mpoly/0000755000176200001440000000000013623427170011417 5ustar liggesusersmpoly/NAMESPACE0000644000176200001440000000557013623305753012647 0ustar liggesusers# Generated by roxygen2: do not edit by hand S3method("!=",mpoly) S3method("*",mpoly) S3method("*",mpolyList) S3method("+",mpoly) S3method("+",mpolyList) S3method("-",mpoly) S3method("-",mpolyList) S3method("==",mpoly) S3method("[",mpoly) S3method("^",mpoly) S3method("^",mpolyList) S3method(as.function,bezier) S3method(as.function,mpoly) S3method(as.function,mpolyList) S3method(as.mpoly,default) S3method(as.mpoly,lm) S3method(as.mpoly,numeric) S3method(as.mpoly,polynomial) S3method(deriv,mpoly) S3method(print,mpoly) S3method(print,mpolyList) S3method(reorder,mpoly) S3method(round,mpoly) S3method(terms,mpoly) export(LC) export(LCM) export(LM) export(LT) export(as.mpoly) export(bernstein) export(bernsteinApprox) export(bernstein_approx) export(bezier) export(bezierFunction) export(bezier_function) export(burst) export(chebyshev) export(chebyshev_roots) export(dehomogenize) export(eq_mp) export(exponents) export(gradient) export(hermite) export(homogeneous_components) export(homogenize) export(insert) export(is.bernstein) export(is.bezier) export(is.chebyshev) export(is.constant) export(is.homogeneous) export(is.linear) export(is.mpoly) export(is.mpolyList) export(is.unipoly) export(is.wholenumber) export(jacobi) export(laguerre) export(legendre) export(lissajous) export(make_indeterminate_list) export(monomials) export(mp) export(mpoly) export(mpolyList) export(multideg) export(partitions) export(permutations) export(plug) export(solve_unipoly) export(swap) export(totaldeg) export(tuples) export(vars) importFrom(ggplot2,aes) importFrom(ggplot2,ggplot) importFrom(orthopolynom,chebyshev.c.polynomials) importFrom(orthopolynom,chebyshev.s.polynomials) importFrom(orthopolynom,chebyshev.t.polynomials) importFrom(orthopolynom,chebyshev.u.polynomials) importFrom(orthopolynom,glaguerre.polynomials) importFrom(orthopolynom,hermite.h.polynomials) importFrom(orthopolynom,hermite.he.polynomials) importFrom(orthopolynom,jacobi.g.polynomials) importFrom(orthopolynom,jacobi.p.polynomials) importFrom(orthopolynom,legendre.polynomials) importFrom(partitions,parts) importFrom(plyr,ddply) importFrom(polynom,polynomial) importFrom(stats,coef) importFrom(stats,deriv) importFrom(stats,reorder) importFrom(stringi,stri_c) importFrom(stringi,stri_length) importFrom(stringi,stri_replace_all_fixed) importFrom(stringi,stri_replace_all_regex) importFrom(stringi,stri_sub) importFrom(stringr,"str_sub<-") importFrom(stringr,fixed) importFrom(stringr,str_c) importFrom(stringr,str_count) importFrom(stringr,str_detect) importFrom(stringr,str_dup) importFrom(stringr,str_extract) importFrom(stringr,str_extract_all) importFrom(stringr,str_locate) importFrom(stringr,str_locate_all) importFrom(stringr,str_remove) importFrom(stringr,str_remove_all) importFrom(stringr,str_replace) importFrom(stringr,str_replace_all) importFrom(stringr,str_split) importFrom(stringr,str_sub) importFrom(stringr,str_trim) importFrom(tidyr,gather) mpoly/tools/0000755000176200001440000000000013522636150012555 5ustar liggesusersmpoly/tools/README-bernstein-approx-1.png0000644000176200001440000026306513522636150017670 0ustar liggesusersPNG  IHDR:"iCCPkCGColorSpaceGenericRGB8U]hU>sg#$Sl4t? 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Polynomials are defined with the `mp()` function: ``` r library("mpoly") # Registered S3 methods overwritten by 'ggplot2': # method from # [.quosures rlang # c.quosures rlang # print.quosures rlang mp("x + y") # x + y mp("(x + 4 y)^2 (x - .25)") # x^3 - 0.25 x^2 + 8 x^2 y - 2 x y + 16 x y^2 - 4 y^2 ``` [Term orders](http://en.wikipedia.org/wiki/Lexicographical_order#Monomials) are available with the reorder function: ``` r (p <- mp("(x + y)^2 (1 + x)")) # x^3 + x^2 + 2 x^2 y + 2 x y + x y^2 + y^2 reorder(p, varorder = c("y","x"), order = "lex") # y^2 x + y^2 + 2 y x^2 + 2 y x + x^3 + x^2 reorder(p, varorder = c("x","y"), order = "glex") # x^3 + 2 x^2 y + x y^2 + x^2 + 2 x y + y^2 ``` Vectors of polynomials (`mpolyList`’s) can be specified in the same way: ``` r mp(c("(x+y)^2", "z")) # x^2 + 2 x y + y^2 # z ``` Polynomial parts ---------------- You can extract pieces of polynoimals using the standard `[` operator, which works on its terms: ``` r p[1] # x^3 p[1:3] # x^3 + x^2 + 2 x^2 y p[-1] # x^2 + 2 x^2 y + 2 x y + x y^2 + y^2 ``` There are also many other functions that can be used to piece apart polynomials, for example the leading term (default lex order): ``` r LT(p) # x^3 LC(p) # [1] 1 LM(p) # x^3 ``` You can also extract information about exponents: ``` r exponents(p) # [[1]] # x y # 3 0 # # [[2]] # x y # 2 0 # # [[3]] # x y # 2 1 # # [[4]] # x y # 1 1 # # [[5]] # x y # 1 2 # # [[6]] # x y # 0 2 multideg(p) # x y # 3 0 totaldeg(p) # [1] 3 monomials(p) # x^3 # x^2 # 2 x^2 y # 2 x y # x y^2 # y^2 ``` Polynomial arithmetic --------------------- Arithmetic is defined for both polynomials (`+`, `-`, `*` and `^`)… ``` r p1 <- mp("x + y") p2 <- mp("x - y") p1 + p2 # 2 x p1 - p2 # 2 y p1 * p2 # x^2 - y^2 p1^2 # x^2 + 2 x y + y^2 ``` … and vectors of polynomials: ``` r (ps1 <- mp(c("x", "y"))) # x # y (ps2 <- mp(c("2 x", "y + z"))) # 2 x # y + z ps1 + ps2 # 3 x # 2 y + z ps1 - ps2 # -1 x # -1 z ps1 * ps2 # 2 x^2 # y^2 + y z ``` Some calculus ------------- You can compute derivatives easily: ``` r p <- mp("x + x y + x y^2") deriv(p, "y") # x + 2 x y gradient(p) # y^2 + y + 1 # 2 y x + x ``` Function coercion ----------------- You can turn polynomials and vectors of polynomials into functions you can evaluate with `as.function()`. Here’s a basic example using a single multivariate polynomial: ``` r f <- as.function(mp("x + 2 y")) # makes a function with a vector argument # f(.) with . = (x, y) f(c(1,1)) # [1] 3 f <- as.function(mp("x + 2 y"), vector = FALSE) # makes a function with all arguments # f(x, y) f(1, 1) # [1] 3 ``` Here’s a basic example with a vector of multivariate polynomials: ``` r (p <- mp(c("x", "2 y"))) # x # 2 y f <- as.function(p) # f(.) with . = (x, y) f(c(1,1)) # [1] 1 2 f <- as.function(p, vector = FALSE) # f(x, y) f(1, 1) # [1] 1 2 ``` Whether you’re working with a single multivariate polynomial or a vector of them (`mpolyList`), if it/they are actually univariate polynomial(s), the resulting function is vectorized. Here’s an example with a single univariate polynomial. ``` r f <- as.function(mp("x^2")) # f(.) with . = x f(1:3) # [1] 1 4 9 (mat <- matrix(1:4, 2)) # [,1] [,2] # [1,] 1 3 # [2,] 2 4 f(mat) # it's vectorized properly over arrays # [,1] [,2] # [1,] 1 9 # [2,] 4 16 ``` Here’s an example with a vector of univariate polynomials: ``` r (p <- mp(c("t", "t^2"))) # t # t^2 f <- as.function(p) f(1) # [1] 1 1 f(1:3) # [,1] [,2] # [1,] 1 1 # [2,] 2 4 # [3,] 3 9 ``` You can use this to visualize a univariate polynomials like this: ``` r library("tidyverse"); theme_set(theme_classic()) ``` ``` r f <- as.function(mp("(x-2) x (x+2)")) # f(.) with . = x x <- seq(-2.5, 2.5, .1) qplot(x, f(x), geom = "line") ``` ![](tools/README-as-function-1.png) For multivariate polynomials, it’s a little more complicated: ``` r f <- as.function(mp("x^2 - y^2")) # f(.) with . = (x, y) s <- seq(-2.5, 2.5, .1) df <- expand.grid(x = s, y = s) df$f <- apply(df, 1, f) qplot(x, y, data = df, geom = "raster", fill = f) ``` ![](tools/README-as-function-multi-1.png) Using [tidyverse-style coding](https://www.tidyverse.org) (install tidyverse packages with `install.packages("tidyverse")`), this looks a bit cleaner: ``` r f <- as.function(mp("x^2 - y^2"), vector = FALSE) # f(x, y) seq(-2.5, 2.5, .1) %>% list("x" = ., "y" = .) %>% cross_df() %>% mutate(f = f(x, y)) %>% ggplot(aes(x, y, fill = f)) + geom_raster() ``` ![](tools/README-as-function-multi-tidy-1.png) Algebraic geometry ------------------ **Grobner bases are no longer implemented in mpoly; they’re now in [m2r](https://github.com/musicman3320/m2r).** ``` r # polys <- mp(c("t^4 - x", "t^3 - y", "t^2 - z")) # grobner(polys) ``` Homogenization and dehomogenization: ``` r (p <- mp("x + 2 x y + y - z^3")) # x + 2 x y + y - z^3 (hp <- homogenize(p)) # x t^2 + 2 x y t + y t^2 - z^3 dehomogenize(hp, "t") # x + 2 x y + y - z^3 homogeneous_components(p) # x + y # 2 x y # -1 z^3 ``` Special polynomials ------------------- **mpoly** can make use of many pieces of the **polynom** and **orthopolynom** packages with `as.mpoly()` methods. In particular, many special polynomials are available. #### [Chebyshev polynomials](http://en.wikipedia.org/wiki/Chebyshev_polynomials) You can construct [Chebyshev polynomials](http://en.wikipedia.org/wiki/Chebyshev_polynomials) as follows: ``` r chebyshev(1) # Loading required package: polynom # # Attaching package: 'polynom' # The following object is masked from 'package:mpoly': # # LCM # x chebyshev(2) # -1 + 2 x^2 chebyshev(0:5) # 1 # x # 2 x^2 - 1 # 4 x^3 - 3 x # 8 x^4 - 8 x^2 + 1 # 16 x^5 - 20 x^3 + 5 x ``` And you can visualize them: ``` r s <- seq(-1, 1, length.out = 201); N <- 5 (chebPolys <- chebyshev(0:N)) # 1 # x # 2 x^2 - 1 # 4 x^3 - 3 x # 8 x^4 - 8 x^2 + 1 # 16 x^5 - 20 x^3 + 5 x df <- as.function(chebPolys)(s) %>% cbind(s, .) %>% as.data.frame() names(df) <- c("x", paste0("T_", 0:N)) mdf <- df %>% gather(degree, value, -x) qplot(x, value, data = mdf, geom = "path", color = degree) ``` ![](tools/README-chebyshev-1.png) #### [Jacobi polynomials](http://en.wikipedia.org/wiki/Jacobi_polynomials) ``` r s <- seq(-1, 1, length.out = 201); N <- 5 (jacPolys <- jacobi(0:N, 2, 2)) # 1 # 5 x # 17.5 x^2 - 2.5 # 52.5 x^3 - 17.5 x # 144.375 x^4 - 78.75 x^2 + 4.375 # 375.375 x^5 - 288.75 x^3 + 39.375 x df <- as.function(jacPolys)(s) %>% cbind(s, .) %>% as.data.frame names(df) <- c("x", paste0("P_", 0:N)) mdf <- df %>% gather(degree, value, -x) qplot(x, value, data = mdf, geom = "path", color = degree) + coord_cartesian(ylim = c(-25, 25)) ``` ![](tools/README-jacobi-1.png) #### [Legendre polynomials](http://en.wikipedia.org/wiki/Legendre_polynomials) ``` r s <- seq(-1, 1, length.out = 201); N <- 5 (legPolys <- legendre(0:N)) # 1 # x # 1.5 x^2 - 0.5 # 2.5 x^3 - 1.5 x # 4.375 x^4 - 3.75 x^2 + 0.375 # 7.875 x^5 - 8.75 x^3 + 1.875 x df <- as.function(legPolys)(s) %>% cbind(s, .) %>% as.data.frame names(df) <- c("x", paste0("P_", 0:N)) mdf <- df %>% gather(degree, value, -x) qplot(x, value, data = mdf, geom = "path", color = degree) ``` ![](tools/README-legendre-1.png) #### [Hermite polynomials](http://en.wikipedia.org/wiki/Hermite_polynomials) ``` r s <- seq(-3, 3, length.out = 201); N <- 5 (hermPolys <- hermite(0:N)) # 1 # x # x^2 - 1 # x^3 - 3 x # x^4 - 6 x^2 + 3 # x^5 - 10 x^3 + 15 x df <- as.function(hermPolys)(s) %>% cbind(s, .) %>% as.data.frame names(df) <- c("x", paste0("He_", 0:N)) mdf <- df %>% gather(degree, value, -x) qplot(x, value, data = mdf, geom = "path", color = degree) ``` ![](tools/README-hermite-1.png) #### [(Generalized) Laguerre polynomials](http://en.wikipedia.org/wiki/Laguerre_polynomials) ``` r s <- seq(-5, 20, length.out = 201); N <- 5 (lagPolys <- laguerre(0:N)) # 1 # -1 x + 1 # 0.5 x^2 - 2 x + 1 # -0.1666667 x^3 + 1.5 x^2 - 3 x + 1 # 0.04166667 x^4 - 0.6666667 x^3 + 3 x^2 - 4 x + 1 # -0.008333333 x^5 + 0.2083333 x^4 - 1.666667 x^3 + 5 x^2 - 5 x + 1 df <- as.function(lagPolys)(s) %>% cbind(s, .) %>% as.data.frame names(df) <- c("x", paste0("L_", 0:N)) mdf <- df %>% gather(degree, value, -x) qplot(x, value, data = mdf, geom = "path", color = degree) + coord_cartesian(ylim = c(-25, 25)) ``` ![](tools/README-laguerre-1.png) #### [Bernstein polynomials](http://en.wikipedia.org/wiki/Bernstein_polynomial) [Bernstein polynomials](http://en.wikipedia.org/wiki/Bernstein_polynomial) are not in **polynom** or **orthopolynom** but are available in **mpoly** with `bernstein()`: ``` r bernstein(0:4, 4) # x^4 - 4 x^3 + 6 x^2 - 4 x + 1 # -4 x^4 + 12 x^3 - 12 x^2 + 4 x # 6 x^4 - 12 x^3 + 6 x^2 # -4 x^4 + 4 x^3 # x^4 s <- seq(0, 1, length.out = 101) N <- 5 # number of bernstein polynomials to plot (bernPolys <- bernstein(0:N, N)) # -1 x^5 + 5 x^4 - 10 x^3 + 10 x^2 - 5 x + 1 # 5 x^5 - 20 x^4 + 30 x^3 - 20 x^2 + 5 x # -10 x^5 + 30 x^4 - 30 x^3 + 10 x^2 # 10 x^5 - 20 x^4 + 10 x^3 # -5 x^5 + 5 x^4 # x^5 df <- as.function(bernPolys)(s) %>% cbind(s, .) %>% as.data.frame names(df) <- c("x", paste0("B_", 0:N)) mdf <- df %>% gather(degree, value, -x) qplot(x, value, data = mdf, geom = "path", color = degree) ``` ![](tools/README-bernstein-1.png) You can use the `bernstein_approx()` function to compute the Bernstein polynomial approximation to a function. Here’s an approximation to the standard normal density: ``` r p <- bernstein_approx(dnorm, 15, -1.25, 1.25) round(p, 4) # -0.1624 x^2 + 0.0262 x^4 - 0.002 x^6 + 0.0001 x^8 + 0.3796 x <- seq(-3, 3, length.out = 101) df <- data.frame( x = rep(x, 2), y = c(dnorm(x), as.function(p)(x)), which = rep(c("actual", "approx"), each = 101) ) # f(.) with . = x qplot(x, y, data = df, geom = "path", color = which) ``` ![](tools/README-bernstein-approx-1.png) [Bezier polynomials and curves](http://en.wikipedia.org/wiki/Bézier_curve) -------------------------------------------------------------------------- You can construct [Bezier polynomials](http://en.wikipedia.org/wiki/Bézier_curve) for a given collection of points with `bezier()`: ``` r points <- data.frame(x = c(-1,-2,2,1), y = c(0,1,1,0)) (bezPolys <- bezier(points)) # -10 t^3 + 15 t^2 - 3 t - 1 # -3 t^2 + 3 t ``` And viewing them is just as easy: ``` r df <- as.function(bezPolys)(s) %>% as.data.frame ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red", size = 4) + geom_path(data = points, color = "red", linetype = 2) + geom_path(size = 2) ``` ![](tools/README-bezier-plot-1.png) Weighting is available also: ``` r points <- data.frame(x = c(1,-2,2,-1), y = c(0,1,1,0)) (bezPolys <- bezier(points)) # -14 t^3 + 