code, and we don't consider that a @c language binding to gmp. @c @end itemize @item Smalltalk @itemize @bullet @item GNU Smalltalk @spaceuref{http://www.smalltalk.org/versions/GNUSmalltalk.html} @end itemize @item Other @itemize @bullet @item Axiom @uref{http://savannah.nongnu.org/projects/axiom} @* Computer algebra using GCL. @item DrGenius @spaceuref{http://drgenius.seul.org/} @* Geometry system and mathematical programming language. @item GiNaC @spaceuref{http://www.ginac.de/} @* C++ computer algebra using CLN. @item GOO @spaceuref{http://www.googoogaga.org/} @* Dynamic object oriented language. @item Maxima @uref{http://www.ma.utexas.edu/users/wfs/maxima.html} @* Macsyma computer algebra using GCL. @item Q @spaceuref{http://q-lang.sourceforge.net/} @* Equational programming system. @item Regina @spaceuref{http://regina.sourceforge.net/} @* Topological calculator. @item Yacas @spaceuref{yacas.sourceforge.net} @* Yet another computer algebra system. @end itemize @end table @node Algorithms, Internals, Language Bindings, Top @chapter Algorithms @cindex Algorithms This chapter is an introduction to some of the algorithms used for various GMP operations. The code is likely to be hard to understand without knowing something about the algorithms. Some GMP internals are mentioned, but applications that expect to be compatible with future GMP releases should take care to use only the documented functions. @menu * Multiplication Algorithms:: * Division Algorithms:: * Greatest Common Divisor Algorithms:: * Powering Algorithms:: * Root Extraction Algorithms:: * Radix Conversion Algorithms:: * Other Algorithms:: * Assembly Coding:: @end menu @node Multiplication Algorithms, Division Algorithms, Algorithms, Algorithms @section Multiplication @cindex Multiplication algorithms N@cross{}N limb multiplications and squares are done using one of seven algorithms, as the size N increases. @quotation @multitable {KaratsubaMMM} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item Algorithm @tab Threshold @item Basecase @tab (none) @item Karatsuba @tab @code{MUL_TOOM22_THRESHOLD} @item Toom-3 @tab @code{MUL_TOOM33_THRESHOLD} @item Toom-4 @tab @code{MUL_TOOM44_THRESHOLD} @item Toom-6.5 @tab @code{MUL_TOOM6H_THRESHOLD} @item Toom-8.5 @tab @code{MUL_TOOM8H_THRESHOLD} @item FFT @tab @code{MUL_FFT_THRESHOLD} @end multitable @end quotation Similarly for squaring, with the @code{SQR} thresholds. N@cross{}M multiplications of operands with different sizes above @code{MUL_TOOM22_THRESHOLD} are currently done by special Toom-inspired algorithms or directly with FFT, depending on operand size (@pxref{Unbalanced Multiplication}). @menu * Basecase Multiplication:: * Karatsuba Multiplication:: * Toom 3-Way Multiplication:: * Toom 4-Way Multiplication:: * Higher degree Toom'n'half:: * FFT Multiplication:: * Other Multiplication:: * Unbalanced Multiplication:: @end menu @node Basecase Multiplication, Karatsuba Multiplication, Multiplication Algorithms, Multiplication Algorithms @subsection Basecase Multiplication Basecase N@cross{}M multiplication is a straightforward rectangular set of cross-products, the same as long multiplication done by hand and for that reason sometimes known as the schoolbook or grammar school method. This is an @m{O(NM),O(N*M)} algorithm. See Knuth section 4.3.1 algorithm M (@pxref{References}), and the @file{mpn/generic/mul_basecase.c} code. Assembly implementations of @code{mpn_mul_basecase} are essentially the same as the generic C code, but have all the usual assembly tricks and obscurities introduced for speed. A square can be done in roughly half the time of a multiply, by using the fact that the cross products above and below the diagonal are the same. A triangle of products below the diagonal is formed, doubled (left shift by one bit), and then the products on the diagonal added. This can be seen in @file{mpn/generic/sqr_basecase.c}. Again the assembly implementations take essentially the same approach. @tex \def\GMPline#1#2#3#4#5#6{% \hbox {% \vrule height 2.5ex depth 1ex \hbox to 2em {\hfil{#2}\hfil}% \vrule \hbox to 2em {\hfil{#3}\hfil}% \vrule \hbox to 2em {\hfil{#4}\hfil}% \vrule \hbox to 2em {\hfil{#5}\hfil}% \vrule \hbox to 2em {\hfil{#6}\hfil}% \vrule}} \GMPdisplay{ \hbox{% \vbox{% \hbox to 1.5em {\vrule height 2.5ex depth 1ex width 0pt}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u0\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u1\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u2\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u3\hfil}% \hbox {\vrule height 2.5ex depth 1ex width 0pt u4\hfil}% \vfill}% \vbox{% \hbox{% \hbox to 2em {\hfil u0\hfil}% \hbox to 2em {\hfil u1\hfil}% \hbox to 2em {\hfil u2\hfil}% \hbox to 2em {\hfil u3\hfil}% \hbox to 2em {\hfil u4\hfil}}% \vskip 0.7ex \hrule \GMPline{u0}{d}{}{}{}{}% \hrule \GMPline{u1}{}{d}{}{}{}% \hrule \GMPline{u2}{}{}{d}{}{}% \hrule \GMPline{u3}{}{}{}{d}{}% \hrule \GMPline{u4}{}{}{}{}{d}% \hrule}}} @end tex @ifnottex @example @group u0 u1 u2 u3 u4 +---+---+---+---+---+ u0 | d | | | | | +---+---+---+---+---+ u1 | | d | | | | +---+---+---+---+---+ u2 | | | d | | | +---+---+---+---+---+ u3 | | | | d | | +---+---+---+---+---+ u4 | | | | | d | +---+---+---+---+---+ @end group @end example @end ifnottex In practice squaring isn't a full 2@cross{} faster than multiplying, it's usually around 1.5@cross{}. Less than 1.5@cross{} probably indicates @code{mpn_sqr_basecase} wants improving on that CPU. On some CPUs @code{mpn_mul_basecase} can be faster than the generic C @code{mpn_sqr_basecase} on some small sizes. @code{SQR_BASECASE_THRESHOLD} is the size at which to use @code{mpn_sqr_basecase}, this will be zero if that routine should be used always. @node Karatsuba Multiplication, Toom 3-Way Multiplication, Basecase Multiplication, Multiplication Algorithms @subsection Karatsuba Multiplication @cindex Karatsuba multiplication The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks. A brief description is given here. The inputs @math{x} and @math{y} are treated as each split into two parts of equal length (or the most significant part one limb shorter if N is odd). @tex % GMPboxwidth used for all the multiplication pictures \global\newdimen\GMPboxwidth \global\GMPboxwidth=5em % GMPboxdepth and GMPboxheight are also used for the float pictures \global\newdimen\GMPboxdepth \global\GMPboxdepth=1ex \global\newdimen\GMPboxheight \global\GMPboxheight=2ex \gdef\GMPvrule{\vrule height \GMPboxheight depth \GMPboxdepth} \def\GMPbox#1#2{% \vbox {% \hrule \hbox to 2\GMPboxwidth{% \GMPvrule \hfil $#1$\hfil \vrule \hfil $#2$\hfil \vrule}% \hrule}} \GMPdisplay{% \vbox{% \hbox to 2\GMPboxwidth {high \hfil low} \vskip 0.7ex \GMPbox{x_1}{x_0} \vskip 0.5ex \GMPbox{y_1}{y_0} }} @end tex @ifnottex @example @group high low +----------+----------+ | x1 | x0 | +----------+----------+ +----------+----------+ | y1 | y0 | +----------+----------+ @end group @end example @end ifnottex Let @math{b} be the power of 2 where the split occurs, i.e.@: if @ms{x,0} is @math{k} limbs (@ms{y,0} the same) then @m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}. With that @m{x=x_1b+x_0,x=x1*b+x0} and @m{y=y_1b+y_0,y=y1*b+y0}, and the following holds, @display @m{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0, x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0} @end display This formula means doing only three multiplies of (N/2)@cross{}(N/2) limbs, whereas a basecase multiply of N@cross{}N limbs is equivalent to four multiplies of (N/2)@cross{}(N/2). The factors @math{(b^2+b)} etc represent the positions where the three products must be added. @tex \def\GMPboxA#1#2{% \vbox{% \hrule \hbox{% \GMPvrule \hbox to 2\GMPboxwidth {\hfil\hbox{$#1$}\hfil}% \vrule \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}% \vrule} \hrule}} \def\GMPboxB#1#2{% \hbox{% \raise \GMPboxdepth \hbox to \GMPboxwidth {\hfil #1\hskip 0.5em}% \vbox{% \hrule \hbox{% \GMPvrule \hbox to 2\GMPboxwidth {\hfil\hbox{$#2$}\hfil}% \vrule}% \hrule}}} \GMPdisplay{% \vbox{% \hbox to 4\GMPboxwidth {high \hfil low} \vskip 0.7ex \GMPboxA{x_1y_1}{x_0y_0} \vskip 0.5ex \GMPboxB{$+$}{x_1y_1} \vskip 0.5ex \GMPboxB{$+$}{x_0y_0} \vskip 0.5ex \GMPboxB{$-$}{(x_1-x_0)(y_1-y_0)} }} @end tex @ifnottex @example @group high low +--------+--------+ +--------+--------+ | x1*y1 | | x0*y0 | +--------+--------+ +--------+--------+ +--------+--------+ add | x1*y1 | +--------+--------+ +--------+--------+ add | x0*y0 | +--------+--------+ +--------+--------+ sub | (x1-x0)*(y1-y0) | +--------+--------+ @end group @end example @end ifnottex The term @m{(x_1-x_0)(y_1-y_0),(x1-x0)*(y1-y0)} is best calculated as an absolute value, and the sign used to choose to add or subtract. Notice the sum @m{\mathop{\rm high}(x_0y_0)+\mathop{\rm low}(x_1y_1), high(x0*y0)+low(x1*y1)} occurs twice, so it's possible to do @m{5k,5*k} limb additions, rather than @m{6k,6*k}, but in GMP extra function call overheads outweigh the saving. Squaring is similar to multiplying, but with @math{x=y} the formula reduces to an equivalent with three squares, @display @m{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2, x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2} @end display The final result is accumulated from those three squares the same way as for the three multiplies above. The middle term @m{(x_1-x_0)^2,(x1-x0)^2} is now always positive. A similar formula for both multiplying and squaring can be constructed with a middle term @m{(x_1+x_0)(y_1+y_0),(x1+x0)*(y1+y0)}. But those sums can exceed @math{k} limbs, leading to more carry handling and additions than the form above. Karatsuba multiplication is asymptotically an @math{O(N^@W{1.585})} algorithm, the exponent being @m{\log3/\log2,log(3)/log(2)}, representing 3 multiplies each @math{1/2} the size of the inputs. This is a big improvement over the basecase multiply at @math{O(N^2)} and the advantage soon overcomes the extra additions Karatsuba performs. @code{MUL_TOOM22_THRESHOLD} can be as little as 10 limbs. The @code{SQR} threshold is usually about twice the @code{MUL}. The basecase algorithm will take a time of the form @m{M(N) = aN^2 + bN + c, M(N) = a*N^2 + b*N + c} and the Karatsuba algorithm @m{K(N) = 3M(N/2) + dN + e, K(N) = 3*M(N/2) + d*N + e}, which expands to @m{K(N) = {3\over4} aN^2 + {3\over2} bN + 3c + dN + e, K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e}. The factor @m{3\over4, 3/4} for @math{a} means per-crossproduct speedups in the basecase code will increase the threshold since they benefit @math{M(N)} more than @math{K(N)}. And conversely the @m{3\over2, 3/2} for @math{b} means linear style speedups of @math{b} will increase the threshold since they benefit @math{K(N)} more than @math{M(N)}. The latter can be seen for instance when adding an optimized @code{mpn_sqr_diagonal} to @code{mpn_sqr_basecase}. Of course all speedups reduce total time, and in that sense the algorithm thresholds are merely of academic interest. @node Toom 3-Way Multiplication, Toom 4-Way Multiplication, Karatsuba Multiplication, Multiplication Algorithms @subsection Toom 3-Way Multiplication @cindex Toom multiplication The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom and FFT algorithms. A description of Toom can be found in Knuth section 4.3.3, with an example 3-way calculation after Theorem A@. The 3-way form used in GMP is described here. The operands are each considered split into 3 pieces of equal length (or the most significant part 1 or 2 limbs shorter than the other two). @tex \def\GMPbox#1#2#3{% \vbox{% \hrule \vfil \hbox to 3\GMPboxwidth {% \GMPvrule \hfil$#1$\hfil \vrule \hfil$#2$\hfil \vrule \hfil$#3$\hfil \vrule}% \vfil \hrule }} \GMPdisplay{% \vbox{% \hbox to 3\GMPboxwidth {high \hfil low} \vskip 0.7ex \GMPbox{x_2}{x_1}{x_0} \vskip 0.5ex \GMPbox{y_2}{y_1}{y_0} \vskip 0.5ex }} @end tex @ifnottex @example @group high low +----------+----------+----------+ | x2 | x1 | x0 | +----------+----------+----------+ +----------+----------+----------+ | y2 | y1 | y0 | +----------+----------+----------+ @end group @end example @end ifnottex @noindent These parts are treated as the coefficients of two polynomials @display @group @m{X(t) = x_2t^2 + x_1t + x_0, X(t) = x2*t^2 + x1*t + x0} @m{Y(t) = y_2t^2 + y_1t + y_0, Y(t) = y2*t^2 + y1*t + y0} @end group @end display Let @math{b} equal the power of 2 which is the size of the @ms{x,0}, @ms{x,1}, @ms{y,0} and @ms{y,1} pieces, i.e.