Math-Calculus-Differentiate-0.3/0040755000176200010010000000000010167102436016004 5ustar JonathanNoneMath-Calculus-Differentiate-0.3/MANIFEST0100644000176200010010000000022210167102434017124 0ustar JonathanNoneChanges Differentiate.pm Makefile.PL MANIFEST README test.pl META.yml Module meta-data (added by MakeMaker) Math-Calculus-Differentiate-0.3/Differentiate.pm0100644000176200010010000005526310160672116021123 0ustar JonathanNone# ######################################################################################## # A CALCULUS DIFFERENTIATION OBJECT # An implementation of algebraic differentiation by Jonathan Worthington. # Copyright (C) Jonathan Worthington 2004 # This module may be used and distributed under the same terms as Perl. # ######################################################################################## package Math::Calculus::Differentiate; use 5.006; use Math::Calculus::Expression; use strict; our $VERSION = '0.3'; our @ISA = qw/Math::Calculus::Expression/; =head1 NAME Math::Calculus::Differentiate - Algebraic Differentiation Engine =head1 SYNOPSIS use Math::Calculus::Differentiate; # Create an object. my $exp = Math::Calculus::Differentiate->new; # Set a variable and expression. $exp->addVariable('x'); $exp->setExpression('x^2 + 5*x') or die $exp->getError; # Differentiate and simplify. $exp->differentiate or die $exp->getError;; $exp->simplify or die $exp->getError;; # Print the result. print $exp->getExpression; # Prints 2*x + 5 =head1 DESCRIPTION This module can take an algebraic expression, parse it into a tree structure, modify the tree to give a representation of the differentiated function, simplify the tree and turn the tree back into an output of the same form as the input. It supports differentiation of expressions including the +, -, *, / and ^ (raise to power) operators, bracketed expressions to enable correct precedence and the functions ln, exp, sin, cos, tan, sec, cosec, cot, sinh, cosh, tanh, sech, cosech, coth, asin, acos, atan, asinh, acosh and atanh. =head1 EXPORT None by default. =head1 METHODS =item new $exp = Math::Calculus::Differentiate->new; Creates a new instance of the differentiation engine, which can hold an individual expression. =item addVariable $exp->addVariable('x'); Sets a certain named value in the expression as being a variable. A named value must be an alphabetic chracter. =item setExpression $exp->setExpression('x^2 + 5*x); Takes an expression in human-readable form and stores it internally as a tree structure, checking it is a valid expression that the module can understand in the process. Note that the engine is strict about syntax. For example, note above that you must write 5*x and not just 5x. Whitespace is allowed in the expression, but does not have any effect on precedence. If you require control of precedence, use brackets; bracketed expressions will always be evaluated first, as you would normally expect. The module follows the BODMAS precedence convention. Returns undef on failure and a true value on success. =item getExpression $expr = $exp->getExpression; Returns a textaul, human readable representation of the expression that is being stored. =cut # Differentiate. # ############## =item differentiate $exp->differentiate('x'); Differentiates the expression that was stored with setExpression with respect to the variable passed as a parameter. Returns undef on failure and a true value on success. =cut sub differentiate { # Get invocant and variable. my ($self, $variable) = @_; # Check variable is in the list of variables. return undef unless grep { $_ eq $variable } @{$self->{'variables'}}; # Clear error and traceback, and pass control to the differentiate routine. $self->{'error'} = $self->{'traceback'} = undef; eval { $self->{'expression'} = $self->differentiateTree($variable, $self->{'expression'}); }; # Return an appropriate value (or lack thereof...). if ($self->{'error'}) { return undef; } else { return 1; } } =item simplify $exp->simplify; Attempts to simplify the expression that is stored internally. It is a very good idea to call this after calling differentiate, as the tree will often not be in the most compact possible form, and this will affect the readability of output from getExpression and the performance of future calls to differentiate if you are intending to obtain higher derivatives. Returns undef on failure and a true value on success. =item getTraceback $exp->getTraceback; When setExpression and differentiate are called, a traceback is generated to describe what these functions did. If an error occurs, this traceback can be extremely useful in helping track down the source of the error. =item getError $exp->getError; When any method other than getTraceback is called, the error message stored is cleared, and then any errors that occur during the execution of the method are stored. If failure occurs, call this method to get a textual representation of the error. =head1 SEE ALSO The author of this module has a website at L, which has the latest news about the module and a web-based frontend to allow you to test the module out for yourself. =head1 AUTHOR Jonathan Worthington, Ejonathan@jwcs.netE =head1 COPYRIGHT AND LICENSE Copyright (C) 2004 by Jonathan Worthington This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.8.1 or, at your option, any later version of Perl 5 you may have available. =cut # ######################################################################################## # Private Methods # ######################################################################################## # Differentiate Tree explores the current expression tree, recursively differentiating # the branches of the tree. # ######################################################################################## sub differentiateTree { # Get invocant, variable and tree. my ($self, $variable, $tree) = @_; # Generate traceback. $self->{'traceback'} .= "Parsing " . $self->prettyPrint($tree) . "\n"; # If we're at a node... unless (ref $tree) { # Is it the variable? if ($tree eq $variable) { # It goes to 1. return 1; # Or - the variable... } elsif ($tree eq "-$variable") { # It goes to -1. return -1; # Otherwise, it's a constant and goes to zero. } else { return 0; } } else { # We've got a complex expression. Our actions from here depend on what the # expression is. # Addition or subtraction - just differentiate each operand. if ($tree->{'operation'} eq '+' || $tree->{'operation'} eq '-') { return { operation => $tree->{'operation'}, operand1 => $self->differentiateTree($variable, $tree->{'operand1'}), operand2 => $self->differentiateTree($variable, $tree->{'operand2'}) }; # Multiplication. } elsif ($tree->{'operation'} eq '*') { # Check if any branches are constant. my $o1c = $self->isConstant($variable, $tree->{'operand1'}); my $o2c = $self->isConstant($variable, $tree->{'operand2'}); # If they're both constant, return the tree as it is. if ($o1c && $o2c) { return $tree; # If the first is constant, only differentiate the second. } elsif ($o1c) { return { operation => $tree->{'operation'}, operand1 => $tree->{'operand1'}, operand2 => $self->differentiateTree($variable, $tree->{'operand2'}) }; # If the second is constant, only differentiate the first. } elsif ($o2c) { return { operation => $tree->{'operation'}, operand1 => $self->differentiateTree($variable, $tree->{'operand1'}), operand2 => $tree->{'operand2'} }; # Otherwise, it's the product rule. d[uv] = udv + vdu } else { return { operation => '+', operand1 => { operation => '*', operand1 => $tree->{'operand1'}, operand2 => $self->differentiateTree($variable, $tree->{'operand2'}) }, operand2 => { operation => '*', operand1 => $tree->{'operand2'}, operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) } }; } # Division. } elsif ($tree->{'operation'} eq '/') { # Check if any branches are constant. my $o1c = $self->isConstant($variable, $tree->{'operand1'}); my $o2c = $self->isConstant($variable, $tree->{'operand2'}); # If they're both constant, return the tree as it is. if ($o1c && $o2c) { return $tree; # If the denominator is constant, just differentiate the top. } elsif ($o2c) { return { operation => '/', operand1 => $self->differentiateTree($variable, $tree->{'operand1'}), operand2 => $tree->{'operand2'} }; # If the numerator is constant, e.g. k/u, then return k * d[u^-1]. } elsif ($o1c) { my $uinv = { operation => '^', operand1 => $tree->{'operand2'}, operand2 => -1 }; return { operation => '*', operand1 => $tree->{'operand1'}, operand2 => $self->differentiateTree($variable, $uinv) } # Otherwise, neither is constant. Use d[u/v] = (vdu - udv) / v^2. } else { my $vdu = { operation => '*', operand2 => $tree->{'operand2'}, operand1 => $self->differentiateTree($variable, $tree->{'operand1'}) }; my $udv = { operation => '*', operand2 => $tree->{'operand1'}, operand1 => $self->differentiateTree($variable, $tree->{'operand2'}) }; return { operation => '/', operand1 => { operation => '-', operand1 => $vdu, operand2 => $udv }, operand2 => { operation => '^', operand1 => $tree->{'operand2'}, operand2 => 2 } }; } # Powers. } elsif ($tree->{'operation'} eq '^') { # Check if any branches are constant. my $o1c = $self->isConstant($variable, $tree->{'operand1'}); my $o2c = $self->isConstant($variable, $tree->{'operand2'}); # If they're both constant, return the tree as it is. if ($o1c && $o2c) { return $tree; # If the power is constant... } elsif ($o2c) { # d[(f(x))^n] = n*f'(x)*f(x)^(n-1) return { operation => '*', operand1 => $tree->{'operand2'}, operand2 => { operation => '*', operand1 => $self->differentiateTree($variable, $tree->{'operand1'}), operand2 => { operation => '^', operand1 => $tree->{'operand1'}, operand2 => { operation => '-', operand1 => $tree->{'operand2'}, operand2 => 1 } } } }; # If the value being raised to a power is constant... } elsif ($o1c) { # d[k^v] = dv * ln(k) * exp(ln(k) * v) my $dv = $self->differentiateTree($variable, $tree->{'operand2'}); my $lnk = { operation => 'ln', operand1 => $tree->{'operand1'}, operand2 => undef }; return { operation => '*', operand1 => $dv, operand2 => { operation => '*', operand1 => $lnk, operand2 => { operation => 'exp', operand1 => { operation => '*', operand1 => $lnk, operand2 => $tree->{'operand2'} }, operand2 => undef } } }; # If it's a function of the variable raised to another function of the variable... } else { # d[u^v] = exp(ln(u) * v) * ((vdu)/u + ln(u)dv) my $lnu = { operation => 'ln', operand1 => $tree->{'operand1'}, operand2 => undef }; my $dv = $self->differentiateTree($variable, $tree->{'operand2'}); my $vdu = { operation => '*', operand1 => $tree->{'operand2'}, operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) }; return { operation => '*', operand1 => { operation => 'exp', operand1 => { operation => '*', operand1 => $lnu, operand2 => $tree->{'operand2'} }, operand2 => undef }, operand2 => { operation => '+', operand1 => { operation => '/', operand1 => $vdu, operand2 => $tree->{'operand1'} }, operand2 => { operation => '*', operand1 => $lnu, operand2 => $dv } } }; } # Natural logarithm } elsif ($tree->{'operation'} =~ /^(\-?)ln$/) { # Stash negativity. my $neg = $1; # d[ln(u)] = du/u my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => "${neg}1", operand2 => { operation => '/', operand1 => $du, operand2 => $tree->{'operand1'} } }; # Exponential (e) } elsif ($tree->{'operation'} =~ /^(\-?)exp$/) { # Stash negativity. my $neg = $1; # d[exp(u)] = exp(u)du my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => $du, operand2 => $tree }; # sin } elsif ($tree->{'operation'} =~ /^(\-?)sin$/) { # Stash negativity. my $neg = $1; # d[sin(u)] = cos(u)du my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => $du, operand2 => { operation => "${neg}cos", operand1 => $tree->{'operand1'}, operand2 => undef } }; # cos } elsif ($tree->{'operation'} =~ /^(\-?)cos$/) { # Stash negativity. my $neg = $1 eq '-' ? '' : '-'; # d[cos(u)] = -sin(u)du my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => $du, operand2 => { operation => "${neg}sin", operand1 => $tree->{'operand1'}, operand2 => undef } }; # tan } elsif ($tree->{'operation'} =~ /^(\-?)tan$/) { # Stash negativity. my $neg = $1; # d[tan(u)] = (sec(u))^2 * du my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => "${neg}1", operand2 => { operation => '*', operand1 => $du, operand2 => { operation => '^', operand1 => { operation => "sec", operand1 => $tree->{'operand1'}, operand2 => undef }, operand2 => 2 } } }; # sec } elsif ($tree->{'operation'} =~ /^(\-?)sec$/) { # Stash negativity. my $neg = $1; # Convert to 1/cos and differentiate. return $self->differentiateTree($variable, { operation => '/', operand1 => "${neg}1", operand2 => { operation => 'cos', operand1 => $tree->{'operand1'}, operand2 => undef } }); # cosec } elsif ($tree->{'operation'} =~ /^(\-?)cosec$/) { # Stash negativity. my $neg = $1; # Convert to 1/sin and differentiate. return $self->differentiateTree($variable, { operation => '/', operand1 => "${neg}1", operand2 => { operation => 'sin', operand1 => $tree->{'operand1'}, operand2 => undef } }); # cot } elsif ($tree->{'operation'} =~ /^(\-?)cot$/) { # Stash negativity. my $neg = $1; # Convert to 1/tan and differentiate. return $self->differentiateTree($variable, { operation => '/', operand1 => "${neg}1", operand2 => { operation => 'tan', operand1 => $tree->{'operand1'}, operand2 => undef } }); # sinh } elsif ($tree->{'operation'} =~ /^(\-?)sinh$/) { # Stash negativity. my $neg = $1; # d[sinh(u)] = cosh(u)du my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => $du, operand2 => { operation => "${neg}cosh", operand1 => $tree->{'operand1'}, operand2 => undef } }; # cosh } elsif ($tree->{'operation'} =~ /^(\-?)cosh$/) { # Stash negativity. my $neg = $1; # d[cosh(u)] = sinh(u)du my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => $du, operand2 => { operation => "${neg}sinh", operand1 => $tree->{'operand1'}, operand2 => undef } }; # tanh } elsif ($tree->{'operation'} =~ /^(\-?)tanh$/) { # Stash negativity. my $neg = $1; # d[tanh(u)] = (sech(u))^2 * du my $du = $self->differentiateTree($variable, $tree->{'operand1'}); return { operation => '*', operand1 => "${neg}1", operand2 => { operation => '*', operand1 => $du, operand2 => { operation => '^', operand1 => { operation => "sech", operand1 => $tree->{'operand1'}, operand2 => undef }, operand2 => 2 } } }; # sech } elsif ($tree->{'operation'} =~ /^(\-?)sech$/) { # Stash negativity. my $neg = $1; # Convert to 1/cosh and differentiate. return $self->differentiateTree($variable, { operation => '/', operand1 => "${neg}1", operand2 => { operation => 'cosh', operand1 => $tree->{'operand1'}, operand2 => undef } }); # cosech } elsif ($tree->{'operation'} =~ /^(\-?)cosech$/) { # Stash negativity. my $neg = $1; # Convert to 1/sinh and differentiate. return $self->differentiateTree($variable, { operation => '/', operand1 => "${neg}1", operand2 => { operation => 'sinh', operand1 => $tree->{'operand1'}, operand2 => undef } }); # coth } elsif ($tree->{'operation'} =~ /^(\-?)coth$/) { # Stash negativity. my $neg = $1; # Convert to 1/tanh and differentiate. return $self->differentiateTree($variable, { operation => '/', operand1 => "${neg}1", operand2 => { operation => 'tanh', operand1 => $tree->{'operand1'}, operand2 => undef } }); # asin } elsif ($tree->{'operation'} =~ /^(\-?)asin$/) { # Stash negativity. my $neg = $1; # d[asin(u)] = du / (1 - u^2)^0.5 my $du; if ($neg) { $du = { operation => '-', operand1 => '0', operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) }; } else { $du = $self->differentiateTree($variable, $tree->{'operand1'}); } return { operation => '/', operand1 => $du, operand2 => { operation => '^', operand1 => { operation => '-', operand1 => 1, operand2 => { operation => '^', operand1 => $tree->{'operand1'}, operand2 => 2 } }, operand2 => 0.5 } }; # acos } elsif ($tree->{'operation'} =~ /^(\-?)acos$/) { # Stash negativity. my $neg = $1; # d[acos(u)] = -du / (1 - u^2)^0.5 my $du; if ($neg) { $du = $self->differentiateTree($variable, $tree->{'operand1'}); } else { $du = { operation => '-', operand1 => '0', operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) }; } return { operation => '/', operand1 => $du, operand2 => { operation => '^', operand1 => { operation => '-', operand1 => 1, operand2 => { operation => '^', operand1 => $tree->{'operand1'}, operand2 => 2 } }, operand2 => 0.5 } }; # atan } elsif ($tree->{'operation'} =~ /^(\-?)atan$/) { # Stash negativity. my $neg = $1; # d[atan(u)] = du / (1 + u^2) my $du; if ($neg) { $du = { operation => '-', operand1 => '0', operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) }; } else { $du = $self->differentiateTree($variable, $tree->{'operand1'}); } return { operation => '/', operand1 => $du, operand2 => { operation => '+', operand1 => 1, operand2 => { operation => '^', operand1 => $tree->{'operand1'}, operand2 => 2 } } }; # asinh } elsif ($tree->{'operation'} =~ /^(\-?)asinh$/) { # Stash negativity. my $neg = $1; # d[asinh(u)] = du / (1 + u^2)^0.5 my $du; if ($neg) { $du = { operation => '-', operand1 => '0', operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) }; } else { $du = $self->differentiateTree($variable, $tree->{'operand1'}); } return { operation => '/', operand1 => $du, operand2 => { operation => '^', operand1 => { operation => '+', operand1 => 1, operand2 => { operation => '^', operand1 => $tree->{'operand1'}, operand2 => 2 } }, operand2 => 0.5 } }; # acosh } elsif ($tree->{'operation'} =~ /^(\-?)acosh$/) { # Stash negativity. my $neg = $1; # d[acosh(u)] = du / (u^2 - 1)^0.5 my $du; if ($neg) { $du = { operation => '-', operand1 => '0', operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) }; } else { $du = $self->differentiateTree($variable, $tree->{'operand1'}); } return { operation => '/', operand1 => $du, operand2 => { operation => '^', operand1 => { operation => '-', operand1 => { operation => '^', operand1 => $tree->{'operand1'}, operand2 => 2 }, operand2 => 1 }, operand2 => 0.5 } }; # atanh } elsif ($tree->{'operation'} =~ /^(\-?)atanh$/) { # Stash negativity. my $neg = $1; # d[atanh(u)] = du / (1 - u^2) my $du; if ($neg) { $du = { operation => '-', operand1 => '0', operand2 => $self->differentiateTree($variable, $tree->{'operand1'}) }; } else { $du = $self->differentiateTree($variable, $tree->{'operand1'}); } return { operation => '/', operand1 => $du, operand2 => { operation => '-', operand1 => 1, operand2 => { operation => '^', operand1 => $tree->{'operand1'}, operand2 => 2 } } }; # Otherwise, we don't know what it is. } else { $self->{'error'} = "Could not differentiate " . $self->prettyPrint($tree); die; } } } 1; Math-Calculus-Differentiate-0.3/META.yml0100644000176200010010000000056110167102434017252 0ustar JonathanNone# http://module-build.sourceforge.net/META-spec.html #XXXXXXX This is a prototype!!! It will change in the future!!! XXXXX# name: Math-Calculus-Differentiate version: 0.3 version_from: Differentiate.pm installdirs: site requires: Math::Calculus::Expression: 0.1 distribution_type: module generated_by: ExtUtils::MakeMaker version 6.17 Math-Calculus-Differentiate-0.3/test.pl0100644000176200010010000000355210160673640017325 0ustar JonathanNone# Before `make install' is performed this script should be runnable with # `make test'. After `make install' it should work as `perl test.pl' ######################### use Test; BEGIN { plan tests => 12 } my $res; ## BASIC MODULE TESTS # Check include of module works. use Math::Calculus::Differentiate; ok(1); # Create object. my $exp = Math::Calculus::Differentiate->new; ok($exp); # Add a variable. $res = $exp->addVariable('x'); ok($res); ## REGRESSION TESTS # d[x] = 1 $exp->setExpression('x'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '1'); # d[-x] = -1 $exp->setExpression('-x'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '-1'); # d[x^2 + 4*x + 5] = 2*x + 4 $exp->setExpression('x^2 + 4*x + 5'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '2*x + 4'); # d[sin(-2 * x)] = -2*cos(-2*x) $exp->setExpression('sin(-2 * x)'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '-2*cos(-2*x)'); # d[asin(x)] = 1/(1 - x^2)^0.5 $exp->setExpression('asin(x)'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '1/(1 - x^2)^0.5'); # d[ln(x^2)] = (2*x)/x^2 $exp->setExpression('ln(x^2)'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '(2*x)/x^2'); # d[x^x] = exp(ln(x)*x)*(1 + ln(x)) $exp->setExpression('x^x'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq 'exp(ln(x)*x)*(1 + ln(x))'); # d[x/x] = 0 $exp->setExpression('x/x'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '0'); # d[x - x] = 0 $exp->setExpression('x - x'); $exp->differentiate('x'); $exp->simplify; $res = $exp->getExpression; ok($res eq '0'); Math-Calculus-Differentiate-0.3/Changes0100644000176200010010000000207710167102416017300 0ustar JonathanNoneRevision history for Perl module Math::Calculus::Differentiate ============================================================== VERSION 0.3 * Released to CPAN 5th January 2004 * Abstracted much code out to Math::Calculus::Expression, which this module now inherits. * Fixed an expression parser bug. VERSION 0.2.1 * Released to CPAN on 4th September 2004. * Fixed several bugs relating to handling of negation. * Added support for differentiating inverse trigometric and hyperbolic functions (asin, acos, atan, asinh, acosh and atanh). * Introduced several more rules to the simplifier. * Included a handful of regression tests. VERSION 0.2 * Released to CPAN on 1st September 2004. * Completely re-written to use a tree structure internally. * Precedence problems gone, and the / operator works. * Interface changed from that of the 0.1 module. * Documentation written. I didn't write any for 0.1, just in case somebody tried to use it. ;-) VERSION 0.1 * Released informally, badly written, never made it to CPAN, eventually scrapped completely. Math-Calculus-Differentiate-0.3/README0100644000176200010010000000221310160673736016670 0ustar JonathanNoneMath::Calculus::Differentiate version 0.3.0 =========================================== Math::Calculus::Differentiate takes an algebraic expression (e.g. in the form x^2 + 5*x + sin(x)), differentiates it and returns the derivative in the same format (e.g. 2*x + 5 + cos(x)). In understands the operators +, -, *, / and ^ (raising to a power) as well as the functions sin, cos, tan, sec, cosec, cot, sinh, cosh, tanh, sech, cosech, coth, asin, acos, atan, asinh, acosh, atanh, ln and exp. Functions are valid either side of all operators - that is, you can differentiate a function raised to the power of another function. Partial differentiation is supported to the extent that you can differentiate with respect to a single variable. INSTALLATION To install this module type the following: perl Makefile.PL make make test make install DEPENDENCIES This module requires Perl 5.6.0 or greater and the module Math::Calculus::Expression. COPYRIGHT AND LICENCE Copyright (C) 2004 Jonathan Worthington This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself. Math-Calculus-Differentiate-0.3/Makefile.PL0100644000176200010010000000103710160670204017750 0ustar JonathanNoneuse ExtUtils::MakeMaker; # See lib/ExtUtils/MakeMaker.pm for details of how to influence # the contents of the Makefile that is written. WriteMakefile( 'NAME' => 'Math::Calculus::Differentiate', 'VERSION_FROM' => 'Differentiate.pm', # finds $VERSION 'PREREQ_PM' => {Math::Calculus::Expression => 0.1}, ($] >= 5.005 ? ## Add these new keywords supported since 5.005 (ABSTRACT_FROM => 'Differentiate.pm', # retrieve abstract from module AUTHOR => 'J. Worthington ') : ()), );