21 t^2 - 9 t + 1 # -3 t^2 + 3 t df <- as.function(bezPolys, weights = c(1,5,5,1))(s) %>% as.data.frame ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red", size = 4) + geom_path(data = points, color = "red", linetype = 2) + geom_path(size = 2) ``` ![](tools/README-bezier-weighting-1.png) To make the evaluation of the Bezier polynomials stable, `as.function()` has a special method for Bezier polynomials that makes use of [de Casteljau’s algorithm](http://en.wikipedia.org/wiki/De_Casteljau%27s_algorithm). This allows `bezier()` to be used as a smoother: ``` r s <- seq(0, 1, length.out = 201) df <- as.function(bezier(cars))(s) %>% as.data.frame qplot(speed, dist, data = cars) + geom_path(data = df, color = "red") ``` ![](tools/README-bezier-smooth-1.png) Other stuff ----------- I’m starting to put in methods for some other R functions: ``` r set.seed(1) n <- 101 df <- data.frame(x = seq(-5, 5, length.out = n)) df$y <- with(df, -x^2 + 2*x - 3 + rnorm(n, 0, 2)) mod <- lm(y ~ x + I(x^2), data = df) (p <- mod %>% as.mpoly %>% round) # 1.983 x - 1.01 x^2 - 2.709 qplot(x, y, data = df) + stat_function(fun = as.function(p), colour = 'red') # f(.) with . = x ``` ![](tools/README-lm-1.png) ``` r s <- seq(-5, 5, length.out = n) df <- expand.grid(x = s, y = s) %>% mutate(z = x^2 - y^2 + 3*x*y + rnorm(n^2, 0, 3)) (mod <- lm(z ~ poly(x, y, degree = 2, raw = TRUE), data = df)) # # Call: # lm(formula = z ~ poly(x, y, degree = 2, raw = TRUE), data = df) # # Coefficients: # (Intercept) # -0.070512 # poly(x, y, degree = 2, raw = TRUE)1.0 # -0.004841 # poly(x, y, degree = 2, raw = TRUE)2.0 # 1.005307 # poly(x, y, degree = 2, raw = TRUE)0.1 # 0.001334 # poly(x, y, degree = 2, raw = TRUE)1.1 # 3.003755 # poly(x, y, degree = 2, raw = TRUE)0.2 # -0.999536 as.mpoly(mod) # -0.004840798 x + 1.005307 x^2 + 0.001334122 y + 3.003755 x y - 0.9995356 y^2 - 0.07051218 ``` Installation ------------ - From CRAN: `install.packages("mpoly")` - From Github (dev version): ``` r # install.packages("devtools") devtools::install_github("dkahle/mpoly") ``` Acknowledgements ---------------- This material is based upon work partially supported by the National Science Foundation under Grant No. [1622449](https://nsf.gov/awardsearch/showAward?AWD_ID=1622449). mpoly/man/0000755000176200001440000000000013522636150012170 5ustar liggesusersmpoly/man/gradient.Rd0000644000176200001440000000446613623301001014250 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/deriv.mpoly.R \name{gradient} \alias{gradient} \title{Compute gradient of a multivariate polynomial.} \usage{ gradient(mpoly) } \arguments{ \item{mpoly}{an object of class mpoly} } \value{ An object of class mpoly or mpolyList. } \description{ This is a wrapper for deriv.mpoly. } \examples{ m <- mp("x y + y z + z^2") gradient(m) # gradient descent illustration using the symbolically # computed gradient of the rosenbrock function (shifted) rosenbrock <- mp("(1 - x)^2 + 100 (y - x^2)^2") fn <- as.function(rosenbrock) (rosenbrock_gradient <- gradient(rosenbrock)) gr <- as.function(rosenbrock_gradient) # visualize the function library(ggplot2) s <- seq(-5, 5, .05) df <- expand.grid(x = s, y = s) df$z <- apply(df, 1, fn) ggplot(df, aes(x = x, y = y)) + geom_raster(aes(fill = z + 1e-10)) + scale_fill_continuous(trans = "log10") # run the gradient descent algorithm using line-search # step sizes computed with optimize() current <- steps <- c(-3,-4) change <- 1 tol <- 1e-5 while(change > tol){ last <- current delta <- optimize( function(delta) fn(current - delta*gr(current)), interval = c(1e-10, .1) )$minimum current <- current - delta*gr(current) steps <- unname(rbind(steps, current)) change <- abs(fn(current) - fn(last)) } steps <- as.data.frame(steps) names(steps) <- c("x", "y") # visualize steps, note the optim at c(1,1) # the routine took 5748 steps ggplot(df, aes(x = x, y = y)) + geom_raster(aes(fill = z + 1e-10)) + geom_path(data = steps, color = "red") + geom_point(data = steps, color = "red", size = .5) + scale_fill_continuous(trans = "log10") # it gets to the general region of space quickly # but once it gets on the right arc, it's terrible # here's what the end game looks like last_steps <- tail(steps, 100) rngx <- range(last_steps$x); sx <- seq(rngx[1], rngx[2], length.out = 201) rngy <- range(last_steps$y); sy <- seq(rngy[1], rngy[2], length.out = 201) df <- expand.grid(x = sx, y = sy) df$z <- apply(df, 1, fn) ggplot(df, aes(x = x, y = y)) + geom_raster(aes(fill = z)) + geom_path(data = last_steps, color = "red", size = .25) + geom_point(data = last_steps, color = "red", size = 1) + scale_fill_continuous(trans = "log10") } \seealso{ \code{\link[=deriv.mpoly]{deriv.mpoly()}} } mpoly/man/homogenize.Rd0000644000176200001440000000230713476534026014634 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/homogenize.R \name{homogenize} \alias{homogenize} \alias{dehomogenize} \alias{is.homogeneous} \alias{homogeneous_components} \title{Homogenize a polynomial} \usage{ homogenize(x, indeterminate = "t") dehomogenize(x, indeterminate = "t") is.homogeneous(x) homogeneous_components(x) } \arguments{ \item{x}{an mpoly object, see \code{\link[=mpoly]{mpoly()}}} \item{indeterminate}{name of homogenization} } \value{ a (de/homogenized) mpoly or an mpolyList } \description{ Homogenize a polynomial. } \examples{ x <- mp("x^4 + y + 2 x y^2 - 3 z") is.homogeneous(x) (xh <- homogenize(x)) is.homogeneous(xh) homogeneous_components(x) homogenize(x, "o") xh <- homogenize(x) dehomogenize(xh) # assumes indeterminate = "t" plug(xh, "t", 1) # same effect, but dehomogenize is faster # the functions are vectorized (ps <- mp(c("x + y^2", "x + y^3"))) (psh <- homogenize(ps)) dehomogenize(psh) # demonstrating a leading property of homogeneous polynomials library(magrittr) p <- mp("x^2 + 2 x + 3") (ph <- homogenize(p, "y")) lambda <- 3 (d <- totaldeg(p)) ph \%>\% plug("x", lambda*mp("x")) \%>\% plug("y", lambda*mp("y")) lambda^d * ph } mpoly/man/mpoly.Rd0000644000176200001440000000232513577046366013637 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mpoly-package.R, R/mpoly.R \docType{package} \name{mpoly} \alias{mpoly} \alias{package-mpoly} \title{Multivariate polynomials in R.} \usage{ mpoly(list, varorder) } \arguments{ \item{list}{a list from which to construct an mpoly object} \item{varorder}{(optional) a character vector setting the intrinsic variable order of the polynomial} } \value{ Object of class mpoly. } \description{ A package for symbolic computation and more with multivariate polynomials mpoly is the most basic function used to create objects of class mpoly. However, it is not a general purpose function; for that see \code{\link[=mp]{mp()}}. } \examples{ list <- list( c(x = 1, coef = 1, y = 0), c(x = 0, y = 1, coef = 2), c(y = 1, coef = -6), c(z = 1, coef = -3, x = 2), c(x = 1, coef = 0, x = 3), c(t = 1, coef = 4, t = 2, y = 4), c(x = 1), c(x = 1), c(coef = 5), c(coef = 5), c(coef = -5) ) mpoly(list) # 3 x - 4 y - 3 x^2 z + 4 y^4 t^3 + 5 mpoly(list, varorder = c("y", "z", "t", "x")) list <- list( c(x = 5, x = 2, coef = 5, coef = 6, y = 0) ) mpoly(list) } \seealso{ \code{\link[=mp]{mp()}} } \author{ David Kahle \email{david@kahle.io} } mpoly/man/mpoly-defunct.Rd0000644000176200001440000000050413476534026015253 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mpoly-defunct.R \name{mpoly-defunct} \alias{mpoly-defunct} \title{Defunct mpoly functions} \description{ This is a list of past functions of the mpoly package. } \details{ The following are defunct mpoly functions: \itemize{ \item grobner } } mpoly/man/terms.mpoly.Rd0000644000176200001440000000101713103463706014747 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/terms.mpoly.R \name{terms.mpoly} \alias{terms.mpoly} \title{Extract the terms of a multivariate polynomial.} \usage{ \method{terms}{mpoly}(x, ...) } \arguments{ \item{x}{an object of class mpoly} \item{...}{additional parameters} } \value{ An object of class mpolyList. } \description{ Compute the terms of an mpoly object as a mpolyList. } \examples{ \dontrun{ .Deprecated issues a warning x <- mp("x^2 - y + x y z") terms(x) monomials(x) } } mpoly/man/bezier_function.Rd0000644000176200001440000000365013476534026015657 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/bezier.R \name{bezier_function} \alias{bezier_function} \alias{bezierFunction} \title{Bezier function} \usage{ bezier_function(points, weights = rep(1L, nrow(points))) bezierFunction(...) } \arguments{ \item{points}{a matrix or data frame of numerics. the rows represent points.} \item{weights}{the weights in a weighted Bezier curve} \item{...}{...; used internally} } \value{ function of a single parameter } \description{ Compute the Bezier function of a collection of polynomials. By Bezier function we mean the Bezier curve function, a parametric map running from t = 0, the first point, to t = 1, the last point, where the coordinate mappings are linear combinations of Bernstein polynomials. } \details{ The function returned is vectorized and evaluates the Bezier curve in a numerically stable way with de Castlejau's algorithm (implemented in R). } \examples{ library(ggplot2); theme_set(theme_bw()) t <- seq(0, 1, length.out = 201) points <- data.frame(x = 0:3, y = c(0,1,-1,0)) f <- bezier_function(points) df <- as.data.frame(f(t)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red", size = 8) + geom_path(data = points, color = "red") + geom_path() f <- bezier_function(points, weights = c(1,5,5,1)) df <- as.data.frame(f(t)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red", size = 8) + geom_path(data = points, color = "red") + geom_path() f <- bezier_function(points, weights = c(1,10,10,1)) df <- as.data.frame(f(t)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red", size = 8) + geom_path(data = points, color = "red") + geom_path() } \references{ \url{http://en.wikipedia.org/wiki/Bezier_curve}, \url{http://en.wikipedia.org/wiki/De_Casteljau's_algorithm} } \seealso{ \code{\link[=bezier]{bezier()}} } \author{ David Kahle } mpoly/man/as.function.mpolyList.Rd0000644000176200001440000000576713476536111016724 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/as.function.mpolyList.R, R/bezier.R \name{as.function.mpolyList} \alias{as.function.mpolyList} \alias{as.function.bezier} \title{Change a vector of multivariate polynomials into a function.} \usage{ \method{as.function}{mpolyList}(x, varorder = vars(x), vector = TRUE, silent = FALSE, ..., plus_pad = 1L, times_pad = 1L, squeeze = TRUE) \method{as.function}{bezier}(x, ...) } \arguments{ \item{x}{an object of class mpoly} \item{varorder}{the order of the variables} \item{vector}{whether the function should take a vector argument (TRUE) or a series of arguments (FALSE)} \item{silent}{logical; if TRUE, suppresses output} \item{...}{any additional arguments} \item{plus_pad}{number of spaces to the left and right of plus sign} \item{times_pad}{number of spaces to the left and right of times sign} \item{squeeze}{minify code in the created function} } \description{ Transforms an mpolyList object into a function which can be evaluated. } \examples{ # basic examples (mpolyList <- mp(c("2 x + 1", "x - z^2"))) (f <- as.function(mpolyList)) f(c(1,2)) # -> (2*1 + 1, 1-2^2) = 3 -3 f <- as.function(mpolyList, varorder = c("x","y","z")) f(c(1,0,2)) # -> 3 -3 f(c(1,4,2)) # -> 3 -3 f <- as.function(mpolyList, varorder = c("x","y","z"), vector = FALSE) f(1, 0, 2) # -> 3 -3 f(1, 4, 2) # -> 3 -3 # making a gradient function (useful for optim) mpoly <- mp("x + y^2 + y z") mpolyList <- gradient(mpoly) f <- as.function(mpolyList, varorder = vars(mpoly)) f(c(0,2,3)) # -> 1 7 2 # a univariate mpolyList creates a vectorized function ps <- mp(c("x", "x^2", "x^3")) f <- as.function(ps) f s <- seq(-1, 1, length.out = 11) f(s) # another example ps <- chebyshev(1:3) f <- as.function(ps) f(s) # the binomial pmf as an algebraic (polynomial) map # from [0,1] to [0,1]^size # p |-> {choose(size, x) p^x (1-p)^(size-x)}_{x = 0, ..., size} abinom <- function(size, indet = "p"){ chars4mp <- vapply(as.list(0:size), function(x){ sprintf("\%d \%s^\%d (1-\%s)^\%d", choose(size, x), indet, x, indet, size-x) }, character(1)) mp(chars4mp) } (ps <- abinom(2, "p")) # = mp(c("(1-p)^2", "2 p (1-p)", "p^2")) f <- as.function(ps) f(.5) # P[X = 0], P[X = 1], and P[X = 2] for X ~ Bin(2, .5) dbinom(0:2, 2, .5) f(.75) # P[X = 0], P[X = 1], and P[X = 2] for X ~ Bin(2, .75) dbinom(0:2, 2, .75) # as the degree gets larger, you'll need to be careful when evaluating # the polynomial. as.function() is not currently optimized for # stable numerical evaluation of polynomials; it evaluates them in # the naive way all.equal( as.function(abinom(10))(.5), dbinom(0:10, 10, .5) ) all.equal( as.function(abinom(30))(.5), dbinom(0:30, 20, .5) ) # the function produced is vectorized: number_of_probs <- 11 probs <- seq(0, 1, length.out = number_of_probs) (mat <- f(probs)) colnames(mat) <- sprintf("P[X = \%d]", 0:2) rownames(mat) <- sprintf("p = \%.2f", s) mat } \seealso{ \code{\link[=plug]{plug()}}, \code{\link[=as.function.mpolyList]{as.function.mpolyList()}} } mpoly/man/jacobi.Rd0000644000176200001440000000316413577046366013730 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/jacobi.R \name{jacobi} \alias{jacobi} \title{Jacobi polynomials} \usage{ jacobi( degree, alpha = 1, beta = 1, kind = "p", indeterminate = "x", normalized = FALSE ) } \arguments{ \item{degree}{degree of polynomial} \item{alpha}{the first parameter, also called p} \item{beta}{the second parameter, also called q} \item{kind}{"g" or "p"} \item{indeterminate}{indeterminate} \item{normalized}{provide normalized coefficients} } \value{ a mpoly object or mpolyList object } \description{ Jacobi polynomials as computed by orthopolynom. } \examples{ jacobi(0) jacobi(1) jacobi(2) jacobi(3) jacobi(4) jacobi(5) jacobi(6) jacobi(10, 2, 2, normalized = TRUE) jacobi(0:5) jacobi(0:5, normalized = TRUE) jacobi(0:5, kind = "g") jacobi(0:5, indeterminate = "t") # visualize the jacobi polynomials library(ggplot2); theme_set(theme_classic()) library(tidyr) s <- seq(-1, 1, length.out = 201) N <- 5 # number of jacobi polynomials to plot (jacPolys <- jacobi(0:N, 2, 2)) df <- data.frame(s, as.function(jacPolys)(s)) names(df) <- c("x", paste0("P_", 0:N)) mdf <- gather(df, degree, value, -x) qplot(x, value, data = mdf, geom = "line", color = degree) qplot(x, value, data = mdf, geom = "line", color = degree) + coord_cartesian(ylim = c(-30, 30)) } \seealso{ \code{\link[orthopolynom:jacobi.g.polynomials]{orthopolynom::jacobi.g.polynomials()}}, \code{\link[orthopolynom:jacobi.p.polynomials]{orthopolynom::jacobi.p.polynomials()}}, \url{http://en.wikipedia.org/wiki/Jacobi_polynomials} } \author{ David Kahle calling code from the orthopolynom package } mpoly/man/mp.Rd0000644000176200001440000000246013476534026013104 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mp.R \name{mp} \alias{mp} \alias{make_indeterminate_list} \title{Define a multivariate polynomial.