@: if they're @math{k} limbs each then @m{b=2\GMPraise{$k*$@code{mp\_bits\_per\_limb}}, b=2^(k*mp_bits_per_limb)}. With this @math{x=X(b)} and @math{y=Y(b)}. Let a polynomial @m{W(t)=X(t)Y(t),W(t)=X(t)*Y(t)} and suppose its coefficients are @display @m{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0, W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0} @end display The @m{w_i,w[i]} are going to be determined, and when they are they'll give the final result using @math{w=W(b)}, since @m{xy=X(b)Y(b),x*y=X(b)*Y(b)=W(b)}. The coefficients will be roughly @math{b^2} each, and the final @math{W(b)} will be an addition like, @tex \def\GMPbox#1#2{% \moveright #1\GMPboxwidth \vbox{% \hrule \hbox{% \GMPvrule \hbox to 2\GMPboxwidth {\hfil$#2$\hfil}% \vrule}% \hrule }} \GMPdisplay{% \vbox{% \hbox to 6\GMPboxwidth {high \hfil low}% \vskip 0.7ex \GMPbox{0}{w_4} \vskip 0.5ex \GMPbox{1}{w_3} \vskip 0.5ex \GMPbox{2}{w_2} \vskip 0.5ex \GMPbox{3}{w_1} \vskip 0.5ex \GMPbox{4}{w_0} }} @end tex @ifnottex @example @group high low +-------+-------+ | w4 | +-------+-------+ +--------+-------+ | w3 | +--------+-------+ +--------+-------+ | w2 | +--------+-------+ +--------+-------+ | w1 | +--------+-------+ +-------+-------+ | w0 | +-------+-------+ @end group @end example @end ifnottex The @m{w_i,w[i]} coefficients could be formed by a simple set of cross products, like @m{w_4=x_2y_2,w4=x2*y2}, @m{w_3=x_2y_1+x_1y_2,w3=x2*y1+x1*y2}, @m{w_2=x_2y_0+x_1y_1+x_0y_2,w2=x2*y0+x1*y1+x0*y2} etc, but this would need all nine @m{x_iy_j,x[i]*y[j]} for @math{i,j=0,1,2}, and would be equivalent merely to a basecase multiply. Instead the following approach is used. @math{X(t)} and @math{Y(t)} are evaluated and multiplied at 5 points, giving values of @math{W(t)} at those points. In GMP the following points are used, @quotation @multitable {@m{t=\infty,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item Point @tab Value @item @math{t=0} @tab @m{x_0y_0,x0 * y0}, which gives @ms{w,0} immediately @item @math{t=1} @tab @m{(x_2+x_1+x_0)(y_2+y_1+y_0),(x2+x1+x0) * (y2+y1+y0)} @item @math{t=-1} @tab @m{(x_2-x_1+x_0)(y_2-y_1+y_0),(x2-x1+x0) * (y2-y1+y0)} @item @math{t=2} @tab @m{(4x_2+2x_1+x_0)(4y_2+2y_1+y_0),(4*x2+2*x1+x0) * (4*y2+2*y1+y0)} @item @m{t=\infty,t=inf} @tab @m{x_2y_2,x2 * y2}, which gives @ms{w,4} immediately @end multitable @end quotation At @math{t=-1} the values can be negative and that's handled using the absolute values and tracking the sign separately. At @m{t=\infty,t=inf} the value is actually @m{\lim_{t\to\infty} {X(t)Y(t)\over t^4}, X(t)*Y(t)/t^4 in the limit as t approaches infinity}, but it's much easier to think of as simply @m{x_2y_2,x2*y2} giving @ms{w,4} immediately (much like @m{x_0y_0,x0*y0} at @math{t=0} gives @ms{w,0} immediately). Each of the points substituted into @m{W(t)=w_4t^4+\cdots+w_0,W(t)=w4*t^4+@dots{}+w0} gives a linear combination of the @m{w_i,w[i]} coefficients, and the value of those combinations has just been calculated. @tex \GMPdisplay{% $\matrix{% W(0) & = & & & & & & & & & w_0 \cr W(1) & = & w_4 & + & w_3 & + & w_2 & + & w_1 & + & w_0 \cr W(-1) & = & w_4 & - & w_3 & + & w_2 & - & w_1 & + & w_0 \cr W(2) & = & 16w_4 & + & 8w_3 & + & 4w_2 & + & 2w_1 & + & w_0 \cr W(\infty) & = & w_4 \cr }$} @end tex @ifnottex @example @group W(0) = w0 W(1) = w4 + w3 + w2 + w1 + w0 W(-1) = w4 - w3 + w2 - w1 + w0 W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0 W(inf) = w4 @end group @end example @end ifnottex This is a set of five equations in five unknowns, and some elementary linear algebra quickly isolates each @m{w_i,w[i]}. This involves adding or subtracting one @math{W(t)} value from another, and a couple of divisions by powers of 2 and one division by 3, the latter using the special @code{mpn_divexact_by3} (@pxref{Exact Division}). The conversion of @math{W(t)} values to the coefficients is interpolation. A polynomial of degree 4 like @math{W(t)} is uniquely determined by values known at 5 different points. The points are arbitrary and can be chosen to make the linear equations come out with a convenient set of steps for quickly isolating the @m{w_i,w[i]}. Squaring follows the same procedure as multiplication, but there's only one @math{X(t)} and it's evaluated at the 5 points, and those values squared to give values of @math{W(t)}. The interpolation is then identical, and in fact the same @code{toom_interpolate_5pts} subroutine is used for both squaring and multiplying. Toom-3 is asymptotically @math{O(N^@W{1.465})}, the exponent being @m{\log5/\log3,log(5)/log(3)}, representing 5 recursive multiplies of 1/3 the original size each. This is an improvement over Karatsuba at @math{O(N^@W{1.585})}, though Toom does more work in the evaluation and interpolation and so it only realizes its advantage above a certain size. Near the crossover between Toom-3 and Karatsuba there's generally a range of sizes where the difference between the two is small. @code{MUL_TOOM33_THRESHOLD} is a somewhat arbitrary point in that range and successive runs of the tune program can give different values due to small variations in measuring. A graph of time versus size for the two shows the effect, see @file{tune/README}. At the fairly small sizes where the Toom-3 thresholds occur it's worth remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be expected to make accurate predictions, due of course to the big influence of all sorts of overheads, and the fact that only a few recursions of each are being performed. Even at large sizes there's a good chance machine dependent effects like cache architecture will mean actual performance deviates from what might be predicted. The formula given for the Karatsuba algorithm (@pxref{Karatsuba Multiplication}) has an equivalent for Toom-3 involving only five multiplies, but this would be complicated and unenlightening. An alternate view of Toom-3 can be found in Zuras (@pxref{References}), using a vector to represent the @math{x} and @math{y} splits and a matrix multiplication for the evaluation and interpolation stages. The matrix inverses are not meant to be actually used, and they have elements with values much greater than in fact arise in the interpolation steps. The diagram shown for the 3-way is attractive, but again doesn't have to be implemented that way and for example with a bit of rearrangement just one division by 6 can be done. @node Toom 4-Way Multiplication, Higher degree Toom'n'half, Toom 3-Way Multiplication, Multiplication Algorithms @subsection Toom 4-Way Multiplication @cindex Toom multiplication Karatsuba and Toom-3 split the operands into 2 and 3 coefficients, respectively. Toom-4 analogously splits the operands into 4 coefficients. Using the notation from the section on Toom-3 multiplication, we form two polynomials: @display @group @m{X(t) = x_3t^3 + x_2t^2 + x_1t + x_0, X(t) = x3*t^3 + x2*t^2 + x1*t + x0} @m{Y(t) = y_3t^3 + y_2t^2 + y_1t + y_0, Y(t) = y3*t^3 + y2*t^2 + y1*t + y0} @end group @end display @math{X(t)} and @math{Y(t)} are evaluated and multiplied at 7 points, giving values of @math{W(t)} at those points. In GMP the following points are used, @quotation @multitable {@m{t=-1/2,t=inf}M} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM} @item Point @tab Value @item @math{t=0} @tab @m{x_0y_0,x0 * y0}, which gives @ms{w,0} immediately @item @math{t=1/2} @tab @m{(x_3+2x_2+4x_1+8x_0)(y_3+2y_2+4y_1+8y_0),(x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)} @item @math{t=-1/2} @tab @m{(-x_3+2x_2-4x_1+8x_0)(-y_3+2y_2-4y_1+8y_0),(-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)} @item @math{t=1} @tab @m{(x_3+x_2+x_1+x_0)(y_3+y_2+y_1+y_0),(x3+x2+x1+x0) * (y3+y2+y1+y0)} @item @math{t=-1} @tab @m{(-x_3+x_2-x_1+x_0)(-y_3+y_2-y_1+y_0),(-x3+x2-x1+x0) * (-y3+y2-y1+y0)} @item @math{t=2} @tab @m{(8x_3+4x_2+2x_1+x_0)(8y_3+4y_2+2y_1+y_0),(8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)} @item @m{t=\infty,t=inf} @tab @m{x_3y_3,x3 * y3}, which gives @ms{w,6} immediately @end multitable @end quotation The number of additions and subtractions for Toom-4 is much larger than for Toom-3. But several subexpressions occur multiple times, for example @m{x_2+x_0,x2+x0}, occurs for both @math{t=1} and @math{t=-1}. Toom-4 is asymptotically @math{O(N^@W{1.404})}, the exponent being @m{\log7/\log4,log(7)/log(4)}, representing 7 recursive multiplies of 1/4 the original size each. @node Higher degree Toom'n'half, FFT Multiplication, Toom 4-Way Multiplication, Multiplication Algorithms @subsection Higher degree Toom'n'half @cindex Toom multiplication The Toom algorithms described above (@pxref{Toom 3-Way Multiplication}, @pxref{Toom 4-Way Multiplication}) generalizes to split into an arbitrary number of pieces. In general a split of two equally long operands into @math{r} pieces leads to evaluations and pointwise multiplications done at @m{2r-1,2*r-1} points. To fully exploit symmetries it would be better to have a multiple of 4 points, that's why for higher degree Toom'n'half is used. Toom'n'half means that the existence of one more piece is considered for a single operand. It can be virtual, i.e. zero, or real, when the two operand are not exactly balanced. By chosing an even @math{r}, Toom-@m{r{1\over2},r+1/2} requires @math{2r} points, a multiple of four. The four-plets of points inlcude 0, @m{\infty,inf}, +1, -1 and @m{\pm2^i,+-2^i}, @m{\pm2^{-i},+-2^-i} . Each of them giving shortcuts for the evaluation phase and for some steps in the interpolation phase. Further tricks are used to reduce the memory footprint of the whole multiplication algorithm to a memory buffer equanl in size to the result of the product. Current GMP uses both Toom-6'n'half and Toom-8'n'half. @node FFT Multiplication, Other Multiplication, Higher degree Toom'n'half, Multiplication Algorithms @subsection FFT Multiplication @cindex FFT multiplication @cindex Fast Fourier Transform At large to very large sizes a Fermat style FFT multiplication is used, following Sch@"onhage and Strassen (@pxref{References}). Descriptions of FFTs in various forms can be found in many textbooks, for instance Knuth section 4.3.3 part C or Lipson chapter IX@. A brief description of the form used in GMP is given here. The multiplication done is @m{xy \bmod 2^N+1, x*y mod 2^N+1}, for a given @math{N}. A full product @m{xy,x*y} is obtained by choosing @m{N \ge \mathop{\rm bits}(x)+\mathop{\rm bits}(y), N>=bits(x)+bits(y)} and padding @math{x} and @math{y} with high zero limbs. The modular product is the native form for the algorithm, so padding to get a full product is unavoidable. The algorithm follows a split, evaluate, pointwise multiply, interpolate and combine similar to that described above for Karatsuba and Toom-3. A @math{k} parameter controls the split, with an FFT-@math{k} splitting into @math{2^k} pieces of @math{M=N/2^k} bits each. @math{N} must be a multiple of @m{2^k\times@code{mp\_bits\_per\_limb}, (2^k)*@nicode{mp_bits_per_limb}} so the split falls on limb boundaries, avoiding bit shifts in the split and combine stages. The evaluations, pointwise multiplications, and interpolation, are all done modulo @m{2^{N'}+1, 2^N'+1} where @math{N'} is @math{2M+k+3} rounded up to a multiple of @math{2^k} and of @code{mp_bits_per_limb}. The results of interpolation will be the following negacyclic convolution of the input pieces, and the choice of @math{N'} ensures these sums aren't truncated. @tex $$ w_n = \sum_{{i+j = b2^k+n}\atop{b=0,1}} (-1)^b x_i y_j $$ @end tex @ifnottex @example --- \ b w[n] = / (-1) * x[i] * y[j] --- i+j==b*2^k+n b=0,1 @end example @end ifnottex The points used for the evaluation are @math{g^i} for @math{i=0} to @math{2^k-1} where @m{g=2^{2N'/2^k}, g=2^(2N'/2^k)}. @math{g} is a @m{2^k,2^k'}th root of unity mod @m{2^{N'}+1,2^N'+1}, which produces necessary cancellations at the interpolation stage, and it's also a power of 2 so the fast Fourier transforms used for the evaluation and interpolation do only shifts, adds and negations. The pointwise multiplications are done modulo @m{2^{N'}+1, 2^N'+1} and either recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or basecase), whichever is optimal at the size @math{N'}. The interpolation is an inverse fast Fourier transform. The resulting set of sums of @m{x_iy_j, x[i]*y[j]} are added at appropriate offsets to give the final result. Squaring is the same, but @math{x} is the only input so it's one transform at the evaluate stage and the pointwise multiplies are squares. The interpolation is the same. For a mod @math{2^N+1} product, an FFT-@math{k} is an @m{O(N^{k/(k-1)}), O(N^(k/(k-1)))} algorithm, the exponent representing @math{2^k} recursed modular multiplies each @m{1/2^{k-1},1/2^(k-1)} the size of the original. Each successive @math{k} is an asymptotic improvement, but overheads mean each is only faster at bigger and bigger sizes. In the code, @code{MUL_FFT_TABLE} and @code{SQR_FFT_TABLE} are the thresholds where each @math{k} is used. Each new @math{k} effectively swaps some multiplying for some shifts, adds and overheads. A mod @math{2^N+1} product can be formed with a normal @math{N@cross{}N@rightarrow{}2N} bit multiply plus a subtraction, so an FFT and Toom-3 etc can be compared directly. A @math{k=4} FFT at @math{O(N^@W{1.333})} can be expected to be the first faster than Toom-3 at @math{O(N^@W{1.465})}. In practice this is what's found, with @code{MUL_FFT_MODF_THRESHOLD} and @code{SQR_FFT_MODF_THRESHOLD} being between 300 and 1000 limbs, depending on the CPU@. So far it's been found that only very large FFTs recurse into pointwise multiplies above these sizes. When an FFT is to give a full product, the change of @math{N} to @math{2N} doesn't alter the theoretical complexity for a given @math{k}, but for the purposes of considering where an FFT might be first used it can be assumed that the FFT is recursing into a normal multiply and that on that basis it's doing @math{2^k} recursed multiplies each @m{1/2^{k-2},1/2^(k-2)} the size of the inputs, making it @m{O(N^{k/(k-2)}), O(N^(k/(k-2)))}. This would mean @math{k=7} at @math{O(N^@W{1.4})} would be the first FFT faster than Toom-3. In practice @code{MUL_FFT_THRESHOLD} and @code{SQR_FFT_THRESHOLD} have been found to be in the @math{k=8} range, somewhere between 3000 and 10000 limbs. The way @math{N} is split into @math{2^k} pieces and then @math{2M+k+3} is rounded up to a multiple of @math{2^k} and @code{mp_bits_per_limb} means that when @math{2^k@ge{}@nicode{mp\_bits\_per\_limb}} the effective @math{N} is a multiple of @m{2^{2k-1},2^(2k-1)} bits. The @math{+k+3} means some values of @math{N} just under such a multiple will be rounded to the next. The complexity calculations above assume that a favourable size is used, meaning one which isn't padded through rounding, and it's also assumed that the extra @math{+k+3} bits are negligible at typical FFT sizes. The practical effect of the @m{2^{2k-1},2^(2k-1)} constraint is to introduce a step-effect into measured speeds. For example @math{k=8} will round @math{N} up to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb groups of sizes for which @code{mpn_mul_n} runs at the same speed. Or for @math{k=9} groups of 2048 limbs, @math{k=10} groups of 8192 limbs, etc. In practice it's been found each @math{k} is used at quite small multiples of its size constraint and so the step effect is quite noticeable in a time versus size graph. The threshold determinations currently measure at the mid-points of size steps, but this is sub-optimal since at the start of a new step it can happen that it's better to go back to the previous @math{k} for a while. Something more sophisticated for @code{MUL_FFT_TABLE} and @code{SQR_FFT_TABLE} will be needed. @node Other Multiplication, Unbalanced Multiplication, FFT Multiplication, Multiplication Algorithms @subsection Other Multiplication @cindex Toom multiplication The Toom algorithms described above (@pxref{Toom 3-Way Multiplication}, @pxref{Toom 4-Way Multiplication}) generalizes to split into an arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C@. This is not currently used. The notes here are merely for interest. In general a split into @math{r+1} pieces is made, and evaluations and pointwise multiplications done at @m{2r+1,2*r+1} points. A 4-way split does 7 pointwise multiplies, 5-way does 9, etc. Asymptotically an @math{(r+1)}-way algorithm is @m{O(N^{log(2r+1)/log(r+1)}), O(N^(log(2*r+1)/log(r+1)))}. Only the pointwise multiplications count towards big-@math{O} complexity, but the time spent in the evaluate and interpolate stages grows with @math{r} and has a significant practical impact, with the asymptotic advantage of each @math{r} realized only at bigger and bigger sizes. The overheads grow as @m{O(Nr),O(N*r)}, whereas in an @math{r=2^k} FFT they grow only as @m{O(N \log r), O(N*log(r))}. Knuth algorithm C evaluates at points 0,1,2,@dots{},@m{2r,2*r}, but exercise 4 uses @math{-r},@dots{},0,@dots{},@math{r} and the latter saves some small multiplies in the evaluate stage (or rather trades them for additions), and has a further saving of nearly half the interpolate steps. The idea is to separate odd and even final coefficients and then perform algorithm C steps C7 and C8 on them separately. The divisors at step C7 become @math{j^2} and the multipliers at C8 become @m{2tj-j^2,2*t*j-j^2}. Splitting odd and even parts through positive and negative points can be thought of as using @math{-1} as a square root of unity. If a 4th root of unity was available then a further split and speedup would be possible, but no such root exists for plain integers. Going to complex integers with @m{i=\sqrt{-1}, i=sqrt(-1)} doesn't help, essentially because in Cartesian form it takes three real multiplies to do a complex multiply. The existence of @m{2^k,2^k'}th roots of unity in a suitable ring or field lets the fast Fourier transform keep splitting and get to @m{O(N \log r), O(N*log(r))}. Floating point FFTs use complex numbers approximating Nth roots of unity. Some processors have special support for such FFTs. But these are not used in GMP since it's very difficult to guarantee an exact result (to some number of bits). An occasional difference of 1 in the last bit might not matter to a typical signal processing algorithm, but is of course of vital importance to GMP. @node Unbalanced Multiplication, , Other Multiplication, Multiplication Algorithms @subsection Unbalanced Multiplication @cindex Unbalanced multiplication Multiplication of operands with different sizes, both below @code{MUL_TOOM22_THRESHOLD} are done with plain schoolbook multiplication (@pxref{Basecase Multiplication}). For really large operands, we invoke FFT directly. For operands between these sizes, we use Toom inspired algorithms suggested by Alberto Zanoni and Marco Bodrato. The idea is to split the operands into polynomials of different degree. GMP currently splits the smaller operand onto 2 coefficients, i.e., a polynomial of degree 1, but the larger operand can be split into 2, 3, or 4 coefficients, i.e., a polynomial of degree 1 to 3. @c FIXME: This is mighty ugly, but a cleaner @need triggers texinfo bugs that @c screws up layout here and there in the rest of the manual. @c @tex @c \goodbreak @c @end tex @node Division Algorithms, Greatest Common Divisor Algorithms, Multiplication Algorithms, Algorithms @section Division Algorithms @cindex Division algorithms @menu * Single Limb Division:: * Basecase Division:: * Divide and Conquer Division:: * Block-Wise Barrett Division:: * Exact Division:: * Exact Remainder:: * Small Quotient Division:: @end menu @node Single Limb Division, Basecase Division, Division Algorithms, Division Algorithms @subsection Single Limb Division N@cross{}1 division is implemented using repeated 2@cross{}1 divisions from high to low, either with a hardware divide instruction or a multiplication by inverse, whichever is best on a given CPU. The multiply by inverse follows ``Improved division by invariant integers'' by M@"oller and Granlund (@pxref{References}) and is implemented as @code{udiv_qrnnd_preinv} in @file{gmp-impl.h}. The idea is to have a fixed-point approximation to @math{1/d} (see @code{invert_limb}) and then multiply by the high limb (plus one bit) of the dividend to get a quotient @math{q}. With @math{d} normalized (high bit set), @math{q} is no more than 1 too small. Subtracting @m{qd,q*d} from the dividend gives a remainder, and reveals whether @math{q} or @math{q-1} is correct. The result is a division done with two multiplications and four or five arithmetic operations. On CPUs with low latency multipliers this can be much faster than a hardware divide, though the cost of calculating the inverse at the start may mean it's only better on inputs bigger than say 4 or 5 limbs. When a divisor must be normalized, either for the generic C @code{__udiv_qrnnd_c} or the multiply by inverse, the division performed is actually @m{a2^k,a*2^k} by @m{d2^k,d*2^k} where @math{a} is the dividend and @math{k} is the power necessary to have the high bit of @m{d2^k,d*2^k} set. The bit shifts for the dividend are usually accomplished ``on the fly'' meaning by extracting the appropriate bits at each step. Done this way the quotient limbs come out aligned ready to store. When only the remainder is wanted, an alternative is to take the dividend limbs unshifted and calculate @m{r = a \bmod d2^k, r = a mod d*2^k} followed by an extra final step @m{r2^k \bmod d2^k, r*2^k mod d*2^k}. This can help on CPUs with poor bit shifts or few registers. The multiply by inverse can be done two limbs at a time. The calculation is basically the same, but the inverse is two limbs and the divisor treated as if padded with a low zero limb. This means more work, since the inverse will need a 2@cross{}2 multiply, but the four 1@cross{}1s to do that are independent and can therefore be done partly or wholly in parallel. Likewise for a 2@cross{}1 calculating @m{qd,q*d}. The net effect is to process two limbs with roughly the same two multiplies worth of latency that one limb at a time gives. This extends to 3 or 4 limbs at a time, though the extra work to apply the inverse will almost certainly soon reach the limits of multiplier throughput. A similar approach in reverse can be taken to process just half a limb at a time if the divisor is only a half limb. In this case the 1@cross{}1 multiply for the inverse effectively becomes two @m{{1\over2}\times1, (1/2)x1} for each limb, which can be a saving on CPUs with a fast half limb multiply, or in fact if the only multiply is a half limb, and especially if it's not pipelined. @node Basecase Division, Divide and Conquer Division, Single Limb Division, Division Algorithms @subsection Basecase Division Basecase N@cross{}M division is like long division done by hand, but in base @m{2\GMPraise{@code{mp\_bits\_per\_limb}}, 2^mp_bits_per_limb}. See Knuth section 4.3.1 algorithm D, and @file{mpn/generic/sb_divrem_mn.c}. Briefly stated, while the dividend remains larger than the divisor, a high quotient limb is formed and the N@cross{}1 product @m{qd,q*d} subtracted at the top end of the dividend. With a normalized divisor (most significant bit set), each quotient limb can be formed with a 2@cross{}1 division and a 1@cross{}1 multiplication plus some subtractions. The 2@cross{}1 division is by the high limb of the divisor and is done either with a hardware divide or a multiply by inverse (the same as in @ref{Single Limb Division}) whichever is faster. Such a quotient is sometimes one too big, requiring an addback of the divisor, but that happens rarely. With Q=N@minus{}M being the number of quotient limbs, this is an @m{O(QM),O(Q*M)} algorithm and will run at a speed similar to a basecase Q@cross{}M multiplication, differing in fact only in the extra multiply and divide for each of the Q quotient limbs. @node Divide and Conquer Division, Block-Wise Barrett Division, Basecase Division, Division Algorithms @subsection Divide and Conquer Division For divisors larger than @code{DC_DIV_QR_THRESHOLD}, division is done by dividing. Or to be precise by a recursive divide and conquer algorithm based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler (@pxref{References}). The algorithm consists essentially of recognising that a 2N@cross{}N division can be done with the basecase division algorithm (@pxref{Basecase Division}), but using N/2 limbs as a base, not just a single limb. This way the multiplications that arise are (N/2)@cross{}(N/2) and can take advantage of Karatsuba and higher multiplication algorithms (@pxref{Multiplication Algorithms}). The two ``digits'' of the quotient are formed by recursive N@cross{}(N/2) divisions. If the (N/2)@cross{}(N/2) multiplies are done with a basecase multiplication then the work is about the same as a basecase division, but with more function call overheads and with some subtractions separated from the multiplies. These overheads mean that it's only when N/2 is above @code{MUL_TOOM22_THRESHOLD} that divide and conquer is of use. @code{DC_DIV_QR_THRESHOLD} is based on the divisor size N, so it will be somewhere above twice @code{MUL_TOOM22_THRESHOLD}, but how much above depends on the CPU@. An optimized @code{mpn_mul_basecase} can lower @code{DC_DIV_QR_THRESHOLD} a little by offering a ready-made advantage over repeated @code{mpn_submul_1} calls. Divide and conquer is asymptotically @m{O(M(N)\log N),O(M(N)*log(N))} where @math{M(N)} is the time for an N@cross{}N multiplication done with FFTs. The actual time is a sum over multiplications of the recursed sizes, as can be seen near the end of section 2.2 of Burnikel and Ziegler. For example, within the Toom-3 range, divide and conquer is @m{2.63M(N), 2.63*M(N)}. With higher algorithms the @math{M(N)} term improves and the multiplier tends to @m{\log N, log(N)}. In practice, at moderate to large sizes, a 2N@cross{}N division is about 2 to 4 times slower than an N@cross{}N multiplication. @node Block-Wise Barrett Division, Exact Division, Divide and Conquer Division, Division Algorithms @subsection Block-Wise Barrett Division For the largest divisions, a block-wise Barrett division algorithm is used. Here, the divisor is inverted to a precision determined by the relative size of the dividend and divisor. Blocks of quotient limbs are then generated by multiplying blocks from the dividend by the inverse. Our block-wise algorithm computes a smaller inverse than in the plain Barrett algorithm. For a @math{2n/n} division, the inverse will be just @m{\lceil n/2 \rceil, ceil(n/2)} limbs. @node Exact Division, Exact Remainder, Block-Wise Barrett Division, Division Algorithms @subsection Exact Division A so-called exact division is when the dividend is known to be an exact multiple of the divisor. Jebelean's exact division algorithm uses this knowledge to make some significant optimizations (@pxref{References}). The idea can be illustrated in decimal for example with 368154 divided by 543. Because the low digit of the dividend is 4, the low digit of the quotient must be 8. This is arrived at from @m{4 \mathord{\times} 7 \bmod 10, 4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of the divisor), since @m{3 \mathord{\times} 7 \mathop{\equiv} 1 \bmod 10, 3*7 @equiv{} 1 mod 10}. So @m{8\mathord{\times}543 = 4344,8*543=4344} can be subtracted from the dividend leaving 363810. Notice the low digit has become zero. The procedure is repeated at the second digit, with the next quotient digit 7 (@m{1 \mathord{\times} 7 \bmod 10, 7 @equiv{} 1*7 mod 10}), subtracting @m{7\mathord{\times}543 = 3801,7*543=3801}, leaving 325800. And finally at the third digit with quotient digit 6 (@m{8 \mathord{\times} 7 \bmod 10, 8*7 mod 10}), subtracting @m{6\mathord{\times}543 = 3258,6*543=3258} leaving 0. So the quotient is 678. Notice however that the multiplies and subtractions don't need to extend past the low three digits of the dividend, since that's enough to determine the three quotient digits. For the last quotient digit no subtraction is needed at all. On a 2N@cross{}N division like this one, only about half the work of a normal basecase division is necessary. For an N@cross{}M exact division producing Q=N@minus{}M quotient limbs, the saving over a normal basecase division is in two parts. Firstly, each of the Q quotient limbs needs only one multiply, not a 2@cross{}1 divide and multiply. Secondly, the crossproducts are reduced when @math{Q>M} to @m{QM-M(M+1)/2,Q*M-M*(M+1)/2}, or when @math{Q@le{}M} to @m{Q(Q-1)/2, Q*(Q-1)/2}. Notice the savings are complementary. If Q is big then many divisions are saved, or if Q is small then the crossproducts reduce to a small number. The modular inverse used is calculated efficiently by @code{binvert_limb} in @file{gmp-impl.h}. This does four multiplies for a 32-bit limb, or six for a 64-bit limb. @file{tune/modlinv.c} has some alternate implementations that might suit processors better at bit twiddling than multiplying. The sub-quadratic exact division described by Jebelean in ``Exact Division with Karatsuba Complexity'' is not currently implemented. It uses a rearrangement similar to the divide and conquer for normal division (@pxref{Divide and Conquer Division}), but operating from low to high. A further possibility not currently implemented is ``Bidirectional Exact Integer Division'' by Krandick and Jebelean which forms quotient limbs from both the high and low ends of the dividend, and can halve once more the number of crossproducts needed in a 2N@cross{}N division. A special case exact division by 3 exists in @code{mpn_divexact_by3}, supporting Toom-3 multiplication and @code{mpq} canonicalizations. It forms quotient digits with a multiply by the modular inverse of 3 (which is @code{0xAA..AAB}) and uses two comparisons to determine a borrow for the next limb. The multiplications don't need to be on the dependent chain, as long as the effect of the borrows is applied, which can help chips with pipelined multipliers. @node Exact Remainder, Small Quotient Division, Exact Division, Division Algorithms @subsection Exact Remainder @cindex Exact remainder If the exact division algorithm is done with a full subtraction at each stage and the dividend isn't a multiple of the divisor, then low zero limbs are produced but with a remainder in the high limbs. For dividend @math{a}, divisor @math{d}, quotient @math{q}, and @m{b = 2 \GMPraise{@code{mp\_bits\_per\_limb}}, b = 2^mp_bits_per_limb}, this remainder @math{r} is of the form @tex $$ a = qd + r b^n $$ @end tex @ifnottex @example a = q*d + r*b^n @end example @end ifnottex @math{n} represents the number of zero limbs produced by the subtractions, that being the number of limbs produced for @math{q}. @math{r} will be in the range @math{0@le{}r b \GMPhat r + u_2, v2*q>b*r+u2} condition appropriately relaxed. @need 1000 @node Greatest Common Divisor Algorithms, Powering Algorithms, Division Algorithms, Algorithms @section Greatest Common Divisor @cindex Greatest common divisor algorithms @cindex GCD algorithms @menu * Binary GCD:: * Lehmer's Algorithm:: * Subquadratic GCD:: * Extended GCD:: * Jacobi Symbol:: @end menu @node Binary GCD, Lehmer's Algorithm, Greatest Common Divisor Algorithms, Greatest Common Divisor Algorithms @subsection Binary GCD At small sizes GMP uses an @math{O(N^2)} binary style GCD@. This is described in many textbooks, for example Knuth section 4.5.2 algorithm B@. It simply consists of successively reducing odd operands @math{a} and @math{b} using @quotation @math{a,b = @abs{}(a-b),@min{}(a,b)} @* strip factors of 2 from @math{a} @end quotation The Euclidean GCD algorithm, as per Knuth algorithms E and A, repeatedly computes the quotient @m{q = \lfloor a/b \rfloor, q = floor(a/b)} and replaces @math{a,b} by @math{v, u - q v}. The binary algorithm has so far been found to be faster than the Euclidean algorithm everywhere. One reason the binary method does well is that the implied quotient at each step is usually small, so often only one or two subtractions are needed to get the same effect as a division. Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth section 4.5.3 Theorem E. When the implied quotient is large, meaning @math{b} is much smaller than @math{a}, then a division is worthwhile. This is the basis for the initial @math{a @bmod b} reductions in @code{mpn_gcd} and @code{mpn_gcd_1} (the latter for both N@cross{}1 and 1@cross{}1 cases). But after that initial reduction, big quotients occur too rarely to make it worth checking for them. @sp 1 The final @math{1@cross{}1} GCD in @code{mpn_gcd_1} is done in the generic C code as described above. For two N-bit operands, the algorithm takes about 0.68 iterations per bit. For optimum performance some attention needs to be paid to the way the factors of 2 are stripped from @math{a}. Firstly it may be noted that in twos complement the number of low zero bits on @math{a-b} is the same as @math{b-a}, so counting or testing can begin on @math{a-b} without waiting for @math{@abs{}(a-b)} to be determined. A loop stripping low zero bits tends not to branch predict well, since the condition is data dependent. But on average there's only a few low zeros, so an option is to strip one or two bits arithmetically then loop for more (as done for AMD K6). Or use a lookup table to get a count for several bits then loop for more (as done for AMD K7). An alternative approach is to keep just one of @math{a} or @math{b} odd and iterate @quotation @math{a,b = @abs{}(a-b), @min{}(a,b)} @* @math{a = a/2} if even @* @math{b = b/2} if even @end quotation This requires about 1.25 iterations per bit, but stripping of a single bit at each step avoids any branching. Repeating the bit strip reduces to about 0.9 iterations per bit, which may be a worthwhile tradeoff. Generally with the above approaches a speed of perhaps 6 cycles per bit can be achieved, which is still not terribly fast with for instance a 64-bit GCD taking nearly 400 cycles. It's this sort of time which means it's not usually advantageous to combine a set of divisibility tests into a GCD. Currently, the binary algorithm is used for GCD only when @math{N < 3}. @node Lehmer's Algorithm, Subquadratic GCD, Binary GCD, Greatest Common Divisor Algorithms @comment node-name, next, previous, up @subsection Lehmer's algorithm Lehmer's improvement of the Euclidean algorithms is based on the observation that the initial part of the quotient sequence depends only on the most significant parts of the inputs. The variant of Lehmer's algorithm used in GMP splits off the most significant two limbs, as suggested, e.g., in ``A Double-Digit Lehmer-Euclid Algorithm'' by Jebelean (@pxref{References}). The quotients of two double-limb inputs are collected as a 2 by 2 matrix with single-limb elements. This is done by the function @code{mpn_hgcd2}. The resulting matrix is applied to the inputs using @code{mpn_mul_1} and @code{mpn_submul_1}. Each iteration usually reduces the inputs by almost one limb. In the rare case of a large quotient, no progress can be made by examining just the most significant two limbs, and the quotient is computed using plain division. The resulting algorithm is asymptotically @math{O(N^2)}, just as the Euclidean algorithm and the binary algorithm. The quadratic part of the work are the calls to @code{mpn_mul_1} and @code{mpn_submul_1}. For small sizes, the linear work is also significant. There are roughly @math{N} calls to the @code{mpn_hgcd2} function. This function uses a couple of important optimizations: @itemize @item It uses the same relaxed notion of correctness as @code{mpn_hgcd} (see next section). This means that when called with the most significant two limbs of two large numbers, the returned matrix does not always correspond exactly to the initial quotient sequence for the two large numbers; the final quotient may sometimes be one off. @item It takes advantage of the fact the quotients are usually small. The division operator is not used, since the corresponding assembler instruction is very slow on most architectures. (This code could probably be improved further, it uses many branches that are unfriendly to prediction). @item It switches from double-limb calculations to single-limb calculations half-way through, when the input numbers have been reduced in size from two limbs to one and a half. @end itemize @node Subquadratic GCD, Extended GCD, Lehmer's Algorithm, Greatest Common Divisor Algorithms @subsection Subquadratic GCD For inputs larger than @code{GCD_DC_THRESHOLD}, GCD is computed via the HGCD (Half GCD) function, as a generalization to Lehmer's algorithm. Let the inputs @math{a,b} be of size @math{N} limbs each. Put @m{S=\lfloor N/2 \rfloor + 1, S = floor(N/2) + 1}. Then HGCD(a,b) returns a transformation matrix @math{T} with non-negative elements, and reduced numbers @math{(c;d) = T^{-1} (a;b)}. The reduced numbers @math{c,d} must be larger than @math{S} limbs, while their difference @math{abs(c-d)} must fit in @math{S} limbs. The matrix elements will also be of size roughly @math{N/2}. The HGCD base case uses Lehmer's algorithm, but with the above stop condition that returns reduced numbers and the corresponding transformation matrix half-way through. For inputs larger than @code{HGCD_THRESHOLD}, HGCD is computed recursively, using the divide and conquer algorithm in ``On Sch@"onhage's algorithm and subquadratic integer GCD computation'' by M@"oller (@pxref{References}). The recursive algorithm consists of these main steps. @itemize @item Call HGCD recursively, on the most significant @math{N/2} limbs. Apply the resulting matrix @math{T_1} to the full numbers, reducing them to a size just above @math{3N/2}. @item Perform a small number of division or subtraction steps to reduce the numbers to size below @math{3N/2}. This is essential mainly for the unlikely case of large quotients. @item Call HGCD recursively, on the most significant @math{N/2} limbs of the reduced numbers. Apply the resulting matrix @math{T_2} to the full numbers, reducing them to a size just above @math{N/2}. @item Compute @math{T = T_1 T_2}. @item Perform a small number of division and subtraction steps to satisfy the requirements, and return. @end itemize GCD is then implemented as a loop around HGCD, similarly to Lehmer's algorithm. Where Lehmer repeatedly chops off the top two limbs, calls @code{mpn_hgcd2}, and applies the resulting matrix to the full numbers, the subquadratic GCD chops off the most significant third of the limbs (the proportion is a tuning parameter, and @math{1/3} seems to be more efficient than, e.g, @math{1/2}), calls @code{mpn_hgcd}, and applies the resulting matrix. Once the input numbers are reduced to size below @code{GCD_DC_THRESHOLD}, Lehmer's algorithm is used for the rest of the work. The asymptotic running time of both HGCD and GCD is @m{O(M(N)\log N),O(M(N)*log(N))}, where @math{M(N)} is the time for multiplying two @math{N}-limb numbers. @comment node-name, next, previous, up @node Extended GCD, Jacobi Symbol, Subquadratic GCD, Greatest Common Divisor Algorithms @subsection Extended GCD The extended GCD function, or GCDEXT, calculates @math{@gcd{}(a,b)} and also cofactors @math{x} and @math{y} satisfying @m{ax+by=\gcd(a@C{}b), a*x+b*y=gcd(a@C{}b)}. All the algorithms used for plain GCD are extended to handle this case. The binary algorithm is used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes up to @code{GCDEXT_DC_THRESHOLD}. Above this threshold, GCDEXT is implemented as a loop around HGCD, but with more book-keeping to keep track of the cofactors. This gives the same asymptotic running time as for GCD and HGCD, @m{O(M(N)\log N),O(M(N)*log(N))} One difference to plain GCD is that while the inputs @math{a} and @math{b} are reduced as the algorithm proceeds, the cofactors @math{x} and @math{y} grow in size. This makes the tuning of the chopping-point more difficult. The current code chops off the most significant half of the inputs