} \usage{ make_indeterminate_list(vars) mp(string, varorder, stars_only = FALSE) } \arguments{ \item{vars}{a character vector of indeterminates} \item{string}{a character string containing a polynomial, see examples} \item{varorder}{(optional) order of variables in string} \item{stars_only}{if you format your multiplications using asterisks, setting this to \code{TRUE} will reduce preprocessing time} } \value{ An object of class mpoly. } \description{ mp is a smart function which attempts to create a formal mpoly object from a character string containing the usual representation of a multivariate polynomial. } \examples{ ( m <- mp("x + y + x y") ) is.mpoly( m ) unclass(m) mp("x + 2 y + x^2 y + x y z") mp("x + 2 y + x^2 y + x y z", varorder = c("y", "z", "x")) ( ms <- mp(c("x + y", "2 x")) ) is.mpolyList(ms) gradient( mp("x + 2 y + x^2 y + x y z") ) gradient( mp("(x + y)^10") ) # mp and the print methods are kinds of inverses of each other ( polys <- mp(c("x + y", "x - y")) ) strings <- print(polys, silent = TRUE) strings mp(strings) } \seealso{ \code{\link[=mpoly]{mpoly()}} } \author{ David Kahle \email{david@kahle.io} } mpoly/man/mpolyList.Rd0000644000176200001440000000122713103463706014455 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mpolyList.R \name{mpolyList} \alias{mpolyList} \title{Define a collection of multivariate polynomials.} \usage{ mpolyList(...) } \arguments{ \item{...}{a series of mpoly objects.} } \value{ An object of class mpolyList. } \description{ Combine a series of mpoly objects into a mpolyList. } \examples{ ( p1 <- mp("t^4 - x") ) ( p2 <- mp("t^3 - y") ) ( p3 <- mp("t^2 - z") ) ( ms <- mpolyList(p1, p2, p3) ) is.mpolyList( ms ) mpolyList(mp("x + 1")) p <- mp("x + 1") mpolyList(p) ps <- mp(c("x + 1", "y + 2")) is.mpolyList(ps) f <- function(){ a <- mp("1") mpolyList(a) } f() } mpoly/man/bernstein-approx.Rd0000644000176200001440000000314713476534026015773 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/bernstein_approx.R \name{bernstein-approx} \alias{bernstein-approx} \alias{bernstein_approx} \alias{bernsteinApprox} \title{Bernstein polynomial approximation} \usage{ bernstein_approx(f, n, lower = 0, upper = 1, indeterminate = "x") bernsteinApprox(...) } \arguments{ \item{f}{the function to approximate} \item{n}{Bernstein polynomial degree} \item{lower}{lower bound for approximation} \item{upper}{upper bound for approximation} \item{indeterminate}{indeterminate} \item{...}{...} } \value{ a mpoly object } \description{ Bernstein polynomial approximation } \examples{ \dontrun{ # visualize the bernstein polynomials library(ggplot2); theme_set(theme_bw()) library(reshape2) f <- function(x) sin(2*pi*x) p <- bernstein_approx(f, 20) round(p, 3) x <- seq(0, 1, length.out = 101) df <- data.frame( x = rep(x, 2), y = c(f(x), as.function(p)(x)), which = rep(c("actual", "approx"), each = 101) ) qplot(x, y, data = df, geom = "line", color = which) p <- bernstein_approx(sin, 20, pi/2, 1.5*pi) round(p, 4) x <- seq(0, 2*pi, length.out = 101) df <- data.frame( x = rep(x, 2), y = c(sin(x), as.function(p)(x)), which = rep(c("actual", "approx"), each = 101) ) qplot(x, y, data = df, geom = "line", color = which) p <- bernstein_approx(dnorm, 15, -1.25, 1.25) round(p, 4) x <- seq(-3, 3, length.out = 101) df <- data.frame( x = rep(x, 2), y = c(dnorm(x), as.function(p)(x)), which = rep(c("actual", "approx"), each = 101) ) qplot(x, y, data = df, geom = "line", color = which) } } \author{ David Kahle } mpoly/man/lissajous.Rd0000644000176200001440000000157713522636150014505 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/lissajous.R \name{lissajous} \alias{lissajous} \title{Lissajous polynomials} \usage{ lissajous(m, n, p, q, digits = 3) } \arguments{ \item{m, n, p, q}{Trigonometric coefficients, see examples for description} \item{digits}{The number of digits to round coefficients to, see \code{\link[=round.mpoly]{round.mpoly()}}. This is useful for cleaning terms that are numerically nonzero, but should be.} } \value{ a mpoly object } \description{ The Lissajous polynomials are the implicit (variety) descriptions of the image of the parametric map x = cos(m t + p), y = sin(n t + q). } \examples{ lissajous(3, 2, -pi/2, 0) lissajous(4, 3, -pi/2, 0) } \seealso{ \code{\link[=chebyshev]{chebyshev()}}, Merino, J. C (2003). Lissajous figures and Chebyshev polynomials. The College Mathematics Journal, 34(2), pp. 122-127. } mpoly/man/hermite.Rd0000644000176200001440000000361213527033477014126 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/hermite.R \name{hermite} \alias{hermite} \title{Hermite polynomials} \usage{ hermite(degree, kind = "he", indeterminate = "x", normalized = FALSE) } \arguments{ \item{degree}{degree of polynomial} \item{kind}{"he" (default, probabilists', see Wikipedia article) or "h" (physicists)} \item{indeterminate}{indeterminate} \item{normalized}{provide normalized coefficients} } \value{ a mpoly object or mpolyList object } \description{ Hermite polynomials as computed by orthopolynom. } \examples{ hermite(0) hermite(1) hermite(2) hermite(3) hermite(4) hermite(5) hermite(6) hermite(10) hermite(0:5) hermite(0:5, normalized = TRUE) hermite(0:5, indeterminate = "t") # visualize the hermite polynomials library(ggplot2); theme_set(theme_classic()) library(tidyr) s <- seq(-3, 3, length.out = 201) N <- 5 # number of hermite polynomials to plot (hermPolys <- hermite(0:N)) # see ?bernstein for a better understanding of # how the code below works df <- data.frame(s, as.function(hermPolys)(s)) names(df) <- c("x", paste0("T_", 0:N)) mdf <- gather(df, degree, value, -x) qplot(x, value, data = mdf, geom = "line", color = degree) # hermite polynomials are orthogonal with respect to the gaussian kernel: He2 <- as.function(hermite(2)) He3 <- as.function(hermite(3)) He4 <- as.function(hermite(4)) w <- dnorm integrate(function(x) He2(x) * He3(x) * w(x), lower = -Inf, upper = Inf) integrate(function(x) He2(x) * He4(x) * w(x), lower = -Inf, upper = Inf) integrate(function(x) He3(x) * He4(x) * w(x), lower = -Inf, upper = Inf) } \seealso{ \code{\link[orthopolynom:hermite.h.polynomials]{orthopolynom::hermite.h.polynomials()}}, \code{\link[orthopolynom:hermite.he.polynomials]{orthopolynom::hermite.he.polynomials()}}, \url{http://en.wikipedia.org/wiki/Hermite_polynomials} } \author{ David Kahle calling code from the orthopolynom package } mpoly/man/deriv.mpoly.Rd0000644000176200001440000000135513476534026014742 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/deriv.mpoly.R \name{deriv.mpoly} \alias{deriv.mpoly} \title{Compute partial derivatives of a multivariate polynomial.} \usage{ \method{deriv}{mpoly}(expr, var, ...) } \arguments{ \item{expr}{an object of class mpoly} \item{var}{character - the partial derivative desired} \item{...}{any additional arguments} } \value{ An object of class mpoly or mpolyList. } \description{ This is a deriv method for mpoly objects. It does not call the \code{\link[stats:deriv]{stats::deriv()}}. } \examples{ m <- mp("x y + y z + z^2") deriv(m, "x") deriv(m, "y") deriv(m, "z") deriv(m, c("x","y","z")) deriv(m, "a") is.mpoly(deriv(m, "x")) is.mpolyList( deriv(m, c("x","y","z")) ) } mpoly/man/partitions.Rd0000644000176200001440000000102113476534026014654 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/partitions.R \name{partitions} \alias{partitions} \title{Enumerate the partitions of an integer} \usage{ partitions(n) } \arguments{ \item{n}{an integer} } \value{ a matrix whose rows are the n-tuples } \description{ Determine all unrestricted partitions of an integer. This function is equivalent to the function parts in the partitions package. } \examples{ partitions(5) str(partitions(5)) } \author{ Robin K. S. Hankin, from package partitions } mpoly/man/round.mpoly.Rd0000644000176200001440000000206513476534026014757 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/round.mpoly.R \name{round.mpoly} \alias{round.mpoly} \title{Round the coefficients of a polynomial} \usage{ \method{round}{mpoly}(x, digits = 3) } \arguments{ \item{x}{an mpoly object} \item{digits}{number of digits to round to} } \value{ the rounded mpoly object } \description{ Round the coefficients of an mpoly object. } \examples{ p <- mp("x + 3.14159265")^4 p round(p) round(p, 0) \dontrun{ library(plyr) library(ggplot2) library(stringr) n <- 101 s <- seq(-5, 5, length.out = n) # one dimensional case df <- data.frame(x = s) df <- mutate(df, y = -x^2 + 2*x - 3 + rnorm(n, 0, 2)) qplot(x, y, data = df) mod <- lm(y ~ x + I(x^2), data = df) p <- as.mpoly(mod) qplot(x, y, data = df) + stat_function(fun = as.function(p), colour = 'red') p round(p, 1) qplot(x, y, data = df) + stat_function(fun = as.function(p), colour = 'red') + stat_function(fun = as.function(round(p,1)), colour = 'blue') } } \seealso{ \code{\link[=mp]{mp()}} } \author{ David Kahle \email{david@kahle.io} } mpoly/man/vars.Rd0000644000176200001440000000065413476534026013446 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/vars.R \name{vars} \alias{vars} \title{Determine the variables in a mpoly object.} \usage{ vars(p) } \arguments{ \item{p}{An mpoly or mpolyList object.} } \value{ A character vector of the variable names. } \description{ Determine the variables in a mpoly object. } \examples{ p <- mp("x + y^2") vars(p) p <- mp(c("x + y^2", "y - 2 x")) vars(p) } mpoly/man/is.wholenumber.Rd0000644000176200001440000000074313103463706015424 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/predicates.R \name{is.wholenumber} \alias{is.wholenumber} \title{Test whether an object is a whole number} \usage{ is.wholenumber(x, tol = .Machine$double.eps^0.5) } \arguments{ \item{x}{object to be tested} \item{tol}{tolerance within which a number is said to be whole} } \value{ Vector of logicals. } \description{ Test whether an object is a whole number. } \examples{ is.wholenumber(seq(-3,3,.5)) } mpoly/man/bezier.Rd0000644000176200001440000000614013476534026013747 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/bezier.R \name{bezier} \alias{bezier} \title{Bezier polynomials} \usage{ bezier(..., indeterminate = "t") } \arguments{ \item{...}{either a sequence of points or a matrix/data frame of points, see examples} \item{indeterminate}{the indeterminate of the resulting polynomial} } \value{ a mpoly object } \description{ Compute the Bezier polynomials of a given collection of points. Note that using \code{\link[mpoly:as.function.mpoly]{mpoly::as.function.mpoly()}} on the resulting Bezier polynomials is made numerically stable by taking advantage of de Casteljau's algorithm; it does not use the polynomial that is printed to the screen. See \code{\link[=bezier_function]{bezier_function()}} for details. } \examples{ p1 <- c(0, 0) p2 <- c(1, 1) p3 <- c(2, -1) p4 <- c(3, 0) bezier(p1, p2, p3, p4) points <- data.frame(x = 0:3, y = c(0,1,-1,0)) bezier(points) points <- data.frame(x = 0:2, y = c(0,1,0)) bezier(points) # visualize the bernstein polynomials library(ggplot2); theme_set(theme_bw()) s <- seq(0, 1, length.out = 101) ## example 1 points <- data.frame(x = 0:3, y = c(0,1,-1,0)) (bezPolys <- bezier(points)) f <- as.function(bezPolys) df <- as.data.frame(f(s)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red") + geom_path(data = points, color = "red") + geom_path() ## example 1 with weights f <- as.function(bezPolys, weights = c(1,5,5,1)) df <- as.data.frame(f(s)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red") + geom_path(data = points, color = "red") + geom_path() ## example 2 points <- data.frame(x = 0:2, y = c(0,1,0)) (bezPolys <- bezier(points)) f <- as.function(bezPolys) df <- as.data.frame(f(s)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red") + geom_path(data = points, color = "red") + geom_path() ## example 3 points <- data.frame(x = c(-1,-2,2,1), y = c(0,1,1,0)) (bezPolys <- bezier(points)) f <- as.function(bezPolys) df <- as.data.frame(f(s)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red") + geom_path(data = points, color = "red") + geom_path() ## example 4 points <- data.frame(x = c(-1,2,-2,1), y = c(0,1,1,0)) (bezPolys <- bezier(points)) f <- as.function(bezPolys) df <- as.data.frame(f(s)) ggplot(aes(x = x, y = y), data = df) + geom_point(data = points, color = "red") + geom_path(data = points, color = "red") + geom_path() ## example 5 qplot(speed, dist, data = cars) s <- seq(0, 1, length.out = 201) p <- bezier(cars) f <- as.function(p) df <- as.data.frame(f(s)) qplot(speed, dist, data = cars) + geom_path(data = df, color = "red") # the curve is not invariant to permutations of the points # but it always goes through the first and last points permute_rows <- function(df) df[sample(nrow(df)),] p <- bezier(permute_rows(cars)) f <- as.function(p) df <- as.data.frame(f(s)) qplot(speed, dist, data = cars) + geom_path(data = df, color = "red") } \seealso{ \code{\link[=bezier_function]{bezier_function()}} } \author{ David Kahle } mpoly/man/laguerre.Rd0000644000176200001440000000363613530645634014303 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/laguerre.R \name{laguerre} \alias{laguerre} \title{Generalized Laguerre polynomials} \usage{ laguerre(degree, alpha = 0, indeterminate = "x", normalized = FALSE) } \arguments{ \item{degree}{degree of polynomial} \item{alpha}{generalization constant} \item{indeterminate}{indeterminate} \item{normalized}{provide normalized coefficients} } \value{ a mpoly object or mpolyList object } \description{ Generalized Laguerre polynomials as computed by orthopolynom. } \examples{ laguerre(0) laguerre(1) laguerre(2) laguerre(3) laguerre(4) laguerre(5) laguerre(6) laguerre(2) laguerre(2, normalized = TRUE) laguerre(0:5) laguerre(0:5, normalized = TRUE) laguerre(0:5, indeterminate = "t") # visualize the laguerre polynomials library(ggplot2); theme_set(theme_classic()) library(tidyr) s <- seq(-5, 20, length.out = 201) N <- 5 # number of laguerre polynomials to plot (lagPolys <- laguerre(0:N)) # see ?bernstein for a better understanding of # how the code below works df <- data.frame(s, as.function(lagPolys)(s)) names(df) <- c("x", paste0("L_", 0:N)) mdf <- gather(df, degree, value, -x) qplot(x, value, data = mdf, geom = "line", color = degree) qplot(x, value, data = mdf, geom = "line", color = degree) + coord_cartesian(ylim = c(-25, 25)) # laguerre polynomials are orthogonal with respect to the exponential kernel: L2 <- as.function(laguerre(2)) L3 <- as.function(laguerre(3)) L4 <- as.function(laguerre(4)) w <- dexp integrate(function(x) L2(x) * L3(x) * w(x), lower = 0, upper = Inf) integrate(function(x) L2(x) * L4(x) * w(x), lower = 0, upper = Inf) integrate(function(x) L3(x) * L4(x) * w(x), lower = 0, upper = Inf) } \seealso{ \code{\link[orthopolynom:glaguerre.polynomials]{orthopolynom::glaguerre.polynomials()}}, \url{http://en.wikipedia.org/wiki/Laguerre_polynomials} } \author{ David Kahle calling code from the orthopolynom package } mpoly/man/bernstein.Rd0000644000176200001440000000207313476534026014461 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/bernstein.R \name{bernstein} \alias{bernstein} \title{Bernstein polynomials} \usage{ bernstein(k, n, indeterminate = "x") } \arguments{ \item{k}{Bernstein polynomial k} \item{n}{Bernstein polynomial degree} \item{indeterminate}{indeterminate} } \value{ a mpoly object } \description{ Bernstein polynomials } \examples{ bernstein(0, 0) bernstein(0, 1) bernstein(1, 1) bernstein(0, 1, "t") bernstein(0:2, 2) bernstein(0:3, 3) bernstein(0:3, 3, "t") bernstein(0:4, 4) bernstein(0:10, 10) bernstein(0:10, 10, "t") bernstein(0:20, 20, "t") \dontrun{ # visualize the bernstein polynomials library(ggplot2); theme_set(theme_classic()) library(tidyr) s <- seq(0, 1, length.out = 101) N <- 10 # number of bernstein polynomials to plot (bernPolys <- bernstein(0:N, N)) df <- data.frame(s, as.function(bernPolys)(s)) names(df) <- c("x", paste0("B_", 0:N)) head(df) mdf <- gather(df, degree, value, -x) head(mdf) qplot(x, value, data = mdf, geom = "line", color = degree) } } \author{ David Kahle } mpoly/man/plug.Rd0000644000176200001440000000123413522636150013426 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/plug.R \name{plug} \alias{plug} \title{Switch indeterminates in a polynomial} \usage{ plug(p, indeterminate, value) } \arguments{ \item{p}{a polynomial} \item{indeterminate}{the indeterminate in the polynomial to switch} \item{value}{the value/indeterminate to substitute} } \value{ an mpoly object } \description{ Switch indeterminates in a polynomial } \examples{ # on an mpoly (p <- mp("(x+y)^3")) plug(p, "x", 5) plug(p, "x", "t") plug(p, "x", "y") plug(p, "x", mp("2 y")) plug(p, "x", mp("x + y")) mp("((x+y)+y)^3") # on an mpolyList ps <- mp(c("x+y", "x+1")) plug(ps, "x", 1) } mpoly/man/reorder.mpoly.Rd0000644000176200001440000000271313476534026015272 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/reorder.mpoly.R \name{reorder.mpoly} \alias{reorder.mpoly} \title{Reorder a multivariate polynomial.} \usage{ \method{reorder}{mpoly}(x, varorder = vars(x), order, ...) } \arguments{ \item{x}{an object of class mpoly} \item{varorder}{the order of the variables} \item{order}{a total order used to order the terms, \code{"lex"}, \code{"glex"}, or \code{"grlex"}} \item{...}{additional arguments} } \value{ An object of class mpoly. } \description{ This function is used to set the intrinsic order of a multivariate polynomial. It is used for both the in-term variables and the terms. } \examples{ list <- list( c(x = 1, y = 2, z = 1, coef = 1), c(x = 2, y = 0, z = 2, coef = 1), c(x = 0, y = 0, z = 2, coef = 1), c(x = 3, y = 0, z = 0, coef = 1) ) (p <- mpoly(list)) # -> x y^2 z + x^2 z^2 + z^2 + x^3 reorder(p) # -> x y^2 z + x^2 z^2 + z^2 + x^3 reorder(p, varorder = c("x","y","z"), order = "lex") # -> x^3 + x^2 z^2 + x y^2 z + z^2 reorder(p, varorder = c("x","y","z"), order = "glex") # -> x^2 z^2 + x y^2 z + x^3 + z^2 reorder(p, varorder = c("x","y","z"), order = "grlex") # -> x y^2 z + x^2 z^2 + x^3 + z^2 reorder(mp("x + 1"), varorder = c("y","x","z"), order = "lex") reorder(mp("x + y"), varorder = c("y","x","z"), order = "lex") reorder(mp("x y + y + 2 x y z^2"), varorder = c("y","x","z")) reorder(mp("x^2 + y x + y"), order = "lex") } mpoly/man/insert.Rd0000644000176200001440000000072113103463706013763 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/permutations.R \name{insert} \alias{insert} \title{Insert an element into a vector.} \usage{ insert(elem, slot, v) } \arguments{ \item{elem}{element to insert} \item{slot}{location of insert} \item{v}{vector to insert into} } \value{ vector with element inserted } \description{ Insert an element into a vector. } \examples{ insert(2, 1, 1) insert(2, 2, 1) insert('x', 5, letters) } mpoly/man/mpoly-equal.Rd0000644000176200001440000000152613476534026014737 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mpoly-equal.R \name{mpoly-equal} \alias{mpoly-equal} \alias{==.mpoly} \alias{==} \alias{!=.mpoly} \title{Determine whether two multivariate polynomials are equal.} \usage{ \method{==}{mpoly}(e1, e2) \method{!=}{mpoly}(e1, e2) } \arguments{ \item{e1}{an object of class mpoly} \item{e2}{an object of class mpoly} } \value{ A logical value. } \description{ Determine whether two multivariate polynomials are equal. } \examples{ p1 <- mp("x + y + 2 z") p1 == p1 p2 <- reorder(p1, order = "lex", varorder = c("z","y","x")) p1 == p2 p2 <- reorder(p1, order = "lex", varorder = c("z","w","y","x")) p1 == p2 p1 == ( 2 * p2 ) p1 <- mp("x + 1") p2 <- mp("x + 1") identical(p1, p2) p1 == p2 mp("x + 1") == mp("y + 1") mp("2") == mp("1") mp("1") == mp("1") mp("0") == mp("-0") } mpoly/man/predicates.Rd0000644000176200001440000000201013111443532014565 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/predicates.R \name{predicates} \alias{predicates} \alias{is.constant} \alias{is.mpoly} \alias{is.unipoly} \alias{is.bernstein} \alias{is.bezier} \alias{is.chebyshev} \alias{is.mpolyList} \alias{is.linear} \title{mpoly predicate functions} \usage{ is.constant(x) is.mpoly(x) is.unipoly(x) is.bernstein(x) is.bezier(x) is.chebyshev(x) is.mpolyList(x) is.linear(x) } \arguments{ \item{x}{object to be tested} } \value{ Vector of logicals. } \description{ Various functions to deal with mpoly and mpolyList objects. } \examples{ p <- mp("5") is.mpoly(p) is.constant(p) is.constant(mp(c("x + 1", "7", "y - 2"))) p <- mp("x + y") is.mpoly(p) is.mpolyList(mp(c("x + 1", "y - 1"))) is.linear(mp("0")) is.linear(mp("x + 1")) is.linear(mp("x + y")) is.linear(mp(c("0", "x + y"))) is.linear(mp("x + x y")) is.linear(mp(c("x + x y", "x"))) (p <- bernstein(2, 5)) is.mpoly(p) is.bernstein(p) (p <- chebyshev(5)) is.mpoly(p) is.chebyshev(p) str(p) } mpoly/man/as.function.mpoly.Rd0000644000176200001440000000337113476536111016055 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/as.function.mpoly.R \name{as.function.mpoly} \alias{as.function.mpoly} \title{Change a multivariate polynomial into a function.} \usage{ \method{as.function}{mpoly}(x, varorder = vars(x), vector = TRUE, silent = FALSE, ..., plus_pad = 1L, times_pad = 1L, squeeze = TRUE) } \arguments{ \item{x}{an object of class mpoly} \item{varorder}{the order of the variables} \item{vector}{whether the function should take a vector argument (TRUE) or a series of arguments (FALSE)} \item{silent}{logical; if TRUE, suppresses output} \item{...}{any additional arguments} \item{plus_pad}{number of spaces to the left and right of plus sign} \item{times_pad}{number of spaces to the left and right of times sign} \item{squeeze}{minify code in the created function} } \description{ Transforms an mpoly object into a function which can be evaluated. } \examples{ p <- mp("(x - 1)^2") (f <- as.function(p)) f(1) f(seq(0, 2, .1)) p <- mp("x + 3 x y + z^2 x") (f <- as.function(p)) f(1:3) # -> 16 f(c(1,1,1)) # -> 5 f <- as.function(p, vector = FALSE) f(1, 2, 3) # -> 16 f(1, 1, 1) # -> 5 f <- as.function(p, varorder = c("z","y","x"), vector = FALSE) f(3, 2, 1) # -> 16 f(1, 1, 1) # -> 5 # for univariate mpolys, as.function() returns a vectorized function # that can even apply to arrays p <- mp("x^2") f <- as.function(p) f(1:10) (mat <- matrix(1:4, 2)) f(mat) p <- mp("1 2 3 4") f <- as.function(p) f(10) # -> 24 bernstein(1, 2) s <- seq(0, 1, .01) as.function(bernstein(1, 2))(s) plot( s, as.function(bernstein(1, 2))(s) ) as.function(mp("x + xx")) as.function(mp("x + xx"), squeeze = FALSE) } \seealso{ \code{\link[=plug]{plug()}}, \code{\link[=as.function.mpolyList]{as.function.mpolyList()}} } mpoly/man/tuples.Rd0000644000176200001440000000232613476534026014005 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/tuples.R \name{tuples} \alias{tuples} \title{Determine all n-tuples using the elements of a set.} \usage{ tuples(set, n = length(set), repeats = FALSE, list = FALSE) } \arguments{ \item{set}{a set} \item{n}{length of each tuple} \item{repeats}{if set contains duplicates, should the result?} \item{list}{tuples as list?} } \value{ a matrix whose rows are the n-tuples } \description{ Determine all n-tuples using the elements of a set. This is really just a simple wrapper for expand.grid, so it is not optimized. } \examples{ tuples(1:2, 3) tuples(1:2, 3, list = TRUE) apply(tuples(c("x","y","z"), 3), 1, paste, collapse = "") # multinomial coefficients r <- 2 # number of variables, e.g. x, y n <- 2 # power, e.g. (x+y)^2 apply(burst(n,r), 1, function(v) factorial(n)/ prod(factorial(v))) # x, y, xy mp("x + y")^n r <- 2 # number of variables, e.g. x, y n <- 3 # power, e.g. (x+y)^3 apply(burst(n,r), 1, function(v) factorial(n)/ prod(factorial(v))) mp("x + y")^n r <- 3 # number of variables, e.g. x, y, z n <- 2 # power, e.g. (x+y+z)^2 apply(burst(n,r), 1, function(v) factorial(n)/ prod(factorial(v))) # x, y, z, xy, xz, yz mp("x + y + z")^n } mpoly/man/components.Rd0000644000176200001440000000222713623301460014642 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/components.R \name{components} \alias{components} \alias{[.mpoly} \alias{LT} \alias{LC} \alias{LM} \alias{multideg} \alias{totaldeg} \alias{monomials} \alias{exponents} \title{Polynomial components} \usage{ \method{[}{mpoly}(x, ndx) LT(x, varorder = vars(x), order = "lex") LC(x, varorder = vars(x), order = "lex") LM(x, varorder = vars(x), order = "lex") multideg(x, varorder = vars(x), order = "lex") totaldeg(x) monomials(x) exponents(x, reduced = FALSE) } \arguments{ \item{x}{an object of class mpoly} \item{ndx}{a subsetting index} \item{varorder}{the order of the variables} \item{order}{a total order used to order the terms} \item{reduced}{if TRUE, don't include zero degrees} } \value{ An object of class mpoly or mpolyList, depending on the context } \description{ Compute quantities/expressions related to a multivariate polynomial. } \examples{ (p <- mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")) p[2] p[-2] p[2:3] LT(p) LC(p) LM(p) multideg(p) totaldeg(p) monomials(p) exponents(p) exponents(p, reduce = TRUE) lapply(exponents(p), is.integer) homogeneous_components(p) } mpoly/man/print.mpolyList.Rd0000644000176200001440000000274413577046366015633 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/print.mpolyList.R \name{print.mpolyList} \alias{print.mpolyList} \title{Pretty printing of a list of multivariate polynomials.} \usage{ \method{print}{mpolyList}( x, varorder = vars(x), stars = FALSE, order, silent = FALSE, ..., plus_pad = 2L, times_pad = 1L ) } \arguments{ \item{x}{an object of class mpoly} \item{varorder}{the order of the variables} \item{stars}{print the multivariate polynomial in the more computer-friendly asterisk notation (default FALSE)} \item{order}{a total order used to order the monomials in the printing} \item{silent}{logical; if TRUE, suppresses output} \item{...}{additional parameters to go to \code{\link[base:cat]{base::cat()}}} \item{plus_pad}{number of spaces to the left and right of plus sign} \item{times_pad}{number of spaces to the left and right of times sign} } \value{ Invisible character vector of the printed objects. } \description{ This function iterates print.mpoly on an object of class mpolyList. } \examples{ mL <- mp(c("x + 1", "y - 1", "x y^2 z + x^2 z^2 + z^2 + x^3")) mL print(mL, order = "lex") print(mL, order = "glex") print(mL, order = "grlex") print(mL, order = "glex", varorder = c("z","y","x")) print(mL, order = "grlex", varorder = c("z","y","x")) print(mL, varorder = c("z","y","x")) (print(mL, varorder = c("z","y","x"), plus_pad = 1L, silent = TRUE)) (print(mL, silent = TRUE, stars = TRUE, plus_pad = 1L, times_pad = 0L)) } mpoly/man/mpolyArithmetic.Rd0000644000176200001440000000131113476534026015634 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mpolyArithmetic.R, R/mpolyListArithmetic.R \name{mpolyArithmetic} \alias{mpolyArithmetic} \alias{+.mpoly} \alias{-.mpoly} \alias{*.mpoly} \alias{^.mpoly} \alias{^.mpolyList} \title{Arithmetic with multivariate polynomials} \usage{ \method{+}{mpoly}(e1, e2) \method{-}{mpoly}(e1, e2) \method{*}{mpoly}(e1, e2) \method{^}{mpoly}(e1, e2) \method{^}{mpolyList}(e1, e2) } \arguments{ \item{e1}{an object of class mpoly} \item{e2}{an object of class mpoly} } \value{ object of class mpoly } \description{ Arithmetic with multivariate polynomials } \examples{ p <- mp("x + y") p + p p - p p * p p^2 p^10 mp("(x+1)^10") p + 1 2*p } mpoly/man/burst.Rd0000644000176200001440000000126513476534026013631 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/burst.R \name{burst} \alias{burst} \title{Enumerate integer r-vectors summing to n} \usage{ burst(n, r = n) } \arguments{ \item{n}{integer to sum to} \item{r}{number of components} } \value{ a matrix whose rows are the n-tuples } \description{ Determine all r-vectors with nonnegative integer entries summing to n. Note that this is not intended to be optimized. } \examples{ burst(4) burst(4, 4) burst(4, 3) burst(4, 2) rowSums(burst(4)) rowSums(burst(4, 3)) rowSums(burst(4, 2)) burst(10, 4) # all possible 2x2 contingency tables with n=10 burst(10, 4) / 10 # all possible empirical relative frequencies } mpoly/man/print.mpoly.Rd0000644000176200001440000000304613476534026014764 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/print.mpoly.R \name{print.mpoly} \alias{print.mpoly} \title{Pretty printing of multivariate polynomials.} \usage{ \method{print}{mpoly}(x, varorder, order, stars = FALSE, silent = FALSE, ..., plus_pad = 2L, times_pad = 1L) } \arguments{ \item{x}{an object of class mpoly} \item{varorder}{the order of the variables} \item{order}{a total order used to order the monomials in the printing} \item{stars}{print the multivariate polynomial in the more computer-friendly asterisk notation (default FALSE)} \item{silent}{logical; if TRUE, suppresses output} \item{...}{additional parameters to go to \code{\link[base:cat]{base::cat()}}} \item{plus_pad}{number of spaces to the left and right of plus sign} \item{times_pad}{number of spaces to the left and right of times sign} } \value{ Invisible string of the printed object. } \description{ This is the major function used to view multivariate polynomials. } \examples{ mp("-x^5 - 3 y^2 + x y^3 - 1") (p <- mp("2 x^5 - 3 y^2 + x y^3")) print(p) # same print(p, silent = TRUE) s <- print(p, silent = TRUE) s print(p, order = "lex") # -> 2 x^5 + x y^3 - 3 y^2 print(p, order = "lex", varorder = c("y","x")) # -> y^3 x - 3 y^2 + 2 x^5 print(p, varorder = c("y","x")) # -> y^3 x - 3 y^2 + 2 x^5 # this is mostly used internally print(p, stars = TRUE) print(p, stars = TRUE, times_pad = 0L) print(p, stars = TRUE, times_pad = 0L, plus_pad = 1L) print(p, stars = TRUE, times_pad = 0L, plus_pad = 0L) print(p, plus_pad = 1L) } mpoly/man/solve_unipoly.Rd0000644000176200001440000000074213111443010015352 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/solve_unipoly.R \name{solve_unipoly} \alias{solve_unipoly} \title{Solve a univariate mpoly with polyroot} \usage{ solve_unipoly(mpoly, real_only = FALSE) } \arguments{ \item{mpoly}{an mpoly} \item{real_only}{return only real solutions?} } \description{ Solve a univariate mpoly with polyroot } \examples{ solve_unipoly(mp("x^2 - 1")) # check x = -1, 1 solve_unipoly(mp("x^2 - 1"), real_only = TRUE) } mpoly/man/LCM.Rd0000644000176200001440000000074713476534026013111 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/LCM.R \name{LCM} \alias{LCM} \title{Compute the least common multiple of two numbers.} \usage{ LCM(x, y) } \arguments{ \item{x}{an object of class numeric} \item{y}{an object of class numeric} } \value{ The least common multiple of x and y. } \description{ A simple algorithm to compute the least common multiple of two numbers } \examples{ LCM(5,7) LCM(5,8) LCM(5,9) LCM(5,10) Reduce(LCM, 1:10) # -> 2520 } mpoly/man/swap.Rd0000644000176200001440000000120313103463706013425 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/swap.R \name{swap} \alias{swap} \title{Swap polynomial indeterminates} \usage{ swap(p, variable, replacement) } \arguments{ \item{p}{polynomial} \item{variable}{the variable in the polynomial to replace} \item{replacement}{the replacement variable} } \value{ a mpoly object } \description{ Swap polynomial indeterminates } \examples{ (p <- mp("(x + y)^2")) swap(p, "x", "t") ## the meta data is retained (p <- bernstein(3, 5)) (p2 <- swap(p, "x", "t")) is.bernstein(p2) (p <- chebyshev(3)) (p2 <- swap(p, "x", "t")) is.chebyshev(p2) } \author{ David Kahle } mpoly/man/eq_mp.Rd0000644000176200001440000000121213476534026013563 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/eq_mp.R \name{eq_mp} \alias{eq_mp} \title{Convert an equation to a polynomial} \usage{ eq_mp(string, ...) } \arguments{ \item{string}{a character string containing a polynomial, see examples} \item{...}{arguments to pass to \code{\link[=mpoly]{mpoly()}}} } \value{ An object of class mpoly or mpolyList. } \description{ Convert characters of the form \code{"p1 = p2"} (or similar) to an mpoly object representing \code{p1 - p2}. } \examples{ eq_mp(c("y = x", "y == (x + 2)")) } \seealso{ \code{\link[=mpoly]{mpoly()}} } \author{ David Kahle \email{david@kahle.io} } mpoly/man/permutations.Rd0000644000176200001440000000103613476534026015220 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/permutations.R \name{permutations} \alias{permutations} \title{Determine all permutations of a set.} \usage{ permutations(set) } \arguments{ \item{set}{a set} } \value{ a matrix whose rows are the permutations of set } \description{ An implementation of the Steinhaus-Johnson-Trotter permutation algorithm. } \examples{ permutations(1:3) permutations(c('first','second','third')) permutations(c(1,1,3)) apply(permutations(letters[1:6]), 1, paste, collapse = '') } mpoly/man/as.mpoly.Rd0000644000176200001440000000326513476534026014236 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/as.mpoly.R \name{as.mpoly} \alias{as.mpoly} \title{Convert an object to an mpoly} \usage{ as.mpoly(x, ...) } \arguments{ \item{x}{an object} \item{...}{additional arguments to pass to methods} } \value{ the object formated as a mpoly object. } \description{ mpoly is the most basic function used to create objects of class mpoly. } \examples{ library(ggplot2); theme_set(theme_classic()) library(dplyr) n <- 101 s <- seq(-5, 5, length.out = n) # one dimensional case df <- data.frame(x = seq(-5, 5, length.out = n)) \%>\% mutate(y = -x^2 + 2*x - 3 + rnorm(n, 0, 2)) (mod <- lm(y ~ x + I(x^2), data = df)) (p <- as.mpoly(mod)) qplot(x, y, data = df) + stat_function(fun = as.function(p), colour = "red", size = 1) (mod <- lm(y ~ poly(x, 2, raw = TRUE), data = df)) (p <- as.mpoly(mod)) qplot(x, y, data = df) + stat_function(fun = as.function(p), colour = "red", size = 1) (mod <- lm(y ~ poly(x, 1, raw = TRUE), data = df)) (p <- as.mpoly(mod)) qplot(x, y, data = df) + stat_function(fun = as.function(p), colour = "red", size = 1) # two dimensional case with ggplot2 df <- expand.grid(x = s, y = s) \%>\% mutate(z = x^2 - y^2 + 3*x*y + rnorm(n^2, 0, 3)) qplot(x, y, data = df, geom = "raster", fill = z) (mod <- lm(z ~ x + y + I(x^2) + I(y^2) + I(x*y), data = df)) (mod <- lm(z ~ poly(x, y, degree = 2, raw = TRUE), data = df)) (p <- as.mpoly(mod)) df$fit <- apply(df[,c("x","y")], 1, as.function(p)) qplot(x, y, data = df, geom = "raster", fill = fit) qplot(x, y, data = df, geom = "raster", fill = z - fit) # residuals } \seealso{ \code{\link[=mp]{mp()}} } \author{ David Kahle \email{david@kahle.io} } mpoly/man/legendre.Rd0000644000176200001440000000455413527034720014254 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/legendre.R \name{legendre} \alias{legendre} \title{Legendre polynomials} \usage{ legendre(degree, indeterminate = "x", normalized = FALSE) } \arguments{ \item{degree}{degree of polynomial} \item{indeterminate}{indeterminate} \item{normalized}{provide normalized coefficients} } \value{ a mpoly object or mpolyList object } \description{ Legendre polynomials as computed by orthopolynom. } \examples{ legendre(0) legendre(1) legendre(2) legendre(3) legendre(4) legendre(5) legendre(6) legendre(2) legendre(2, normalized = TRUE) legendre(0:5) legendre(0:5, normalized = TRUE) legendre(0:5, indeterminate = "t") # visualize the legendre polynomials library(ggplot2); theme_set(theme_classic()) library(tidyr) s <- seq(-1, 1, length.out = 201) N <- 5 # number of legendre polynomials to plot (legPolys <- legendre(0:N)) # see ?bernstein for a better understanding of # how the code below works df <- data.frame(s, as.function(legPolys)(s)) names(df) <- c("x", paste0("P_", 0:N)) mdf <- gather(df, degree, value, -x) qplot(x, value, data = mdf, geom = "line", color = degree) # legendre polynomials and the QR decomposition n <- 201 x <- seq(-1, 1, length.out = n) mat <- cbind(1, x, x^2, x^3, x^4, x^5) Q <- qr.Q(qr(mat)) df <- as.data.frame(cbind(x, Q)) names(df) <- c("x", 0:5) mdf <- gather(df, degree, value, -x) qplot(x, value, data = mdf, geom = "line", color = degree) Q <- apply(Q, 2, function(x) x / x[n]) df <- as.data.frame(cbind(x, Q)) names(df) <- c("x", paste0("P_", 0:5)) mdf <- gather(df, degree, value, -x) qplot(x, value, data = mdf, geom = "line", color = degree) # chebyshev polynomials are orthogonal in two ways: P2 <- as.function(legendre(2)) P3 <- as.function(legendre(3)) P4 <- as.function(legendre(4)) integrate(function(x) P2(x) * P3(x), lower = -1, upper = 1) integrate(function(x) P2(x) * P4(x), lower = -1, upper = 1) integrate(function(x) P3(x) * P4(x), lower = -1, upper = 1) n <- 10000L u <- runif(n, -1, 1) 2 * mean(P2(u) * P3(u)) 2 * mean(P2(u) * P4(u)) 2 * mean(P3(u) * P4(u)) (2/n) * sum(P2(u) * P3(u)) (2/n) * sum(P2(u) * P4(u)) (2/n) * sum(P3(u) * P4(u)) } \seealso{ \code{\link[orthopolynom:legendre.polynomials]{orthopolynom::legendre.polynomials()}}, \url{http://en.wikipedia.org/wiki/Legendre_polynomials} } \author{ David Kahle calling code from the orthopolynom package } mpoly/man/mpolyListArithmetic.Rd0000644000176200001440000000132113476534026016471 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mpolyListArithmetic.R \name{mpolyListArithmetic} \alias{mpolyListArithmetic} \alias{+.mpolyList} \alias{-.mpolyList} \alias{*.mpolyList} \title{Element-wise arithmetic with vectors of multivariate polynomials.} \usage{ \method{+}{mpolyList}(e1, e2) \method{-}{mpolyList}(e1, e2) \method{*}{mpolyList}(e1, e2) } \arguments{ \item{e1}{an object of class mpolyList} \item{e2}{an object of class mpolyList} } \value{ An object of class mpolyList. } \description{ Element-wise arithmetic with vectors of multivariate polynomials. } \examples{ ( ms1 <- mp( c("x", 'y') ) ) ( ms2 <- mp( c("y", '2 x^2') ) ) ms1 + ms2 ms1 - ms2 ms1 * ms2 ms1^3 } mpoly/man/chebyshev.Rd0000644000176200001440000000541413577046366014461 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/chebyshev.R \name{chebyshev} \alias{chebyshev} \alias{chebyshev_roots} \title{Chebyshev polynomials} \usage{ chebyshev(degree, kind = "t", indeterminate = "x", normalized = FALSE) chebyshev_roots(k, n) } \arguments{ \item{degree}{degree of polynomial} \item{kind}{\code{"t"} or \code{"u"} (Chebyshev polynomials of the first and second kinds), or \code{"c"} or \code{"s"}} \item{indeterminate}{indeterminate} \item{normalized}{provide normalized coefficients} \item{k, n}{the k'th root of the n'th chebyshev polynomial} } \value{ a mpoly object or mpolyList object } \description{ Chebyshev polynomials as computed by orthopolynom. } \examples{ chebyshev(0) chebyshev(1) chebyshev(2) chebyshev(3) chebyshev(4) chebyshev(5) chebyshev(6) chebyshev(10) chebyshev(0:5) chebyshev(0:5, normalized = TRUE) chebyshev(0:5, kind = "u") chebyshev(0:5, kind = "c") chebyshev(0:5, kind = "s") chebyshev(0:5, indeterminate = "t") # visualize the chebyshev polynomials library(ggplot2); theme_set(theme_classic()) library(tidyr) s <- seq(-1, 1, length.out = 201) N <- 5 # number of chebyshev polynomials to plot (cheb_polys <- chebyshev(0:N)) # see ?bernstein for a better understanding of # how the code below works df <- data.frame(s, as.function(cheb_polys)(s)) names(df) <- c("x", paste0("T_", 0:N)) mdf <- gather(df, degree, value, -x) qplot(x, value, data = mdf, geom = "line", color = degree) # roots of chebyshev polynomials N <- 5 cheb_roots <- chebyshev_roots(1:N, N) cheb_fun <- as.function(chebyshev(N)) cheb_fun(cheb_roots) # chebyshev polynomials are orthogonal in two ways: T2 <- as.function(chebyshev(2)) T3 <- as.function(chebyshev(3)) T4 <- as.function(chebyshev(4)) w <- function(x) 1 / sqrt(1 - x^2) integrate(function(x) T2(x) * T3(x) * w(x), lower = -1, upper = 1) integrate(function(x) T2(x) * T4(x) * w(x), lower = -1, upper = 1) integrate(function(x) T3(x) * T4(x) * w(x), lower = -1, upper = 1) (cheb_roots <- chebyshev_roots(1:4, 4)) sum(T2(cheb_roots) * T3(cheb_roots) * w(cheb_roots)) sum(T2(cheb_roots) * T4(cheb_roots) * w(cheb_roots)) sum(T3(cheb_roots) * T4(cheb_roots) * w(cheb_roots)) sum(T2(cheb_roots) * T3(cheb_roots)) sum(T2(cheb_roots) * T4(cheb_roots)) sum(T3(cheb_roots) * T4(cheb_roots)) } \seealso{ \code{\link[orthopolynom:chebyshev.t.polynomials]{orthopolynom::chebyshev.t.polynomials()}}, \code{\link[orthopolynom:chebyshev.u.polynomials]{orthopolynom::chebyshev.u.polynomials()}}, \code{\link[orthopolynom:chebyshev.c.polynomials]{orthopolynom::chebyshev.c.polynomials()}}, \code{\link[orthopolynom:chebyshev.s.polynomials]{orthopolynom::chebyshev.s.polynomials()}}, \url{http://en.wikipedia.org/wiki/Chebyshev_polynomials} } \author{ David Kahle calling code from the orthopolynom package } mpoly/DESCRIPTION0000644000176200001440000000152213623427170013125 0ustar liggesusersPackage: mpoly Type: Package Title: Symbolic Computation and More with Multivariate Polynomials Version: 1.1.1 URL: https://github.com/dkahle/mpoly BugReports: https://github.com/dkahle/mpoly/issues Authors@R: person("David", "Kahle", email = "david@kahle.io", role = c("aut", "cre"), comment = c(ORCID = "0000-0002-9999-1558")) Description: Symbolic computing with multivariate polynomials in R. Imports: stringr (>= 1.0.0), stringi, partitions, plyr, stats, ggplot2, polynom, orthopolynom, tidyr Suggests: testthat (>= 2.1.0), magrittr, dplyr, covr License: GPL-2 RoxygenNote: 7.0.2 Encoding: UTF-8 NeedsCompilation: no Packaged: 2020-02-19 19:28:27 UTC; david_kahle Author: David Kahle [aut, cre] () Maintainer: David Kahle Repository: CRAN Date/Publication: 2020-02-20 07:00:08 UTC mpoly/tests/0000755000176200001440000000000013623306133012554 5ustar liggesusersmpoly/tests/testthat/0000755000176200001440000000000013623427170014421 5ustar liggesusersmpoly/tests/testthat/test-as-function-mpolylist.R0000644000176200001440000000144113476534026022006 0ustar liggesuserscontext("as.function.mpolyList()") test_that("basic", { expect_equal( as.function(mp(c("2 x + 1", "x - z^2")), silent = TRUE)(c(1,2)), c(3L, -3L) ) }) test_that("vector", { expect_equal( as.function(mp(c("2 x + 1", "x - z^2")), silent = TRUE, vector = FALSE)(1, 2), c(3L, -3L) ) }) test_that("univariate", { expect_equal( as.function(mp(c("x", "x^2", "x^3")), silent = TRUE)(2), c(2L, 4L, 8L) ) expect_equal( as.function(mp(c("x", "x^2", "x^3")), silent = TRUE)(1:3), structure(c(1, 2, 3, 1, 4, 9, 1, 8, 27), .Dim = c(3L, 3L)) ) }) test_that("errors when should", { expect_error( as.function(mp(c("x", "y", "z")), silent = TRUE, varorder = c("x", "y")), "varorder must contain all of the variables." ) })mpoly/tests/testthat/test-hermite.R0000644000176200001440000000514213476534026017165 0ustar liggesuserscontext("hermite()") test_that("basic", { expect_equal( hermite(3), structure( list( c("x" = 1, "coef" = -3), c("x" = 3, "coef" = 1) ), "class" = c("hermite", "mpoly"), "hermite" = list( "degree" = 0, "kind" = "he", "indeterminate" = "x", "normalized" = FALSE ) ) ) }) test_that("normalized", { expect_equal( hermite(3, normalized = TRUE), structure( list( c("x" = 1, "coef" = -0.7735719), c("x" = 3, "coef" = 0.2578573) ), "class" = c("hermite", "mpoly"), "hermite" = list( "degree" = 0, "kind" = "he", "indeterminate" = "x", "normalized" = TRUE ) ), tolerance = 1e-5 ) }) test_that("indeterminate", { expect_equal( hermite(3, indeterminate = "t"), structure( list( c("t" = 1, "coef" = -3), c("t" = 3, "coef" = 1) ), "class" = c("hermite", "mpoly"), "hermite" = list( "degree" = 0, "kind" = "he", "indeterminate" = "t", "normalized" = FALSE ) ), tolerance = 1e-5 ) }) test_that("kind = h", { expect_equal( hermite(3, kind = "h"), structure( list( c("x" = 1, "coef" = -12), c("x" = 3, "coef" = 8) ), "class" = c("hermite", "mpoly"), "hermite" = list( "degree" = 0, "kind" = "h", "indeterminate" = "x", "normalized" = FALSE ) ), tolerance = 1e-5 ) }) test_that("vectorized", { expect_equal( hermite(0:2), structure( list( structure( list( c("coef" = 1) ), "class" = c("hermite", "mpoly"), "hermite" = list( "degree" = 0, "kind" = "he", "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("x" = 1, "coef" = 1) ), "class" = c("hermite", "mpoly"), "hermite" = list( "degree" = 1, "kind" = "he", "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("coef" = -1), c("x" = 2, "coef" = 1) ), "class" = c("hermite", "mpoly"), "hermite" = list( "degree" = 2, "kind" = "he", "indeterminate" = "x", "normalized" = FALSE ) ) ), class = "mpolyList" ) ) }) mpoly/tests/testthat/test-components.R0000644000176200001440000000401513476534026017713 0ustar liggesuserscontext("Polynomial component extraction") test_that("[.mpoly", { expect_equal(mp("x + y")[1], mp("x")) expect_equal(mp("x + y + z")[1:2], mp("x + y")) expect_equal(mp("x + y + z")[-1], mp("y + z")) }) test_that("LT, LC, and LM work - lex", { expect_equal( LT(mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")), mp("3 x^10") ) expect_equal( LC(mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")), 3 ) expect_equal( LM(mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")), mp("x^10") ) }) test_that("multideg works - lex", { expect_equal(multideg(mp("1")), 0L) expect_equal( multideg(mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")), c(x = 10, y = 0, z = 0) ) }) test_that("totaldeg works - lex", { expect_equal( totaldeg(mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")), 10 ) expect_equal( totaldeg(mp(c("x y", "y z^2", "0"))), c(2L, 3L, 0L) ) expect_error( totaldeg("5 x"), "totaldeg requires an mpoly or mpolyList object.", fixed = TRUE ) }) test_that("monomials works", { expect_equal( monomials(mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")), mp(c("x y^2", "x^4 z", "3 x^3 z", "2 x^2 z", "3 x^10")) ) expect_equal( monomials(mp("x y^2 + 3 y^3 x^2")), mp(c("x y^2", "3 x^2 y^3"), varorder = c("x", "y")) ) expect_error( monomials("5 x"), "monomials requires an mpoly or mpolyList object.", fixed = TRUE ) expect_equal( suppressWarnings(terms(mp("x y^2 + 3 y^3 x^2"))), mp(c("x y^2", "3 x^2 y^3"), varorder = c("x", "y")) ) expect_warning( terms(mp("x y^2 + 3 y^3 x^2")), "'terms.mpoly' is deprecated.", fixed = TRUE ) }) test_that("exponents works", { expect_equal( exponents(mp("x y^2 + 3 y^3 x^2")), list(structure(1:2, .Names = c("x", "y")), structure(2:3, .Names = c("x", "y"))) ) expect_equal( exponents(mp("x y^2 + 3 y^3 x^2 z"), reduced = TRUE), list(structure(1:2, .Names = c("x", "y")), c(x = 2L, y = 3L, z = 1L)) ) }) mpoly/tests/testthat/test-mpolylist.R0000644000176200001440000000101013476534026017552 0ustar liggesuserscontext("mpolyList()") test_that("basic", { expect_equal( mpolyList(mp("x"), mp("x^2"), mp("x^3 t")), structure( list( structure(list(c(x = 1, coef = 1)), class = "mpoly"), structure(list(c(x = 2, coef = 1)), class = "mpoly"), structure(list(c(x = 3, t = 1, coef = 1)), class = "mpoly") ), class = "mpolyList" ) ) expect_error( mpolyList( mp("x"), "y" ), "Each argument must be of class mpoly.", fixed = TRUE ) }) mpoly/tests/testthat/test-lcm.R0000644000176200001440000000031213476534026016275 0ustar liggesuserscontext("LCM() is working properly") test_that("LCM works", { expect_equal(LCM(5L, 7L), 35L) expect_equal(LCM(5L, 8L), 40L) expect_equal(LCM(5L, 9L), 45L) expect_equal(LCM(5L, 10L), 10L) }) mpoly/tests/testthat/test-reorder-mpoly.R0000644000176200001440000000306213476534026020327 0ustar liggesuserscontext("reorder.mpoly()") # examples from cox little and o'shea, 3e, p.59 test_that("lex works", { expect_equal( reorder.mpoly(mp("4 x y^2 z + 4 z^2 - 5 x^3 + 7 x^2 z^2"), order = "lex", varorder = c("x", "y", "z")), structure( list( c(x = 3, coef = -5), c(x = 2, z = 2, coef = 7), c(x = 1, y = 2, z = 1, coef = 4), c(z = 2, coef = 4) ), class = "mpoly" ) ) }) test_that("glex works", { expect_equal( reorder.mpoly(mp("4 x y^2 z + 4 z^2 - 5 x^3 + 7 x^2 z^2"), order = "glex", varorder = c("x", "y", "z")), structure( list( c(x = 2, z = 2, coef = 7), c(x = 1, y = 2, z = 1, coef = 4), c(x = 3, coef = -5), c(z = 2, coef = 4) ), class = "mpoly" ) ) }) test_that("grlex works", { expect_equal( reorder.mpoly(mp("4 x y^2 z + 4 z^2 - 5 x^3 + 7 x^2 z^2"), order = "grlex", varorder = c("x", "y", "z")), structure( list( c(x = 1, y = 2, z = 1, coef = 4), c(x = 2, z = 2, coef = 7), c(x = 3, coef = -5), c(z = 2, coef = 4) ), class = "mpoly" ) ) }) test_that("message when no order specified but varorder not", { expect_message( reorder.mpoly(mp("x + y^2"), order = "glex"), "Using variable ordering - x, y", fixed = TRUE ) }) test_that("all vars in varorder", { expect_error( reorder.mpoly(mp("x + y^2"), varorder = "x"), "If specified, varorder must contain all computed vars - x, y", fixed = TRUE ) }) mpoly/tests/testthat/test-jacobi.R0000644000176200001440000000502613476534026016760 0ustar liggesuserscontext("jacobi()") test_that("basic", { expect_equal( jacobi(3), structure( list( c("x" = 1, "coef" = -7.5), c("x" = 3, "coef" = 17.5) ), "class" = c("jacobi", "mpoly"), "jacobi" = list( "degree" = 0, "kind" = "p", "indeterminate" = "x", "normalized" = FALSE, "alpha" = 1, "beta" = 1 ) ) ) }) test_that("normalized", { expect_equal( jacobi(3, normalized = TRUE), structure( list( c("x" = 1, "coef" = -3.557562), c("x" = 3, "coef" = 8.300979) ), "class" = c("jacobi", "mpoly"), "jacobi" = list( "degree" = 0, "kind" = "p", "indeterminate" = "x", "normalized" = TRUE, "alpha" = 1, "beta" = 1 ) ), tolerance = 1e-5 ) }) test_that("alpha and beta", { expect_equal( jacobi(2, alpha = 2, beta = 3), structure( list( c("coef" = -1), c("x" = 1, "coef" = -2), c("x" = 2, "coef" = 9) ), "class" = c("jacobi", "mpoly"), "jacobi" = list( "degree" = 0, "kind" = "p", "indeterminate" = "x", "normalized" = FALSE, "alpha" = 2, "beta" = 3 ) ) ) }) test_that("kind = g", { expect_equal( jacobi(2, kind = "g"), structure( list( c("coef" = 1/6), c("x" = 1, "coef" = -1), c("x" = 2, "coef" = 1) ), "class" = c("jacobi", "mpoly"), "jacobi" = list( "degree" = 0, "kind" = "g", "indeterminate" = "x", "normalized" = FALSE, "alpha" = 1, "beta" = 1 ) ) ) }) test_that("degree is vectorized", { expect_equal( jacobi(0:1), structure( list( structure( list( c("coef" = 1) ), "class" = c("jacobi", "mpoly"), "jacobi" = list( "degree" = 0, "kind" = "p", "indeterminate" = "x", "normalized" = FALSE, "alpha" = 1, "beta" = 1 ) ), structure( list( c("x" = 1, "coef" = 3) ), "class" = c("jacobi", "mpoly"), "jacobi" = list( "degree" = 1, "kind" = "p", "indeterminate" = "x", "normalized" = FALSE, "alpha" = 1, "beta" = 1 ) ) ), class = "mpolyList" ) ) }) mpoly/tests/testthat/test-plug.R0000644000176200001440000000137113476534026016477 0ustar liggesuserscontext("plug()") test_that("plug() inserts numbers properly", { expect_equal( plug(mp("(x + y)^3"), "y", 5), mp("(x + 5)^3") ) }) test_that("plug() changes variables properly", { expect_equal( plug(mp("(x + y)^3"), "x", "t"), mp("(t + y)^3") ) expect_equal( plug(mp("(x + y)^3"), "x", "t_2"), mp("(t_2 + y)^3") ) expect_equal( plug(mp("(x + y)^3"), "x", "xx"), mp("(xx + y)^3") ) }) test_that("plug() inserts mpoly's properly", { expect_equal( plug(mp("(x + y)^3"), "x", mp("x^2")), mp("(x^2 + y)^3") ) }) test_that("plug() works on mpolyLists", { expect_equal( plug(mp(c("(x + y)^3", "2 x")), "x", mp("x^2")), mp(c("(x^2 + y)^3", "2 x^2")) ) }) mpoly/tests/testthat/test-chebyshev.R0000644000176200001440000000574413522636160017512 0ustar liggesuserscontext("chebyshev()") test_that("basic", { expect_equal( chebyshev(3), structure( list( c("x" = 1, "coef" = -3), c("x" = 3, "coef" = 4) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 0, "kind" = "t", "indeterminate" = "x", "normalized" = FALSE ) ) ) expect_equal( chebyshev_roots(1:3, 3), cos((1:3 - .5) * pi / 3) ) }) test_that("different kinds", { expect_equal( chebyshev(3, kind = "u"), structure( list( c("x" = 1, "coef" = -4), c("x" = 3, "coef" = 8) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 0, "kind" = "u", "indeterminate" = "x", "normalized" = FALSE ) ) ) expect_equal( chebyshev(3, kind = "c"), structure( list( c("x" = 1, "coef" = -1.5), c("x" = 3, "coef" = .5) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 0, "kind" = "c", "indeterminate" = "x", "normalized" = FALSE ) ) ) expect_equal( chebyshev(3, kind = "s"), structure( list( c("x" = 1, "coef" = -2), c("x" = 3, "coef" = 1) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 0, "kind" = "s", "indeterminate" = "x", "normalized" = FALSE ) ) ) }) test_that("normalized", { expect_equal( chebyshev(3, normalized = TRUE), structure( list( c("x" = 1, "coef" = -2.393654), c("x" = 3, "coef" = 3.191538) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 0, "kind" = "t", "indeterminate" = "x", "normalized" = TRUE ) ), tolerance = 1e-5 ) }) test_that("vectorized", { expect_equal( chebyshev(0:2), structure( list( structure( list( c("coef" = 1) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 0, "kind" = "t", "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("x" = 1, "coef" = 1) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 1, "kind" = "t", "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("coef" = -1), c("x" = 2, "coef" = 2) ), "class" = c("chebyshev", "mpoly"), "chebyshev" = list( "degree" = 2, "kind" = "t", "indeterminate" = "x", "normalized" = FALSE ) ) ), "class" = "mpolyList" ) ) }) mpoly/tests/testthat/test-tuples.R0000644000176200001440000000154413476534026017046 0ustar liggesuserscontext("tuples()") test_that("basic tuples() functionality works", { expect_equal( tuples(1:2, 3), structure(c(1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L), .Dim = c(8L, 3L)) ) }) test_that("tuples() list = TRUEs properly", { expect_equal( tuples(1:2, 3, list = TRUE), list( c(1L, 1L, 1L), c(1L, 1L, 2L), c(1L, 2L, 1L), c(1L, 2L, 2L), c(2L, 1L, 1L), c(2L, 1L, 2L), c(2L, 2L, 1L), c(2L, 2L, 2L) ) ) }) test_that("tuples() repeats argument works properly", { expect_equal( tuples(c(1, 2, 2), 2, repeats = TRUE), structure(c(1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2), .Dim = c(9L, 2L)) ) expect_equal( tuples(c(1, 2, 2), 2, repeats = FALSE), structure(c(1, 1, 2, 2, 1, 2, 1, 2), .Dim = c(4L, 2L)) ) }) mpoly/tests/testthat/test-predicates.R0000644000176200001440000000372313476534026017656 0ustar liggesuserscontext("predicate functions") test_that("is.constant() works - single", { expect_equal(is.constant(mp("5")), TRUE) expect_equal(is.constant(mp("5 + x")), FALSE) }) test_that("is.constant() works - vector", { expect_equal(is.constant(mp(c("5", "5 + x"))), c(TRUE, FALSE)) }) test_that("is.mpoly() works", { expect_equal(is.mpoly(mp("5")), TRUE) expect_equal(is.mpoly(mp("5 + x")), TRUE) expect_equal(is.mpoly(mp(c("5", "5 + x"))), FALSE) }) test_that("is.unipoly() works", { expect_equal(is.unipoly(mp("1")), TRUE) expect_equal(is.unipoly(mp("x+1")), TRUE) expect_equal(is.unipoly(mp("x+1+x^2")), TRUE) expect_equal(is.unipoly(mp(c("x+1+x^2", "y"))), c(TRUE, TRUE)) expect_equal(is.unipoly(mp(c("x+1+x^2", "y", "x+y"))), c(TRUE, TRUE, FALSE)) }) test_that("is.bernstein() works", { expect_equal(is.bernstein(bernstein(0, 0)), TRUE) expect_equal(is.bernstein(1), FALSE) expect_equal(is.bernstein(mp("1")), FALSE) }) test_that("is.bezier() works", { expect_equal(is.bezier(bezier(c(0, 0), c(1, 1), c(2, 0))), TRUE) expect_equal(is.bezier(mp(c("2 t", "-2 t^2 + 2 t"))), FALSE) }) test_that("is.chebyshev() works", { expect_equal(is.chebyshev(chebyshev(1)), TRUE) expect_equal(is.chebyshev(mp("x")), FALSE) }) test_that("is.mpolyList() works", { expect_equal(is.mpolyList(mp(c("x", "y"))), TRUE) expect_equal(is.mpolyList( c("x", "y") ), FALSE) expect_equal(is.mpolyList(mp("x")), FALSE) }) test_that("is.linear() works", { expect_equal(is.linear(mp("x + z")), TRUE) expect_equal(is.linear(mp(c("x + z", "y"))), c(TRUE, TRUE)) expect_equal(is.linear(mp("x y")), FALSE) expect_equal(is.linear(mp("x^2")), FALSE) expect_equal(is.linear(mp(c("x + z", "y", "x^2"))), c(TRUE, TRUE, FALSE)) }) test_that("is.wholenumber() works", { expect_equal(is.wholenumber(1L), TRUE) expect_equal(is.wholenumber(1), TRUE) expect_equal(is.wholenumber(1 + 1e-5), FALSE) expect_equal(is.wholenumber(1 + 1e-10), TRUE) }) mpoly/tests/testthat/test-permutations.R0000644000176200001440000000123013476534026020254 0ustar liggesuserscontext("permutations()") test_that("permutations of a single integer works", { expect_equal( permutations(3L), structure(c(1, 1, 2, 2, 3, 3, 2, 3, 1, 3, 1, 2, 3, 2, 3, 1, 2, 1), .Dim = c(6L, 3L)) ) }) test_that("permutations of a vector works", { expect_equal( permutations(letters[1:3]), structure( c("a", "a", "b", "b", "c", "c", "b", "c", "a", "c", "a", "b", "c", "b", "c", "a", "b", "a"), .Dim = c(6L, 3L) ) ) }) test_that("permutations with duplicated entries works", { expect_equal( permutations(c(1, 1, 2)), structure(c(1, 1, 2, 1, 2, 1, 2, 1, 1), .Dim = c(3L, 3L)) ) }) mpoly/tests/testthat/test-lissajous.R0000644000176200001440000000027413476534026017545 0ustar liggesuserscontext("lissajous()") test_that("lissajous() works properly", { expect_equal( lissajous(3, 2, -pi/2, 0), mp("-4 x^2 + 4 x^4 + 9 y^2 - 24 y^4 + 16 y^6") ) }) mpoly/tests/testthat/test-as-function-mpoly.R0000644000176200001440000000175213476534026021117 0ustar liggesuserscontext("as.function.mpoly()") test_that("univariate", { f <- as.function(mp("(3 - x)^2"), silent = TRUE) x <- 2 expect_equal(f(x), (3 - x)^2) x <- 5 expect_equal(f(x), (3 - x)^2) f <- as.function(mp("(3 - x)^2"), silent = TRUE) x <- 2:10 expect_equal(f(x), (3 - x)^2) }) test_that("vector = TRUE", { f <- as.function(mp("3 x + y^2"), vector = TRUE, silent = TRUE) expect_equal(f(c(1,2)), 3*(1) + (2)^2) }) test_that("vector = FALSE", { f <- as.function(mp("3 x + y^2"), vector = FALSE, silent = TRUE) expect_equal(f(1, 2), 3*(1) + (2)^2) }) test_that("constant mpoly", { f <- as.function(mp("3"), silent = TRUE) expect_equal(f(1), 3) expect_equal(f(1:4), rep(3, 4)) }) test_that("bernstein", { s <- seq(0, 1, .01) expect_equal( as.function(bernstein(1, 2))(s), 2*s - 2*s^2 ) s <- seq(0, 1, .01) expect_equal( as.function(bernstein(1, 2))(-3:3), 2*(-3:3) - 2*(-3:3)^2 ) }) mpoly/tests/testthat/test-bernstein-approx.R0000644000176200001440000000043513476534026021030 0ustar liggesuserscontext("bernstein_approx()") test_that("bernstein_approx() works", { expect_equal( bernstein_approx(function(x) x^4, 2L), mp(".125 x + .875 x^2") ) expect_equal( suppressWarnings(bernsteinApprox(function(x) x^4, 2L)), mp(".125 x + .875 x^2") ) }) mpoly/tests/testthat/test-mp.r0000644000176200001440000004016613476534026016211 0ustar liggesuserscontext("mpoly()") test_that("mpoly() flags non-list arguments",{ expect_error(mpoly(1:5), "input to mpoly must be a list.") }) context("mp() is working properly") test_that("mp() splices *'s properly", { expect_equal( mp("x y"), mpoly(list(c(x = 1, y = 1, coef = 1))) ) expect_equal( mp("x y z"), mpoly(list(c(x = 1, y = 1, z = 1, coef = 1))) ) expect_equal( mp("a b c d e"), mpoly(list(c(a = 1, b = 1, c = 1, d = 1, e = 1, coef = 1))) ) expect_equal( mp("x1 x2"), mpoly(list(c(x1 = 1, x2 = 1, coef = 1))) ) expect_equal( mp("x1a x2a"), mpoly(list(c(x1a = 1, x2a = 1, coef = 1))) ) expect_equal( mp("x1a^2 x2a^3"), mpoly(list(c(x1a = 2, x2a = 3, coef = 1))) ) }) test_that("mp() parses character vectors properly", { expect_equal( mp(c("2 x y", "-3 x y z")), structure( list( structure(list(c(x = 1, y = 1, coef = 2)), class = "mpoly"), structure(list(c(x = 1, y = 1, z = 1, coef = -3)), class = "mpoly") ), class = "mpolyList" ) ) expect_equal( mp(c("x y", "x y z", "a^3 b c^2 d")), structure( list( structure(list(c(x = 1, y = 1, coef = 1)), class = "mpoly"), structure(list(c(x = 1, y = 1, z = 1, coef = 1)), class = "mpoly"), structure(list(c(a = 3, b = 1, c = 2, d = 1, coef = 1)), class = "mpoly") ), class = "mpolyList" ) ) }) test_that("mp() obeys varorder", { expect_equal( mp("3 y^2 x^3", varorder = c("x", "y")), structure(list(c(x = 3, y = 2, coef = 3)), class = "mpoly") ) }) # # # # # # # # test_that("mpoly only allows named whole number exponents",{ # # expect_error( # mpoly(list(c(x = "a", coef = 1))), # "each element of list must be of type numeric." # ) # # expect_error( # mpoly(list(c(x = 1, 1, coef = 2))), # "each element of list must be named for every element." # ) # # expect_error( # mpoly(list(c(x = 1, y = 1.5, coef = 2))), # "degrees must be nonnegative integers." # ) # # expect_error( # mpoly(list(c(x = 1, y = -2, coef = 2))), # "degrees must be nonnegative integers." # ) # # expect_error( # mpoly(list(c(x = 1, y = 2, coef = 2)), varorder = "x"), # "if specified varorder must be a permutation of" # ) # # }) # # # # # # # # # # # # # # # # # # # context("Quality of polynomial parsing") # # test_that("parse_nonparenthetical_term works", { # # f <- parse_nonparenthetical_term # m <- mpoly # # expect_equal(f("0"), m(list(c(coef = 0)))) # expect_equal(f("1"), m(list(c(coef = 1)))) # expect_equal(f("+1"), m(list(c(coef = 1)))) # expect_equal(f("-1"), m(list(c(coef = -1)))) # # expect_equal(f("1.5"), m(list(c(coef = 1.5)))) # expect_equal(f("+1.5"), m(list(c(coef = 1.5)))) # expect_equal(f("-1.5"), m(list(c(coef = -1.5)))) # # expect_equal(f(".5"), m(list(c(coef = .5)))) # expect_equal(f("+.5"), m(list(c(coef = .5)))) # expect_equal(f("-.5"), m(list(c(coef = -.5)))) # # expect_equal(f("2 2"), m(list(c(coef = 4)))) # expect_equal(f("-2 2"), m(list(c(coef = -4)))) # expect_error(f("-2 -2"), "Negative signs are only allowed at the beginning of terms.") # # expect_equal(f(".5 .5"), m(list(c(coef = .25)))) # expect_equal(f(".5 2"), m(list(c(coef = 1)))) # expect_equal(f("-.5 .5"), m(list(c(coef = -.25)))) # expect_equal(f("-.5 2"), m(list(c(coef = -1)))) # expect_equal(f("-2 .5"), m(list(c(coef = -1)))) # # expect_equal(f("2^2"), m(list(c(coef = 4)))) # expect_equal(f("-2^2"), m(list(c(coef = -4)))) # expect_equal(f(".5^2"), m(list(c(coef = .25)))) # expect_equal(f("-.5^2"), m(list(c(coef = -.25)))) # # expect_equal(f("0x"), m(list(c(coef = 0)))) # expect_equal(f("-0x"), m(list(c(coef = 0)))) # # expect_equal(f("0 x"), m(list(c(coef = 0)))) # expect_equal(f("-0 x"), m(list(c(coef = 0)))) # # expect_equal(f("x"), m(list(c(coef = 1, x = 1)))) # # expect_equal(f("-x"), m(list(c(coef = -1, x = 1)))) # # # expect_equal(f("- x"), m(list(c(coef = -1, x = 1)))) # # expect_equal(f("2x"), m(list(c(coef = 2, x = 1)))) # expect_equal(f("-2x"), m(list(c(coef = -2, x = 1)))) # expect_equal(f("- 2x"), m(list(c(coef = -2, x = 1)))) # # expect_equal(f(".2x"), m(list(c(coef = .2, x = 1)))) # expect_equal(f("-.2x"), m(list(c(coef = -.2, x = 1)))) # expect_equal(f("- .2x"), m(list(c(coef = -.2, x = 1)))) # # expect_equal(f("2 x"), m(list(c(coef = 2, x = 1)))) # expect_equal(f("-2 x"), m(list(c(coef = -2, x = 1)))) # expect_equal(f("- 2 x"), m(list(c(coef = -2, x = 1)))) # # expect_equal(f(".2 x"), m(list(c(coef = .2, x = 1)))) # expect_equal(f("-.2 x"), m(list(c(coef = -.2, x = 1)))) # expect_equal(f("- .2 x"), m(list(c(coef = -.2, x = 1)))) # # expect_equal(f("x[1]^2"), m(list(c(coef = 1, "x[1]" = 2)))) # expect_equal(f("x[1] ^ 2"), m(list(c(coef = 1, "x[1]" = 2)))) # expect_equal(f("2 x[1] ^ 2"), m(list(c(coef = 2, "x[1]" = 2)))) # expect_equal(f("-2 x[1] ^ 2"), m(list(c(coef = -2, "x[1]" = 2)))) # # expect_equal(f("x1^2"), m(list(c(coef = 1, x1 = 2)))) # expect_equal(f("x1 ^ 2"), m(list(c(coef = 1, x1 = 2)))) # expect_equal(f("2 x1 ^ 2"), m(list(c(coef = 2, x1 = 2)))) # expect_equal(f("-2 x1 ^ 2"), m(list(c(coef = -2, x1 = 2)))) # # expect_equal(f("x^2"), m(list(c(coef = 1, x = 2)))) # expect_equal(f("x ^ 2"), m(list(c(coef = 1, x = 2)))) # expect_equal(f("x ^2"), m(list(c(coef = 1, x = 2)))) # expect_equal(f("x^ 2"), m(list(c(coef = 1, x = 2)))) # # expect_equal(f("x^0"), m(list(c(coef = 1)))) # # expect_equal(f("-x^0"), m(list(c(coef = -1)))) # # expect_equal(f("2x^2"), m(list(c(coef = 2, x = 2)))) # expect_equal(f("-2x^2"), m(list(c(coef = -2, x = 2)))) # # expect_equal(f("5^2x"), m(list(c(coef = 25, x = 1)))) # expect_equal(f("-5^2x"), m(list(c(coef = -25, x = 1)))) # expect_equal(f("5^2x^2"), m(list(c(coef = 25, x = 2)))) # expect_equal(f("-5^2x^2"), m(list(c(coef = -25, x = 2)))) # expect_equal(f("5^2x^2"), m(list(c(coef = 25, x = 2)))) # expect_equal(f(".5^2x^2"), m(list(c(coef = .25, x = 2)))) # # expect_equal(f("12xx 2 y 2x"), m(list(c(coef = 48, xx = 1, y = 1, x = 1)))) # expect_equal(f("2 x y^2 3 2 3^2"), m(list(c(coef = 108, x = 1, y = 2)))) # expect_equal(f("x y .2"), m(list(c(coef = .2, x = 1, y = 1)))) # }) # # # # # # # # # # # # # # # # # # # # # # # # # test_that("parse_nonparenthetical_polynomial works", { # # f <- parse_nonparenthetical_polynomial # m <- mpoly # # ## the ones from before parse_nonparenthetical_term # expect_equal(f("0"), m(list(c(coef = 0)))) # expect_equal(f("1"), m(list(c(coef = 1)))) # expect_equal(f("-1"), m(list(c(coef = -1)))) # # expect_equal(f("1.5"), m(list(c(coef = 1.5)))) # expect_equal(f("-1.5"), m(list(c(coef = -1.5)))) # # expect_equal(f(".5"), m(list(c(coef = .5)))) # expect_equal(f("-.5"), m(list(c(coef = -.5)))) # # expect_equal(f("2 2"), m(list(c(coef = 4)))) # expect_equal(f("-2 2"), m(list(c(coef = -4)))) # # expect_error(f("-2 -2"), "negative signs only allowed at the beginning of terms") # # expect_equal(f(".5 .5"), m(list(c(coef = .25)))) # expect_equal(f(".5 2"), m(list(c(coef = 1)))) # expect_equal(f("-.5 .5"), m(list(c(coef = -.25)))) # expect_equal(f("-.5 2"), m(list(c(coef = -1)))) # expect_equal(f("-2 .5"), m(list(c(coef = -1)))) # # expect_equal(f("2^2"), m(list(c(coef = 4)))) # expect_equal(f("-2^2"), m(list(c(coef = -4)))) # expect_equal(f(".5^2"), m(list(c(coef = .25)))) # expect_equal(f("-.5^2"), m(list(c(coef = -.25)))) # # expect_equal(f("0x"), m(list(c(coef = 0)))) # expect_equal(f("-0x"), m(list(c(coef = 0)))) # # expect_equal(f("0 x"), m(list(c(coef = 0)))) # expect_equal(f("-0 x"), m(list(c(coef = 0)))) # # expect_equal(f("x"), m(list(c(coef = 1, x = 1)))) # # expect_equal(f("-x"), m(list(c(coef = -1, x = 1)))) # # # expect_equal(f("- x"), m(list(c(coef = -1, x = 1)))) # # expect_equal(f("2x"), m(list(c(coef = 2, x = 1)))) # expect_equal(f("-2x"), m(list(c(coef = -2, x = 1)))) # expect_equal(f("- 2x"), m(list(c(coef = -2, x = 1)))) # # expect_equal(f(".2x"), m(list(c(coef = .2, x = 1)))) # expect_equal(f("-.2x"), m(list(c(coef = -.2, x = 1)))) # expect_equal(f("- .2x"), m(list(c(coef = -.2, x = 1)))) # # expect_equal(f("2 x"), m(list(c(coef = 2, x = 1)))) # expect_equal(f("-2 x"), m(list(c(coef = -2, x = 1)))) # expect_equal(f("- 2 x"), m(list(c(coef = -2, x = 1)))) # # expect_equal(f(".2 x"), m(list(c(coef = .2, x = 1)))) # expect_equal(f("-.2 x"), m(list(c(coef = -.2, x = 1)))) # expect_equal(f("- .2 x"), m(list(c(coef = -.2, x = 1)))) # # expect_equal(f("x[1]^2"), m(list(c(coef = 1, "x[1]" = 2)))) # expect_equal(f("x[1] ^ 2"), m(list(c(coef = 1, "x[1]" = 2)))) # expect_equal(f("2 x[1] ^ 2"), m(list(c(coef = 2, "x[1]" = 2)))) # expect_equal(f("-2 x[1] ^ 2"), m(list(c(coef = -2, "x[1]" = 2)))) # # expect_equal(f("x1^2"), m(list(c(coef = 1, x1 = 2)))) # expect_equal(f("x1 ^ 2"), m(list(c(coef = 1, x1 = 2)))) # expect_equal(f("2 x1 ^ 2"), m(list(c(coef = 2, x1 = 2)))) # expect_equal(f("-2 x1 ^ 2"), m(list(c(coef = -2, x1 = 2)))) # # expect_equal(f("x^2"), m(list(c(coef = 1, x = 2)))) # expect_equal(f("x ^ 2"), m(list(c(coef = 1, x = 2)))) # expect_equal(f("x ^2"), m(list(c(coef = 1, x = 2)))) # expect_equal(f("x^ 2"), m(list(c(coef = 1, x = 2)))) # # expect_equal(f("x^0"), m(list(c(coef = 1)))) # # expect_equal(f("-x^0"), m(list(c(coef = -1)))) # # expect_equal(f("2x^2"), m(list(c(coef = 2, x = 2)))) # expect_equal(f("-2x^2"), m(list(c(coef = -2, x = 2)))) # # expect_equal(f("5^2x"), m(list(c(coef = 25, x = 1)))) # expect_equal(f("-5^2x"), m(list(c(coef = -25, x = 1)))) # expect_equal(f("5^2x^2"), m(list(c(coef = 25, x = 2)))) # expect_equal(f("-5^2x^2"), m(list(c(coef = -25, x = 2)))) # expect_equal(f("5^2x^2"), m(list(c(coef = 25, x = 2)))) # expect_equal(f(".5^2x^2"), m(list(c(coef = .25, x = 2)))) # # expect_equal(f("12xx 2 y 2x"), m(list(c(coef = 48, xx = 1, y = 1, x = 1)))) # expect_equal(f("2 x y^2 3 2 3^2"), m(list(c(coef = 108, x = 1, y = 2)))) # expect_equal(f("x y .2"), m(list(c(coef = .2, x = 1, y = 1)))) # # # # ## the new ones # expect_equal(f("1+1"), m(list(c(coef = 2)))) # expect_equal(f("1 + 1"), m(list(c(coef = 2)))) # expect_equal(f("1+ 1"), m(list(c(coef = 2)))) # expect_equal(f("1 +1"), m(list(c(coef = 2)))) # # expect_equal(f("1-1"), m(list(c(coef = 0)))) # expect_equal(f("1 - 1"), m(list(c(coef = 0)))) # expect_equal(f("1- 1"), m(list(c(coef = 0)))) # # expect_equal(f("1 -1"), m(list(c(coef = 