for the call to HGCD in the first iteration, and the most significant two thirds for the remaining calls. This strategy could surely be improved. Also the stop condition for the loop, where Lehmer's algorithm is invoked once the inputs are reduced below @code{GCDEXT_DC_THRESHOLD}, could maybe be improved by taking into account the current size of the cofactors. @node Jacobi Symbol, , Extended GCD, Greatest Common Divisor Algorithms @subsection Jacobi Symbol @cindex Jacobi symbol algorithm [This section is obsolete. The current Jacobi code actually uses a very efficient algorithm.] @code{mpz_jacobi} and @code{mpz_kronecker} are currently implemented with a simple binary algorithm similar to that described for the GCDs (@pxref{Binary GCD}). They're not very fast when both inputs are large. Lehmer's multi-step improvement or a binary based multi-step algorithm is likely to be better. When one operand fits a single limb, and that includes @code{mpz_kronecker_ui} and friends, an initial reduction is done with either @code{mpn_mod_1} or @code{mpn_modexact_1_odd}, followed by the binary algorithm on a single limb. The binary algorithm is well suited to a single limb, and the whole calculation in this case is quite efficient. In all the routines sign changes for the result are accumulated using some bit twiddling, avoiding table lookups or conditional jumps. @need 1000 @node Powering Algorithms, Root Extraction Algorithms, Greatest Common Divisor Algorithms, Algorithms @section Powering Algorithms @cindex Powering algorithms @menu * Normal Powering Algorithm:: * Modular Powering Algorithm:: @end menu @node Normal Powering Algorithm, Modular Powering Algorithm, Powering Algorithms, Powering Algorithms @subsection Normal Powering Normal @code{mpz} or @code{mpf} powering uses a simple binary algorithm, successively squaring and then multiplying by the base when a 1 bit is seen in the exponent, as per Knuth section 4.6.3. The ``left to right'' variant described there is used rather than algorithm A, since it's just as easy and can be done with somewhat less temporary memory. @node Modular Powering Algorithm, , Normal Powering Algorithm, Powering Algorithms @subsection Modular Powering Modular powering is implemented using a @math{2^k}-ary sliding window algorithm, as per ``Handbook of Applied Cryptography'' algorithm 14.85 (@pxref{References}). @math{k} is chosen according to the size of the exponent. Larger exponents use larger values of @math{k}, the choice being made to minimize the average number of multiplications that must supplement the squaring. The modular multiplies and squarings use either a simple division or the REDC method by Montgomery (@pxref{References}). REDC is a little faster, essentially saving N single limb divisions in a fashion similar to an exact remainder (@pxref{Exact Remainder}). @node Root Extraction Algorithms, Radix Conversion Algorithms, Powering Algorithms, Algorithms @section Root Extraction Algorithms @cindex Root extraction algorithms @menu * Square Root Algorithm:: * Nth Root Algorithm:: * Perfect Square Algorithm:: * Perfect Power Algorithm:: @end menu @node Square Root Algorithm, Nth Root Algorithm, Root Extraction Algorithms, Root Extraction Algorithms @subsection Square Root @cindex Square root algorithm @cindex Karatsuba square root algorithm Square roots are taken using the ``Karatsuba Square Root'' algorithm by Paul Zimmermann (@pxref{References}). An input @math{n} is split into four parts of @math{k} bits each, so with @math{b=2^k} we have @m{n = a_3b^3 + a_2b^2 + a_1b + a_0, n = a3*b^3 + a2*b^2 + a1*b + a0}. Part @ms{a,3} must be ``normalized'' so that either the high or second highest bit is set. In GMP, @math{k} is kept on a limb boundary and the input is left shifted (by an even number of bits) to normalize. The square root of the high two parts is taken, by recursive application of the algorithm (bottoming out in a one-limb Newton's method), @tex $$ s',r' = \mathop{\rm sqrtrem} \> (a_3b + a_2) $$ @end tex @ifnottex @example s1,r1 = sqrtrem (a3*b + a2) @end example @end ifnottex This is an approximation to the desired root and is extended by a division to give @math{s},@math{r}, @tex $$\eqalign{ q,u &= \mathop{\rm divrem} \> (r'b + a_1, 2s') \cr s &= s'b + q \cr r &= ub + a_0 - q^2 }$$ @end tex @ifnottex @example q,u = divrem (r1*b + a1, 2*s1) s = s1*b + q r = u*b + a0 - q^2 @end example @end ifnottex The normalization requirement on @ms{a,3} means at this point @math{s} is either correct or 1 too big. @math{r} is negative in the latter case, so @tex $$\eqalign{ \mathop{\rm if} \; r &< 0 \; \mathop{\rm then} \cr r &\leftarrow r + 2s - 1 \cr s &\leftarrow s - 1 }$$ @end tex @ifnottex @example if r < 0 then r = r + 2*s - 1 s = s - 1 @end example @end ifnottex The algorithm is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method, or as a variation on the schoolboy method (no longer taught) for square roots two digits at a time. If the remainder @math{r} is not required then usually only a few high limbs of @math{r} and @math{u} need to be calculated to determine whether an adjustment to @math{s} is required. This optimization is not currently implemented. In the Karatsuba multiplication range this algorithm is @m{O({3\over2} M(N/2)),O(1.5*M(N/2))}, where @math{M(n)} is the time to multiply two numbers of @math{n} limbs. In the FFT multiplication range this grows to a bound of @m{O(6 M(N/2)),O(6*M(N/2))}. In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range. The algorithm does all its calculations in integers and the resulting @code{mpn_sqrtrem} is used for both @code{mpz_sqrt} and @code{mpf_sqrt}. The extended precision given by @code{mpf_sqrt_ui} is obtained by padding with zero limbs. @node Nth Root Algorithm, Perfect Square Algorithm, Square Root Algorithm, Root Extraction Algorithms @subsection Nth Root @cindex Root extraction algorithm @cindex Nth root algorithm Integer Nth roots are taken using Newton's method with the following iteration, where @math{A} is the input and @math{n} is the root to be taken. @tex $$a_{i+1} = {1\over n} \left({A \over a_i^{n-1}} + (n-1)a_i \right)$$ @end tex @ifnottex @example 1 A a[i+1] = - * ( --------- + (n-1)*a[i] ) n a[i]^(n-1) @end example @end ifnottex The initial approximation @m{a_1,a[1]} is generated bitwise by successively powering a trial root with or without new 1 bits, aiming to be just above the true root. The iteration converges quadratically when started from a good approximation. When @math{n} is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized. @node Perfect Square Algorithm, Perfect Power Algorithm, Nth Root Algorithm, Root Extraction Algorithms @subsection Perfect Square @cindex Perfect square algorithm A significant fraction of non-squares can be quickly identified by checking whether the input is a quadratic residue modulo small integers. @code{mpz_perfect_square_p} first tests the input mod 256, which means just examining the low byte. Only 44 different values occur for squares mod 256, so 82.8% of inputs can be immediately identified as non-squares. On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17, for a total 99.25% of inputs identified as non-squares. On a 64-bit system 97 is tested too, for a total 99.62%. These moduli are chosen because they're factors of @math{2^@W{24}-1} (or @math{2^@W{48}-1} for 64-bits), and such a remainder can be quickly taken just using additions (see @code{mpn_mod_34lsub1}). When nails are in use moduli are instead selected by the @file{gen-psqr.c} program and applied with an @code{mpn_mod_1}. The same @math{2^@W{24}-1} or @math{2^@W{48}-1} could be done with nails using some extra bit shifts, but this is not currently implemented. In any case each modulus is applied to the @code{mpn_mod_34lsub1} or @code{mpn_mod_1} remainder and a table lookup identifies non-squares. By using a ``modexact'' style calculation, and suitably permuted tables, just one multiply each is required, see the code for details. Moduli are also combined to save operations, so long as the lookup tables don't become too big. @file{gen-psqr.c} does all the pre-calculations. A square root must still be taken for any value that passes these tests, to verify it's really a square and not one of the small fraction of non-squares that get through (i.e.@: a pseudo-square to all the tested bases). Clearly more residue tests could be done, @code{mpz_perfect_square_p} only uses a compact and efficient set. Big inputs would probably benefit from more residue testing, small inputs might be better off with less. The assumed distribution of squares versus non-squares in the input would affect such considerations. @node Perfect Power Algorithm, , Perfect Square Algorithm, Root Extraction Algorithms @subsection Perfect Power @cindex Perfect power algorithm Detecting perfect powers is required by some factorization algorithms. Currently @code{mpz_perfect_power_p} is implemented using repeated Nth root extractions, though naturally only prime roots need to be considered. (@xref{Nth Root Algorithm}.) If a prime divisor @math{p} with multiplicity @math{e} can be found, then only roots which are divisors of @math{e} need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked. @node Radix Conversion Algorithms, Other Algorithms, Root Extraction Algorithms, Algorithms @section Radix Conversion @cindex Radix conversion algorithms Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation. @menu * Binary to Radix:: * Radix to Binary:: @end menu @node Binary to Radix, Radix to Binary, Radix Conversion Algorithms, Radix Conversion Algorithms @subsection Binary to Radix Conversions from binary to a power-of-2 radix use a simple and fast @math{O(N)} bit extraction algorithm. Conversions from binary to other radices use one of two algorithms. Sizes below @code{GET_STR_PRECOMPUTE_THRESHOLD} use a basic @math{O(N^2)} method. Repeated divisions by @math{b^n} are made, where @math{b} is the radix and @math{n} is the biggest power that fits in a limb. But instead of simply using the remainder @math{r} from such divisions, an extra divide step is done to give a fractional limb representing @math{r/b^n}. The digits of @math{r} can then be extracted using multiplications by @math{b} rather than divisions. Special case code is provided for decimal, allowing multiplications by 10 to optimize to shifts and adds. Above @code{GET_STR_PRECOMPUTE_THRESHOLD} a sub-quadratic algorithm is used. For an input @math{t}, powers @m{b^{n2^i},b^(n*2^i)} of the radix are calculated, until a power between @math{t} and @m{\sqrt{t},sqrt(t)} is reached. @math{t} is then divided by that largest power, giving a quotient which is the digits above that power, and a remainder which is those below. These two parts are in turn divided by the second highest power, and so on recursively. When a piece has been divided down to less than @code{GET_STR_DC_THRESHOLD} limbs, the basecase algorithm described above is used. The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (@pxref{Divide and Conquer Division}), and big divisions tend to have less overheads than lots of separate single limb divisions anyway. But in any case the cost of calculating the powers @m{b^{n2^i},b^(n*2^i)} must first be overcome. @code{GET_STR_PRECOMPUTE_THRESHOLD} and @code{GET_STR_DC_THRESHOLD} represent the same basic thing, the point where it becomes worth doing a big division to cut the input in half. @code{GET_STR_PRECOMPUTE_THRESHOLD} includes the cost of calculating the radix power required, whereas @code{GET_STR_DC_THRESHOLD} assumes that's already available, which is the case when recursing. Since the base case produces digits from least to most significant but they want to be stored from most to least, it's necessary to calculate in advance how many digits there will be, or at least be sure not to underestimate that. For GMP the number of input bits is multiplied by @code{chars_per_bit_exactly} from @code{mp_bases}, rounding up. The result is either correct or one too big. Examining some of the high bits of the input could increase the chance of getting the exact number of digits, but an exact result every time would not be practical, since in general the difference between numbers 100@dots{} and 99@dots{} is only in the last few bits and the work to identify 99@dots{} might well be almost as much as a full conversion. @code{mpf_get_str} doesn't currently use the algorithm described here, it multiplies or divides by a power of @math{b} to move the radix point to the just above the highest non-zero digit (or at worst one above that location), then multiplies by @math{b^n} to bring out digits. This is @math{O(N^2)} and is certainly not optimal. The @math{r/b^n} scheme described above for using multiplications to bring out digits might be useful for more than a single limb. Some brief experiments with it on the base case when recursing didn't give a noticeable improvement, but perhaps that was only due to the implementation. Something similar would work for the sub-quadratic divisions too, though there would be the cost of calculating a bigger radix power. Another possible improvement for the sub-quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced, i.e.@: the highest power would be an @m{b^{nk},b^(n*k)} approximately equal to @m{\sqrt{t},sqrt(t)}, not restricted to a @math{2^i} factor. That ought to smooth out a graph of times against sizes, but may or may not be a net speedup. @node Radix to Binary, , Binary to Radix, Radix Conversion Algorithms @subsection Radix to Binary @strong{This section needs to be rewritten, it currently describes the algorithms used before GMP 4.3.} Conversions from a power-of-2 radix into binary use a simple and fast @math{O(N)} bitwise concatenation algorithm. Conversions from other radices use one of two algorithms. Sizes below @code{SET_STR_PRECOMPUTE_THRESHOLD} use a basic @math{O(N^2)} method. Groups of @math{n} digits are converted to limbs, where @math{n} is the biggest power of the base @math{b} which will fit in a limb, then those groups are accumulated into the result by multiplying by @math{b^n} and adding. This saves multi-precision operations, as per Knuth section 4.4 part E (@pxref{References}). Some special case code is provided for decimal, giving the compiler a chance to optimize multiplications by 10. Above @code{SET_STR_PRECOMPUTE_THRESHOLD} a sub-quadratic algorithm is used. First groups of @math{n} digits are converted into limbs. Then adjacent limbs are combined into limb pairs with @m{xb^n+y,x*b^n+y}, where @math{x} and @math{y} are the limbs. Adjacent limb pairs are combined into quads similarly with @m{xb^{2n}+y,x*b^(2n)+y}. This continues until a single block remains, that being the result. The advantage of this method is that the multiplications for each @math{x} are big blocks, allowing Karatsuba and higher algorithms to be used. But the cost of calculating the powers @m{b^{n2^i},b^(n*2^i)} must be overcome. @code{SET_STR_PRECOMPUTE_THRESHOLD} usually ends up quite big, around 5000 digits, and on some processors much bigger still. @code{SET_STR_PRECOMPUTE_THRESHOLD} is based on the input digits (and tuned for decimal), though it might be better based on a limb count, so as to be independent of the base. But that sort of count isn't used by the base case and so would need some sort of initial calculation or estimate. The main reason @code{SET_STR_PRECOMPUTE_THRESHOLD} is so much bigger than the corresponding @code{GET_STR_PRECOMPUTE_THRESHOLD} is that @code{mpn_mul_1} is much faster than @code{mpn_divrem_1} (often by a factor of 5, or more). @need 1000 @node Other Algorithms, Assembly Coding, Radix Conversion Algorithms, Algorithms @section Other Algorithms @menu * Prime Testing Algorithm:: * Factorial Algorithm:: * Binomial Coefficients Algorithm:: * Fibonacci Numbers Algorithm:: * Lucas Numbers Algorithm:: * Random Number Algorithms:: @end menu @node Prime Testing Algorithm, Factorial Algorithm, Other Algorithms, Other Algorithms @subsection Prime Testing @cindex Prime testing algorithms The primality testing in @code{mpz_probab_prime_p} (@pxref{Number Theoretic Functions}) first does some trial division by small factors and then uses the Miller-Rabin probabilistic primality testing algorithm, as described in Knuth section 4.5.4 algorithm P (@pxref{References}). For an odd input @math{n}, and with @math{n = q@GMPmultiply{}2^k+1} where @math{q} is odd, this algorithm selects a random base @math{x} and tests whether @math{x^q @bmod{} n} is 1 or @math{-1}, or an @m{x^{q2^j} \bmod n, x^(q*2^j) mod n} is @math{1}, for @math{1@le{}j@le{}k}. If so then @math{n} is probably prime, if not then @math{n} is definitely composite. Any prime @math{n} will pass the test, but some composites do too. Such composites are known as strong pseudoprimes to base @math{x}. No @math{n} is a strong pseudoprime to more than @math{1/4} of all bases (see Knuth exercise 22), hence with @math{x} chosen at random there's no more than a @math{1/4} chance a ``probable prime'' will in fact be composite. In fact strong pseudoprimes are quite rare, making the test much more powerful than this analysis would suggest, but @math{1/4} is all that's proven for an arbitrary @math{n}. @node Factorial Algorithm, Binomial Coefficients Algorithm, Prime Testing Algorithm, Other Algorithms @subsection Factorial @cindex Factorial algorithm Factorials are calculated by a combination of two algorithms. An idea is shared among them: to compute the odd part of the factorial; a final step takes account of the power of @math{2} term, by shifting. For small @math{n}, the odd factor of @math{n!} is computed with the simple observation that it is equal to the product of all positive odd numbers smaller than @math{n} times the odd factor of @m{\lfloor n/2\rfloor!, [n/2]!}, where @m{\lfloor x\rfloor, [x]} is the integer part of @math{x}, and so on recursively. The procedure can be best illustrated with an example, @quotation @math{23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19}} @end quotation Current code collects all the factors in a single list, with a loop and no recursion, and compute the product, with no special care for repeated chunks. When @math{n} is larger, computation pass trough prime sieving. An helper function is used, as suggested by Peter Luschny: @tex $$\mathop{\rm msf}(n) = {n!\over\lfloor n/2\rfloor!^2\cdot2^k} = \prod_{p=3}^{n} p^{\mathop{\rm L}(p,n)} $$ @end tex @ifnottex @example n ----- n! | | L(p,n) msf(n) = -------------- = | | p [n/2]!^2.2^k p=3 @end example @end ifnottex Where @math{p} ranges on odd prime numbers. The exponent @math{k} is chosen to obtain an odd integer number: @math{k} is the number of 1 bits in the binary representation of @m{\lfloor n/2\rfloor, [n/2]}. The function L@math{(p,n)} can be defined as zero when @math{p} is composite, and, for any prime @math{p}, it is computed with: @tex $$\mathop{\rm L}(p,n) = \sum_{i>0}\left\lfloor{n\over p^i}\right\rfloor\bmod2 \leq\log_p(n)$$ @end tex @ifnottex @example --- \ n L(p,n) = / [---] mod 2 <= log (n) . --- p^i p i>0 @end example @end ifnottex With this helper function, we are able to compute the odd part of @math{n!} using the recursion implied by @m{n!=\lfloor n/2\rfloor!^2\cdot\mathop{\rm msf}(n)\cdot2^k , n!=[n/2]!^2*msf(n)*2^k}. The recursion stops using the small-@math{n} algorithm on some @m{\lfloor n/2^i\rfloor, [n/2^i]}. Both the above algorithms use binary splitting to compute the product of many small factors. At first as many products as possible are accumulated in a single register, generating a list of factors that fit in a machine word. This list is then split into halves, and the product is computed recursively. Such splitting is more efficient than repeated N@cross{}1 multiplies since it forms big multiplies, allowing Karatsuba and higher algorithms to be used. And even below the Karatsuba threshold a big block of work can be more efficient for the basecase algorithm. @node Binomial Coefficients Algorithm, Fibonacci Numbers Algorithm, Factorial Algorithm, Other Algorithms @subsection Binomial Coefficients @cindex Binomial coefficient algorithm Binomial coefficients @m{\left({n}\atop{k}\right), C(n@C{}k)} are calculated by first arranging @math{k @le{} n/2} using @m{\left({n}\atop{k}\right) = \left({n}\atop{n-k}\right), C(n@C{}k) = C(n@C{}n-k)} if necessary, and then evaluating the following product simply from @math{i=2} to @math{i=k}. @tex $$ \left({n}\atop{k}\right) = (n-k+1) \prod_{i=2}^{k} {{n-k+i} \over i} $$ @end tex @ifnottex @example k (n-k+i) C(n,k) = (n-k+1) * prod ------- i=2 i @end example @end ifnottex It's easy to show that each denominator @math{i} will divide the product so far, so the exact division algorithm is used (@pxref{Exact Division}). The numerators @math{n-k+i} and denominators @math{i} are first accumulated into as many fit a limb, to save multi-precision operations, though for @code{mpz_bin_ui} this applies only to the divisors, since @math{n} is an @code{mpz_t} and @math{n-k+i} in general won't fit in a limb at all. @node Fibonacci Numbers Algorithm, Lucas Numbers Algorithm, Binomial Coefficients Algorithm, Other Algorithms @subsection Fibonacci Numbers @cindex Fibonacci number algorithm The Fibonacci functions @code{mpz_fib_ui} and @code{mpz_fib2_ui} are designed for calculating isolated @m{F_n,F[n]} or @m{F_n,F[n]},@m{F_{n-1},F[n-1]} values efficiently. For small @math{n}, a table of single limb values in @code{__gmp_fib_table} is used. On a 32-bit limb this goes up to @m{F_{47},F[47]}, or on a 64-bit limb up to @m{F_{93},F[93]}. For convenience the table starts at @m{F_{-1},F[-1]}. Beyond the table, values are generated with a binary powering algorithm, calculating a pair @m{F_n,F[n]} and @m{F_{n-1},F[n-1]} working from high to low across the bits of @math{n}. The formulas used are @tex $$\eqalign{ F_{2k+1} &= 4F_k^2 - F_{k-1}^2 + 2(-1)^k \cr F_{2k-1} &= F_k^2 + F_{k-1}^2 \cr F_{2k} &= F_{2k+1} - F_{2k-1} }$$ @end tex @ifnottex @example F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k F[2k-1] = F[k]^2 + F[k-1]^2 F[2k] = F[2k+1] - F[2k-1] @end example @end ifnottex At each step, @math{k} is the high @math{b} bits of @math{n}. If the next bit of @math{n} is 0 then @m{F_{2k},F[2k]},@m{F_{2k-1},F[2k-1]} is used, or if it's a 1 then @m{F_{2k+1},F[2k+1]},@m{F_{2k},F[2k]} is used, and the process repeated until all bits of @math{n} are incorporated. Notice these formulas require just two squares per bit of @math{n}. It'd be possible to handle the first few @math{n} above the single limb table with simple additions, using the defining Fibonacci recurrence @m{F_{k+1} = F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually turns out to be faster for only about 10 or 20 values of @math{n}, and including a block of code for just those doesn't seem worthwhile. If they really mattered it'd be better to extend the data table. Using a table avoids lots of calculations on small numbers, and makes small @math{n} go fast. A bigger table would make more small @math{n} go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm. @code{mpz_fib2_ui} returns both @m{F_n,F[n]} and @m{F_{n-1},F[n-1]}, but @code{mpz_fib_ui} is only interested in @m{F_n,F[n]}. In this case the last step of the algorithm can become one multiply instead of two squares. One of the following two formulas is used, according as @math{n} is odd or even. @tex $$\eqalign{ F_{2k} &= F_k (F_k + 2F_{k-1}) \cr F_{2k+1} &= (2F_k + F_{k-1}) (2F_k - F_{k-1}) + 2(-1)^k }$$ @end tex @ifnottex @example F[2k] = F[k]*(F[k]+2F[k-1]) F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k @end example @end ifnottex @m{F_{2k+1},F[2k+1]} here is the same as above, just rearranged to be a multiply. For interest, the @m{2(-1)^k, 2*(-1)^k} term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into further limbs, which saves some code size. See comments with @code{mpz_fib_ui} and the internal @code{mpn_fib2_ui} for how this is done. @node Lucas Numbers Algorithm, Random Number Algorithms, Fibonacci Numbers Algorithm, Other Algorithms @subsection Lucas Numbers @cindex Lucas number algorithm @code{mpz_lucnum2_ui} derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas. @tex $$\eqalign{ L_k &= F_k + 2F_{k-1} \cr L_{k-1} &= 2F_k - F_{k-1} }$$ @end tex @ifnottex @example L[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1] @end example @end ifnottex @code{mpz_lucnum_ui} is only interested in @m{L_n,L[n]}, and some work can be saved. Trailing zero bits on @math{n} can be handled with a single square each. @tex $$ L_{2k} = L_k^2 - 2(-1)^k $$ @end tex @ifnottex @example L[2k] = L[k]^2 - 2*(-1)^k @end example @end ifnottex And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers, similar to what @code{mpz_fib_ui} does. @tex $$ L_{2k+1} = 5F_{k-1} (2F_k + F_{k-1}) - 4(-1)^k $$ @end tex @ifnottex @example L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k @end example @end ifnottex @node Random Number Algorithms, , Lucas Numbers Algorithm, Other Algorithms @subsection Random Numbers @cindex Random number algorithms For the @code{urandomb} functions, random numbers are generated simply by concatenating bits produced by the generator. As long as the generator has good randomness properties this will produce well-distributed @math{N} bit numbers. For the @code{urandomm} functions, random numbers in a range @math{0@le{}R= ABS(_mp_size)}. When an @code{mpz} routine is about to (or might be about to) increase @code{_mp_size}, it checks @code{_mp_alloc} to see whether there's enough space, and reallocates if not. @code{MPZ_REALLOC} is generally used for this. @end table The various bitwise logical functions like @code{mpz_and} behave as if negative values were twos complement. But sign and magnitude is always used internally, and necessary adjustments are made during the calculations. Sometimes this isn't pretty, but sign and magnitude are best for other routines. Some internal temporary variables are setup with @code{MPZ_TMP_INIT} and these have @code{_mp_d} space obtained from @code{TMP_ALLOC} rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have unpredictable consequences). @code{_mp_size} and @code{_mp_alloc} are @code{int}, although @code{mp_size_t} is usually a @code{long}. This is done to make the fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of range. @node Rational Internals, Float Internals, Integer Internals, Internals @section Rational Internals @cindex Rational internals @code{mpq_t} variables represent rationals using an @code{mpz_t} numerator and denominator (@pxref{Integer Internals}). The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1. It's believed that casting out common factors at each stage of a calculation is best in general. A GCD is an @math{O(N^2)} operation so it's better to do a few small ones immediately than to delay and have to do a big one later. Knowing the numerator and denominator have no common factors can be used for example in @code{mpq_mul} to make only two cross GCDs necessary, not four. This general approach to common factors is badly sub-optimal in the presence of simple factorizations or little prospect for cancellation, but GMP has no way to know when this will occur. As per @ref{Efficiency}, that's left to applications. The @code{mpq_t} framework might still suit, with @code{mpq_numref} and @code{mpq_denref} for direct access to the numerator and denominator, or of course @code{mpz_t} variables can be used directly. @node Float Internals, Raw Output Internals, Rational Internals, Internals @section Float Internals @cindex Float internals Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this. @code{mpf_t} floats have a variable precision mantissa and a single machine word signed exponent. The mantissa is represented using sign and magnitude. @c FIXME: The arrow heads don't join to the lines exactly. @tex \global\newdimen\GMPboxwidth \GMPboxwidth=5em \global\newdimen\GMPboxheight \GMPboxheight=3ex \def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}} \GMPdisplay{% \vbox{% \hbox to 5\GMPboxwidth {most significant limb \hfil least significant limb} \vskip 0.7ex \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}} \hbox { \hbox to 3\GMPboxwidth {% \setbox 0 = \hbox{@code{\_mp\_exp}}% \dimen0=3\GMPboxwidth \advance\dimen0 by -\wd0 \divide\dimen0 by 2 \advance\dimen0 by -1em \setbox1 = \hbox{$\rightarrow$}% \dimen1=\dimen0 \advance\dimen1 by -\wd1 \GMPcentreline{\dimen0}% \hfil \box0% \hfil \GMPcentreline{\dimen1{}}% \box1} \hbox to 2\GMPboxwidth {\hfil @code{\_mp\_d}}} \vskip 0.5ex \vbox {% \hrule \hbox{% \vrule height 2ex depth 1ex \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule \hbox to \GMPboxwidth {}% \vrule} \hrule } \hbox {% \hbox to 0.8 pt {} \hbox to 3\GMPboxwidth {% \hfil $\cdot$} \hbox {$\leftarrow$ radix point\hfil}} \hbox to 5\GMPboxwidth{% \setbox 0 = \hbox{@code{\_mp\_size}}% \dimen0 = 5\GMPboxwidth \advance\dimen0 by -\wd0 \divide\dimen0 by 2 \advance\dimen0 by -1em \dimen1 = \dimen0 \setbox1 = \hbox{$\leftarrow$}% \setbox2 = \hbox{$\rightarrow$}% \advance\dimen0 by -\wd1 \advance\dimen1 by -\wd2 \hbox to 0.3 em {}% \box1 \GMPcentreline{\dimen0}% \hfil \box0 \hfil \GMPcentreline{\dimen1}% \box2} }} @end tex @ifnottex @example most least significant significant limb limb _mp_d |---- _mp_exp ---> | _____ _____ _____ _____ _____ |_____|_____|_____|_____|_____| . <------------ radix point <-------- _mp_size ---------> @sp 1 @end example @end ifnottex @noindent The fields are as follows. @table @asis @item @code{_mp_size} The number of limbs currently in use, or the negative of that when representing a negative value. Zero is represented by @code{_mp_size} and @code{_mp_exp} both set to zero, and in that case the @code{_mp_d} data is unused. (In the future @code{_mp_exp} might be undefined when representing zero.) @item @code{_mp_prec} The precision of the mantissa, in limbs. In any calculation the aim is to produce @code{_mp_prec} limbs of result (the most significant being non-zero). @item @code{_mp_d} A pointer to the array of limbs which is the absolute value of the mantissa. These are stored ``little endian'' as per the @code{mpn} functions, so @code{_mp_d[0]} is the least significant limb and @code{_mp_d[ABS(_mp_size)-1]} the most significant. The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb. @code{_mp_prec+1} limbs are allocated to @code{_mp_d}, the extra limb being for convenience (see below). There are no reallocations during a calculation, only in a change of precision with @code{mpf_set_prec}. @item @code{_mp_exp} The exponent, in limbs, determining the location of the implied radix point. Zero means the radix point is just above the most significant limb. Positive values mean a radix point offset towards the lower limbs and hence a value @math{@ge{} 1}, as for example in the diagram above. Negative exponents mean a radix point further above the highest limb. Naturally the exponent can be any value, it doesn't have to fall within the limbs as the diagram shows, it can be a long way above or a long way below. Limbs other than those included in the @code{@{_mp_d,_mp_size@}} data are treated as zero. @end table The @code{_mp_size} and @code{_mp_prec} fields are @code{int}, although the @code{mp_size_t} type is usually a @code{long}. The @code{_mp_exp} field is usually @code{long}. This is done to make some fields just 32 bits on some 64 bits systems, thereby saving a few bytes of data space but still providing plenty of precision and a very large range. @sp 1 @noindent The following various points should be noted. @table @asis @item Low Zeros The least significant limbs @code{_mp_d[0]} etc can be zero, though such low zeros can always be ignored. Routines likely to produce low zeros check and avoid them to save time in subsequent calculations, but for most routines they're quite unlikely and aren't checked. @item Mantissa Size Range The @code{_mp_size} count of limbs in use can be less than @code{_mp_prec} if the value can be represented in less. This means low precision values or small integers stored in a high precision @code{mpf_t} can still be operated on efficiently. @code{_mp_size} can also be greater than @code{_mp_prec}. Firstly a value is allowed to use all of the @code{_mp_prec+1} limbs available at @code{_mp_d}, and secondly when @code{mpf_set_prec_raw} lowers @code{_mp_prec} it leaves @code{_mp_size} unchanged and so the size can be arbitrarily bigger than @code{_mp_prec}. @item Rounding All rounding is done on limb boundaries. Calculating @code{_mp_prec} limbs with the high non-zero will ensure the application requested minimum precision is obtained. The use of simple ``trunc'' rounding towards zero is efficient, since there's no need to examine extra limbs and increment or decrement. @item Bit Shifts Since the exponent is in limbs, there are no bit shifts in basic operations like @code{mpf_add} and @code{mpf_mul}. When differing exponents are encountered all that's needed is to adjust pointers to line up the relevant limbs. Of course @code{mpf_mul_2exp} and @code{mpf_div_2exp} will require bit shifts, but the choice is between an exponent in limbs which requires shifts there, or one in bits which requires them almost everywhere else. @item Use of @code{_mp_prec+1} Limbs The extra limb on @code{_mp_d} (@code{_mp_prec+1} rather than just @code{_mp_prec}) helps when an @code{mpf} routine might get a carry from its operation. @code{mpf_add} for instance will do an @code{mpn_add} of @code{_mp_prec} limbs. If there's no carry then that's the result, but if there is a carry then it's stored in the extra limb of space and @code{_mp_size} becomes @code{_mp_prec+1}. Whenever @code{_mp_prec+1} limbs are held in a variable, the low limb is not needed for the intended precision, only the @code{_mp_prec} high limbs. But zeroing it out or moving the rest down is unnecessary. Subsequent routines reading the value will simply take the high limbs they need, and this will be @code{_mp_prec} if their target has that same precision. This is no more than a pointer adjustment, and must be checked anyway since the destination precision can be different from the sources. Copy functions like @code{mpf_set} will retain a full @code{_mp_prec+1} limbs if available. This ensures that a variable which has @code{_mp_size} equal to @code{_mp_prec+1} will get its full exact value copied. Strictly speaking this is unnecessary since only @code{_mp_prec} limbs are needed for the application's requested precision, but it's considered that an @code{mpf_set} from one variable into another of the same precision ought to produce an exact copy. @item Application Precisions @code{__GMPF_BITS_TO_PREC} converts an application requested precision to an @code{_mp_prec}. The value in bits is rounded up to a whole limb then an extra limb is added since the most significant limb of @code{_mp_d} is only non-zero and therefore might contain only one bit. @code{__GMPF_PREC_TO_BITS} does the reverse conversion, and removes the extra limb from @code{_mp_prec} before converting to bits. The net effect of reading back with @code{mpf_get_prec} is simply the precision rounded up to a multiple of @code{mp_bits_per_limb}. Note that the extra limb added here for the high only being non-zero is in addition to the extra limb allocated to @code{_mp_d}. For example with a 32-bit limb, an application request for 250 bits will be rounded up to 8 limbs, then an extra added for the high being only non-zero, giving an @code{_mp_prec} of 9. @code{_mp_d} then gets 10 limbs allocated. Reading back with @code{mpf_get_prec} will take @code{_mp_prec} subtract 1 limb and multiply by 32, giving 256 bits. Strictly speaking, the fact the high limb has at least one bit means that a float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but for the purposes of @code{mpf_t} it's considered simply to be 64 bits, a nice multiple of the limb size. @end table @node Raw Output Internals, C++ Interface Internals, Float Internals, Internals @section Raw Output Internals @cindex Raw output internals @noindent @code{mpz_out_raw} uses the following format. @tex \global\newdimen\GMPboxwidth \GMPboxwidth=5em \global\newdimen\GMPboxheight \GMPboxheight=3ex \def\centreline{\hbox{\raise 0.8ex \vbox{\hrule \hbox{\hfil}}}} \GMPdisplay{% \vbox{% \def\GMPcentreline#1{\hbox{\raise 0.5 ex \vbox{\hrule \hbox to #1 {}}}} \vbox {% \hrule \hbox{% \vrule height 2.5ex depth 1.5ex \hbox to \GMPboxwidth {\hfil size\hfil}% \vrule \hbox to 3\GMPboxwidth {\hfil data bytes\hfil}% \vrule} \hrule} }} @end tex @ifnottex @example +------+------------------------+ | size | data bytes | +------+------------------------+ @end example @end ifnottex The size is 4 bytes written most significant byte first, being the number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first. The most significant data byte is always non-zero, so the output is the same on all systems, irrespective of limb size. In GMP 1, leading zero bytes were written to pad the data bytes to a multiple of the limb size. @code{mpz_inp_raw} will still accept this, for compatibility. The use of ``big endian'' for both the size and data fields is deliberate, it makes the data easy to read in a hex dump of a file. Unfortunately it also means that the limb data must be reversed when reading or writing, so neither a big endian nor little endian system can just read and write @code{_mp_d}. @node C++ Interface Internals, , Raw Output Internals, Internals @section C++ Interface Internals @cindex C++ interface internals A system of expression templates is used to ensure something like @code{a=b+c} turns into a simple call to @code{mpz_add} etc. For @code{mpf_class} the scheme also ensures the precision of the final destination is used for any temporaries within a statement like @code{f=w*x+y*z}. These are important features which a naive implementation cannot provide. A simplified description of the scheme follows. The true scheme is complicated by the fact that expressions have different return types. For detailed information, refer to the source code. To perform an operation, say, addition, we first define a ``function object'' evaluating it, @example struct __gmp_binary_plus @{ static void eval(mpf_t f, mpf_t g, mpf_t h) @{ mpf_add(f, g, h); @} @}; @end example @noindent And an ``additive expression'' object, @example __gmp_expr<__gmp_binary_expr > operator+(const mpf_class &f, const mpf_class &g) @{ return __gmp_expr <__gmp_binary_expr >(f, g); @} @end example The seemingly redundant @code{__gmp_expr<__gmp_binary_expr<@dots{}>>} is used to encapsulate any possible kind of expression into a single template type. In fact even @code{mpf_class} etc are @code{typedef} specializations of @code{__gmp_expr}. Next we define assignment of @code{__gmp_expr} to @code{mpf_class}. @example template mpf_class & mpf_class::operator=(const __gmp_expr &expr) @{ expr.eval(this->get_mpf_t(), this->precision()); return *this; @} template void __gmp_expr<__gmp_binary_expr >::eval (mpf_t f, mp_bitcnt_t precision) @{ Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); @} @end example where @code{expr.val1} and @code{expr.val2} are references to the expression's operands (here @code{expr} is the @code{__gmp_binary_expr} stored within the @code{__gmp_expr}). This way, the expression is actually evaluated only at the time of assignment, when the required precision (that of @code{f}) is known. Furthermore the target @code{mpf_t} is now available, thus we can call @code{mpf_add} directly with @code{f} as the output argument. Compound expressions are handled by defining operators taking subexpressions as their arguments, like this: @example template __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > operator+(const __gmp_expr &expr1, const __gmp_expr &expr2) @{ return __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > (expr1, expr2); @} @end example And the corresponding specializations of @code{__gmp_expr::eval}: @example template void __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, Op> >::eval (mpf_t f, mp_bitcnt_t precision) @{ // declare two temporaries mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); @} @end example The expression is thus recursively evaluated to any level of complexity and all subexpressions are evaluated to the precision of @code{f}. @node Contributors, References, Internals, Top @comment node-name, next, previous, up @appendix Contributors @cindex Contributors Torbj@"orn Granlund wrote the original GMP library and is still the main developer. Code not explicitly attributed to others, was contributed by Torbj@"orn. Several other individuals and organizations have contributed GMP. Here is a list in chronological order on first contribution: Gunnar Sj@"odin and Hans Riesel helped with mathematical problems in early versions of the library. Richard Stallman helped with the interface design and revised the first version of this manual. Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions. John Amanatides of York University in Canada contributed the function @code{mpz_probab_prime_p}. Paul Zimmermann wrote the REDC-based mpz_powm code, the Sch@"onhage-Strassen FFT multiply code, and the Karatsuba square root code. He also improved the Toom3 code for GMP 4.2. Paul sparked the development of GMP 2, with his comparisons between bignum packages. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth root code (with Torbj@"orn). Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul) contributed now defunct versions of @code{mpz_gcd}, @code{mpz_divexact}, @code{mpn_gcd}, and @code{mpn_bdivmod}, partially supported by CNPq (Brazil) grant 301314194-2. Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure. He has also made valuable suggestions and tested numerous intermediary releases. Joachim Hollman was involved in the design of the @code{mpf} interface, and in the @code{mpz} design revisions for version 2. Bennet Yee contributed the initial versions of @code{mpz_jacobi} and @code{mpz_legendre}. Andreas Schwab contributed the files @file{mpn/m68k/lshift.S} and @file{mpn/m68k/rshift.S} (now in @file{.asm} form). Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count. Robert also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3, and contributed the ARM assembly code. Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several algorithms. Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions. Kevin Ryde worked on a large number of things: optimized x86 code, m4 asm macros, parameter tuning, speed measuring, the configure system, function inlining, divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number functions, printf and scanf functions, perl interface, demo expression parser, the algorithms chapter in the manual, @file{gmpasm-mode.el}, and various miscellaneous improvements elsewhere. Kent Boortz made the Mac OS 9 port. Steve Root helped write the optimized alpha 21264 assembly code. Gerardo Ballabio wrote the @file{gmpxx.h} C++ class interface and the C++ @code{istream} input routines. Jason Moxham rewrote @code{mpz_fac_ui}. Pedro Gimeno implemented the Mersenne Twister and made other random number improvements. Niels M@"oller wrote the sub-quadratic GCD, extended GCD and jacobi code, the quadratic Hensel division code, and (with Torbj@"orn) the new divide and conquer division code for GMP 4.3. Niels also helped implement the new Toom multiply code for GMP 4.3 and implemented helper functions to simplify Toom evaluations for GMP 5.0. He wrote the original version of mpn_mulmod_bnm1, and he is the main author of the mini-gmp package used for gmp bootstrapping. Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply strategy, and found the optimal strategies for evaluation and interpolation in Toom multiplication. Marco Bodrato helped implement the new Toom multiply code for GMP 4.3 and implemented most of the new Toom multiply and squaring code for 5.0. He is the main author of the current mpn_mulmod_bnm1 and mpn_mullo_n. Marco also wrote the functions mpn_invert and mpn_invertappr. He is the author of the current combinatorial functions: binomial, factorial, multifactorial, primorial. David Harvey suggested the internal function @code{mpn_bdiv_dbm1}, implementing division relevant to Toom multiplication. He also worked on fast assembly sequences, in particular on a fast AMD64 @code{mpn_mul_basecase}. He wrote the internal middle product functions @code{mpn_mulmid_basecase}, @code{mpn_toom42_mulmid}, @code{mpn_mulmid_n} and related helper routines. Martin Boij wrote @code{mpn_perfect_power_p}. Marc Glisse improved @file{gmpxx.h}: use fewer temporaries (faster), specializations of @code{numeric_limits} and @code{common_type}, C++11 features (move constructors, explicit bool conversion, UDL), make the conversion from @code{mpq_class} to @code{mpz_class} explicit, optimize operations where one argument is a small compile-time constant, replace some heap allocations by stack allocations. He also fixed the eofbit handling of C++ streams, and removed one division from @file{mpq/aors.c}. (This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell @email{gmp-devel@@gmplib.org} about the omission!) The development of floating point functions of GNU MP 2, were supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial System SOlving). The development of GMP 2, 3, and 4 was supported in part by the IDA Center for Computing Sciences. Thanks go to Hans Thorsen for donating an SGI system for the GMP test system environment. @node References, GNU Free Documentation License, Contributors, Top @comment node-name, next, previous, up @appendix References @cindex References @c FIXME: In tex, the @uref's are unhyphenated, which is good for clarity, @c but being long words they upset paragraph formatting (the preceding line @c can get badly stretched). Would like an conditional @* style line break @c if the uref is too long to fit on the last line of the paragraph, but it's @c not clear how to do that. For now explicit @texlinebreak{}s are used on @c paragraphs that come out bad. @section Books @itemize @bullet @item Jonathan M. Borwein and Peter B. Borwein, ``Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity'', Wiley, 1998. @item Richard Crandall and Carl Pomerance, ``Prime Numbers: A Computational Perspective'', 2nd edition, Springer-Verlag, 2005. @texlinebreak{} @uref{http://www.math.dartmouth.edu/~carlp/} @item Henri Cohen, ``A Course in Computational Algebraic Number Theory'', Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. @texlinebreak{} @uref{http://www.math.u-bordeaux.fr/~cohen/} @item Donald E. Knuth, ``The Art of Computer Programming'', volume 2, ``Seminumerical Algorithms'', 3rd edition, Addison-Wesley, 1998. @texlinebreak{} @uref{http://www-cs-faculty.stanford.edu/~knuth/taocp.html} @item John D. Lipson, ``Elements of Algebra and Algebraic Computing'', The Benjamin Cummings Publishing Company Inc, 1981. @item Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, ``Handbook of Applied Cryptography'', @uref{http://www.cacr.math.uwaterloo.ca/hac/} @item Richard M. Stallman and the GCC Developer Community, ``Using the GNU Compiler Collection'', Free Software Foundation, 2008, available online @uref{http://gcc.gnu.org/onlinedocs/}, and in the GCC package @uref{ftp://ftp.gnu.org/gnu/gcc/} @end itemize @section Papers @itemize @bullet @item Yves Bertot, Nicolas Magaud and Paul Zimmermann, ``A Proof of GMP Square Root'', Journal of Automated Reasoning, volume 29, 2002, pp.@: 225-252. Also available online as INRIA Research Report 4475, June 2002, @uref{http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf} @item Christoph Burnikel and Joachim Ziegler, ``Fast Recursive Division'', Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, @texlinebreak{} @uref{http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022} @item Torbj@"orn Granlund and Peter L. Montgomery, ``Division by Invariant Integers using Multiplication'', in Proceedings of the SIGPLAN PLDI'94 Conference, June 1994. Also available @uref{http://gmplib.org/~tege/divcnst-pldi94.pdf}. @item Niels M@"oller and Torbj@"orn Granlund, ``Improved division by invariant integers'', IEEE Transactions on Computers, 11 June 2010. @uref{http://gmplib.org/~tege/division-paper.pdf} @item Torbj@"orn Granlund and Niels M@"oller, ``Division of integers large and small'', to appear. @item Tudor Jebelean, ``An algorithm for exact division'', Journal of Symbolic Computation, volume 15, 1993, pp.@: 169-180. Research report version available @texlinebreak{} @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz} @item Tudor Jebelean, ``Exact Division with Karatsuba Complexity - Extended Abstract'', RISC-Linz technical report 96-31, @texlinebreak{} @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz} @item Tudor Jebelean, ``Practical Integer Division with Karatsuba Complexity'', ISSAC 97, pp.@: 339-341. Technical report available @texlinebreak{} @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz} @item Tudor Jebelean, ``A Generalization of the Binary GCD Algorithm'', ISSAC 93, pp.@: 111-116. Technical report version available @texlinebreak{} @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz} @item Tudor Jebelean, ``A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD of Long Integers'', Journal of Symbolic Computation, volume 19, 1995, pp.@: 145-157. Technical report version also available @texlinebreak{} @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz} @item Werner Krandick and Tudor Jebelean, ``Bidirectional Exact Integer Division'', Journal of Symbolic Computation, volume 21, 1996, pp.@: 441-455. Early technical report version also available @uref{ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz} @item Makoto Matsumoto and Takuji Nishimura, ``Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator'', ACM Transactions on Modelling and Computer Simulation, volume 8, January 1998, pp.@: 3-30. Available online @texlinebreak{} @uref{http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz} (or .pdf) @item R. Moenck and A. Borodin, ``Fast Modular Transforms via Division'', Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, October 1972, pp.@: 90-96. Reprinted as ``Fast Modular Transforms'', Journal of Computer and System Sciences, volume 8, number 3, June 1974, pp.@: 366-386. @item Niels M@"oller, ``On Sch@"onhage's algorithm and subquadratic integer GCD computation'', in Mathematics of Computation, volume 77, January 2008, pp.@: 589-607. @item Peter L. Montgomery, ``Modular Multiplication Without Trial Division'', in Mathematics of Computation, volume 44, number 170, April 1985. @item Arnold Sch@"onhage and Volker Strassen, ``Schnelle Multiplikation grosser Zahlen'', Computing 7, 1971, pp.@: 281-292. @item Kenneth Weber, ``The accelerated integer GCD algorithm'', ACM Transactions on Mathematical Software, volume 21, number 1, March 1995, pp.@: 111-122. @item Paul Zimmermann, ``Karatsuba Square Root'', INRIA Research Report 3805, November 1999, @uref{http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf} @item Paul Zimmermann, ``A Proof of GMP Fast Division and Square Root Implementations'', @texlinebreak{} @uref{http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz} @item Dan Zuras, ``On Squaring and Multiplying Large Integers'', ARITH-11: IEEE Symposium on Computer Arithmetic, 1993, pp.@: 260 to 271. Reprinted as ``More on Multiplying and Squaring Large Integers'', IEEE Transactions on Computers, volume 43, number 8, August 1994, pp.@: 899-908. @end itemize @node GNU Free Documentation License, Concept Index, References, Top @appendix GNU Free Documentation License @cindex GNU Free Documentation License @cindex Free Documentation License @cindex Documentation license @include fdl-1.3.texi @node Concept Index, Function Index, GNU Free Documentation License, Top @comment node-name, next, previous, up @unnumbered Concept Index @printindex cp @node Function Index, , Concept Index, Top @comment node-name, next, previous, up @unnumbered Function and Type Index @printindex fn @bye @c Local variables: @c fill-column: 78 @c compile-command: "make gmp.info" @c End: gmp-doc-5.1.2/doc/version.texi0000644000175000000620000000013312146435201015173 0ustar stevestaff@set UPDATED 20 May 2013 @set UPDATED-MONTH May 2013 @set EDITION 5.1.2 @set VERSION 5.1.2