0)))) # # expect_equal(f("1--1"), m(list(c(coef = 2)))) # expect_equal(f("1-- 1"), m(list(c(coef = 2)))) # expect_equal(f("1- -1"), m(list(c(coef = 2)))) # expect_equal(f("1 --1"), m(list(c(coef = 2)))) # expect_equal(f("1 -- 1"), m(list(c(coef = 2)))) # expect_equal(f("1-- 1"), m(list(c(coef = 2)))) # expect_equal(f("1- - 1"), m(list(c(coef = 2)))) # expect_equal(f("1- -1"), m(list(c(coef = 2)))) # expect_equal(f("1 -- 1"), m(list(c(coef = 2)))) # expect_equal(f("1 - -1"), m(list(c(coef = 2)))) # expect_equal(f("1 --1"), m(list(c(coef = 2)))) # # expect_equal(f("1+-1"), m(list(c(coef = 0)))) # expect_equal(f("1+- 1"), m(list(c(coef = 0)))) # expect_equal(f("1+ -1"), m(list(c(coef = 0)))) # expect_equal(f("1 +-1"), m(list(c(coef = 0)))) # expect_equal(f("1 +- 1"), m(list(c(coef = 0)))) # expect_equal(f("1+- 1"), m(list(c(coef = 0)))) # expect_equal(f("1+ - 1"), m(list(c(coef = 0)))) # expect_equal(f("1+ -1"), m(list(c(coef = 0)))) # expect_equal(f("1 +- 1"), m(list(c(coef = 0)))) # expect_equal(f("1 + -1"), m(list(c(coef = 0)))) # expect_equal(f("1 +-1"), m(list(c(coef = 0)))) # # expect_equal(f("-1+1"), m(list(c(coef = 0)))) # expect_equal(f("- 1+1"), m(list(c(coef = 0)))) # # # expect_equal(f("x+1"), m(list(c(coef = 1, x = 1), c(coef = 1)))) # expect_equal(f("x + 1"), m(list(c(coef = 1, x = 1), c(coef = 1)))) # expect_equal(f("x +1"), m(list(c(coef = 1, x = 1), c(coef = 1)))) # expect_equal(f("x+ 1"), m(list(c(coef = 1, x = 1), c(coef = 1)))) # # expect_equal(f("x-1"), m(list(c(coef = 1, x = 1), c(coef = -1)))) # expect_equal(f("x - 1"), m(list(c(coef = 1, x = 1), c(coef = -1)))) # expect_equal(f("x -1"), m(list(c(coef = 1, x = 1), c(coef = -1)))) # expect_equal(f("x- 1"), m(list(c(coef = 1, x = 1), c(coef = -1)))) # # expect_equal(f("-x+1"), m(list(c(coef = -1, x = 1), c(coef = 1)))) # expect_equal(f("-x + 1"), m(list(c(coef = -1, x = 1), c(coef = 1)))) # expect_equal(f("-x +1"), m(list(c(coef = -1, x = 1), c(coef = 1)))) # expect_equal(f("-x+ 1"), m(list(c(coef = -1, x = 1), c(coef = 1)))) # # expect_equal(f("-x-1"), m(list(c(coef = -1, x = 1), c(coef = -1)))) # expect_equal(f("-x - 1"), m(list(c(coef = -1, x = 1), c(coef = -1)))) # expect_equal(f("-x -1"), m(list(c(coef = -1, x = 1), c(coef = -1)))) # expect_equal(f("-x- 1"), m(list(c(coef = -1, x = 1), c(coef = -1)))) # # expect_equal(f("-1 -1 -1"), m(list(c(coef = -3)))) # # expect_equal(f("-x+1"), m(list(c(coef = -1, x = 1), c(coef = 1)))) # expect_equal(f("-x-1"), m(list(c(coef = -1, x = 1), c(coef = -1)))) # expect_equal(f("- x - 1"), m(list(c(coef = -1, x = 1), c(coef = -1)))) # # expect_equal(f("- x^2 - 1^2"), m(list(c(coef = -1, x = 2), c(coef = -1)))) # expect_equal(f("x^2 + x^2 y"), m(list(c(coef = 1, x = 2), c(coef = 1, x = 2, y = 1)))) # expect_equal(f("x-x"), m(list(c(coef = 0)))) # expect_equal(f("x - x"), m(list(c(coef = 0)))) # expect_equal(f("x - 4 x"), m(list(c(coef = -3, x = 1)))) # expect_equal(f("x y^2 - y^2 x"), m(list(c(coef = 0)))) # # expect_equal(f("2^2 x + 5^2 + 3^2"), m(list(c(coef = 4, x = 1), c(coef = 34)))) # expect_equal(f("1 3 5^2 + 2 3^4 x"), m(list(c(coef = 75), c(coef = 162, x = 1)))) # expect_equal(f("x - 2 x -3"), m(list(c(coef = -1, x = 1), c(coef = -3)))) # # # # # # expect_equal(f("5-2x"), m(list(c(coef = 5), c(coef = -2, x = 1)))) # expect_equal(f("5 + -2x"), m(list(c(coef = 5), c(coef = -2, x = 1)))) # expect_equal(f("5^2-x"), m(list(c(coef = 25), c(coef = -1, x = 1)))) # expect_equal(f("1+-x-x"), m(list(c(coef = 1), c(coef = -2, x = 1)))) # # expect_equal(f("-x + 2y - 4x - -4"), m(list( # c(coef = -5, x = 1), # c(coef = 2, y = 1), # c(coef = 4) # ))) # # expect_equal(f("x + 2 y + x^2 y + x y z"), m(list( # c(coef = 1, x = 1), # c(coef = 2, y = 1), # c(coef = 1, x = 2, y = 1), # c(coef = 1, x = 1, y = 1, z = 1) # ))) # # expect_equal(f("10 x + 2 y 3 + x^2 5 y"), m(list( # c(coef = 10, x = 1), # c(coef = 6, y = 1), # c(coef = 5, x = 2, y = 1))) # ) # # expect_equal(f("-4 + 2+2 x + 1 x y^4 -3 prq^3 -y - 3 x 2 - 3 y -2"), m(list( # c(coef = -4), # c(coef = -4, x = 1), # c(coef = 1, x = 1, y = 4), # c(coef = -3, prq = 3), # c(coef = -4, y = 1) # ))) # # }) # # # # test_that("Scientific notation is parsed properly", { # # m <- mpoly # # expect_equal(mp("1e+2 x"), m(list(c(coef = 100, x = 1)))) # expect_equal(mp("1e-2 x"), m(list(c(coef = .01, x = 1)))) # expect_equal(mp("e-a"), m(list(c(coef = 1, e = 1), c(coef = -1, a = 1)))) # expect_equal(mp("e+a"), m(list(c(coef = 1, e = 1), c(coef = 1, a = 1)))) # expect_equal(mp("a-e"), m(list(c(coef = 1, a = 1), c(coef = -1, e = 1)))) # expect_equal(mp("a+e"), m(list(c(coef = 1, a = 1), c(coef = 1, e = 1)))) # }) # # # # mp("(x)") # mp("((((x))))") # mp("(x + 0)") # mp("(x + 1)") # mp("(x - 1)") # mp("(-1 x - 1)") # mp("2 (x + 1)") # mp("-1 (x + 1)") # # mp("-2 x + -1 x (3 x - (7 - 2 x)) 7 (x + 1) -3") # # mp("(x-.25)^2") # mp("(x + 1) (x - 1)") # mp("(x + 1) (x + 2)") # mp("(x + 1)^5") # mp("(x+1)^5") # mp("3 (x + 1) (x + 2)^2") # mp("(x + y)^10") # # mp("(x + y) (x - y)") # mp("((x + y) (x - y))^2") # mp("((x + y) (x - y)^2)") # mp("((x + y) (x - y)^2 3)^2") # # mp(c("x","x + y"))mpoly/tests/testthat/test-print-mpoly.R0000644000176200001440000001537713476534026020035 0ustar liggesuserscontext("print_term()") test_that("print_term() prints constants properly", { expect_equal(print_term(c("coef" = 5)), "5") expect_equal(print_term(c("coef" = -5)), "-5") expect_equal(print_term(c("coef" = 1)), "1") expect_equal(print_term(c("coef" = -1)), "-1") expect_equal(print_term(c("coef" = 0)), "0") expect_equal(print_term(c("coef" = .1)), "0.1") expect_equal(print_term(c("coef" = -.1)), "-0.1") expect_equal(print_term(c("coef" = 0.1)), "0.1") expect_equal(print_term(c("coef" = -0.1)), "-0.1") expect_equal(print_term(c("coef" = 1.2)), "1.2") expect_equal(print_term(c("coef" = -1.2)), "-1.2") # scientific notation expect_equal(print_term(c("coef" = 1e2)), "100") expect_equal(print_term(c("coef" = -1e2)), "-100") expect_equal(print_term(c("coef" = 1e-2)), "0.01") expect_equal(print_term(c("coef" = -1e-2)), "-0.01") # longer digit examples - rounding to 7 digits expect_equal(print_term(c("coef" = 12.3456789)), "12.34568") expect_equal(print_term(c("coef" = -12.3456789)), "-12.34568") }) test_that("print_term() prints univariate terms properly", { expect_equal(print_term(c("x" = 2L, "coef" = 5)), "5 x^2") expect_equal(print_term(c("x" = 2L, "coef" = 1)), "x^2") expect_equal(print_term(c("x" = 2L, "coef" = -5)), "-5 x^2") expect_equal(print_term(c("x" = 2L, "coef" = -1)), "-1 x^2") expect_equal(print_term(c("x" = 1L, "coef" = 5)), "5 x") expect_equal(print_term(c("x" = 1L, "coef" = 1)), "x") expect_equal(print_term(c("x" = 1L, "coef" = -5)), "-5 x") expect_equal(print_term(c("x" = 1L, "coef" = -1)), "-1 x") expect_equal(print_term(c("x" = 2L, "coef" = 5), times = " * ", expo = "**"), "5 * x**2") expect_equal(print_term(c("x" = 2L, "coef" = 1), times = " * ", expo = "**"), "x**2") expect_equal(print_term(c("x" = 2L, "coef" = -5), times = " * ", expo = "**"), "-5 * x**2") expect_equal(print_term(c("x" = 2L, "coef" = -1), times = " * ", expo = "**"), "-1 * x**2") expect_equal(print_term(c("x" = 1L, "coef" = 5)), "5 x") expect_equal(print_term(c("x" = 1L, "coef" = 1)), "x") expect_equal(print_term(c("x" = 1L, "coef" = -5)), "-5 x") expect_equal(print_term(c("x" = 1L, "coef" = -1)), "-1 x") # sci notation expect_equal(print_term(c("x" = 2L, "coef" = 1e2)), "100 x^2") expect_equal(print_term(c("x" = 2L, "coef" = -1e2)), "-100 x^2") expect_equal(print_term(c("x" = 2L, "coef" = 1e-2)), "0.01 x^2") expect_equal(print_term(c("x" = 2L, "coef" = -1e-2)), "-0.01 x^2") expect_equal(print_term(c("x" = 2L, "coef" = 1e2), times = "*", expo = "**"), "100*x**2") expect_equal(print_term(c("x" = 2L, "coef" = -1e2), times = "*", expo = "**"), "-100*x**2") expect_equal(print_term(c("x" = 2L, "coef" = 1e-2), times = "*", expo = "**"), "0.01*x**2") expect_equal(print_term(c("x" = 2L, "coef" = -1e-2), times = "*", expo = "**"), "-0.01*x**2") }) test_that("print_term() prints multivariate terms properly", { expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = 5)), "5 x^2 y^3") expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = -5)), "-5 x^2 y^3") expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = 1)), "x^2 y^3") expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = -1)), "-1 x^2 y^3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = 5)), "5 x y^3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = -5)), "-5 x y^3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = 1)), "x y^3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = -1)), "-1 x y^3") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = 5)), "5 x^2 y") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = -5)), "-5 x^2 y") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = 1)), "x^2 y") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = -1)), "-1 x^2 y") expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = 5), times = " * ", expo = "**"), "5 * x**2 * y**3") expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = -5), times = " * ", expo = "**"), "-5 * x**2 * y**3") expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = 1), times = " * ", expo = "**"), "x**2 * y**3") expect_equal(print_term(c("x" = 2L, "y" = 3L, "coef" = -1), times = " * ", expo = "**"), "-1 * x**2 * y**3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = 5), times = " * ", expo = "**"), "5 * x * y**3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = -5), times = " * ", expo = "**"), "-5 * x * y**3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = 1), times = " * ", expo = "**"), "x * y**3") expect_equal(print_term(c("x" = 1L, "y" = 3L, "coef" = -1), times = " * ", expo = "**"), "-1 * x * y**3") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = 5), times = " * ", expo = "**"), "5 * x**2 * y") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = -5), times = " * ", expo = "**"), "-5 * x**2 * y") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = 1), times = " * ", expo = "**"), "x**2 * y") expect_equal(print_term(c("x" = 2L, "y" = 1L, "coef" = -1), times = " * ", expo = "**"), "-1 * x**2 * y") }) context("print.mpoly()") test_that("constants", { test_coef <- function(number) { p <- structure(list(c("coef" = number)), class = "mpoly") expect_equal( print.mpoly(p, silent = TRUE), as.character(number) ) } test_coef(0) test_coef(1) test_coef(-1) test_coef(1.1) test_coef(-1.1) test_coef(1e-3) test_coef(-1e-3) }) test_that("sums: no coefs", { p <- structure( list( c("x" = 1, "coef" = 1), c("y" = 2, "coef" = 1) ), class = "mpoly" ) expect_equal( print.mpoly(p, silent = TRUE), "x + y^2" ) }) test_that("sums: + coefs", { p <- structure( list( c("x" = 2, "coef" = 2), c("y" = 3, "coef" = 3) ), class = "mpoly" ) expect_equal( print.mpoly(p, silent = TRUE), "2 x^2 + 3 y^3" ) }) test_that("subtraction: ordinary", { p <- structure( list( c("x" = 1, "coef" = 2), c("y" = 3, "coef" = -3) ), class = "mpoly" ) expect_equal( print.mpoly(p, silent = TRUE), "2 x - 3 y^3" ) }) test_that("subtraction: 2 x - y^2", { p <- structure( list( c("x" = 1, "coef" = 2), c("y" = 2, "coef" = -1) ), class = "mpoly" ) expect_equal( print.mpoly(p, silent = TRUE), "2 x - y^2" ) }) test_that("errors", { expect_error( print.mpoly( structure(list(c("x" = 1, "y" = 2, "coef" = 1)), class = "mpoly"), varorder = "x", silent = TRUE ), "if specified, varorder must contain all computed vars - x, y", fixed = TRUE ) }) mpoly/tests/testthat/test-round-mpoly.R0000644000176200001440000000070113476534026020011 0ustar liggesuserscontext("round_mpoly()") test_that("basic", { expect_equal( mp("(x + 3.14159265)^4"), mp("x^4 + 12.56637 x^3 + 59.21763 x^2 + 124.0251 x + 97.40909"), tolerance = 1e-5 ) expect_equal( round(mp("(x + 3.14159265)^4")), mp("x^4 + 12.566 x^3 + 59.218 x^2 + 124.025 x + 97.409") ) expect_equal( round(mp("(x + 3.14159265)^4"), 0), mp("x^4 + 13 x^3 + 59 x^2 + 124 x + 97") ) })mpoly/tests/testthat/test-bernstein.R0000644000176200001440000000204013476534026017513 0ustar liggesuserscontext("bernstein()") #' bernstein(0, 0) #' #' bernstein(0, 1) #' bernstein(1, 1) test_that("bernstein() works", { expect_equal( bernstein(0, 0), structure( list(c("coef" = 1)), class = c("bernstein", "mpoly"), bernstein = list("k" = 0, "n" = 0, "indeterminate" = "x") ) ) expect_equal( bernstein(0, 1), structure( list(c("coef" = 1), c("x" = 1, "coef" = -1)), class = c("bernstein", "mpoly"), bernstein = list("k" = 0, "n" = 1, "indeterminate" = "x") ) ) expect_equal( bernstein(1, 1), structure( list(c("x" = 1, "coef" = 1)), class = c("bernstein", "mpoly"), bernstein = list("k" = 1, "n" = 1, "indeterminate" = "x") ) ) }) test_that("bernstein() chooses variables correctly", { expect_equal( bernstein(0, 1, "t"), structure( list(c("coef" = 1), c("t" = 1, "coef" = -1)), class = c("bernstein", "mpoly"), bernstein = list("k" = 0, "n" = 1, "indeterminate" = "t") ) ) }) mpoly/tests/testthat/test-as-mpoly.R0000644000176200001440000000406013476534026017267 0ustar liggesuserscontext("as.mpoly()") test_that("numeric", { expect_equal( as.mpoly(3), structure(list(c("coef" = 3)), class = "mpoly") ) expect_equal( as.mpoly(1:3), structure( list( c("coef" = 1), c("x" = 1, "coef" = 2), c("x" = 2, "coef" = 3) ), class = "mpoly" ) ) }) test_that("polynomial", { expect_equal( as.mpoly(hermite.h.polynomials(3)[[4]]), structure( list( c("x" = 1, "coef" = -12), c("x" = 3, "coef" = 8) ), class = "mpoly" ) ) }) test_that("lm", { expect_equal( as.mpoly(lm(y ~ x, data = data.frame("x" = c(0,1), "y" = c(1,3)))), structure( list( c("x" = 1, "coef" = 2), c("coef" = 1) ), class = "mpoly" ) ) expect_equal( as.mpoly( lm( y ~ poly(x, 1, raw = TRUE), data = data.frame("x" = c(0,1), "y" = c(1,3)) ) ), structure( list( c("x" = 1, "coef" = 2), c("coef" = 1) ), class = "mpoly" ) ) expect_equal( as.mpoly( lm( y ~ poly(x, 2, raw = TRUE), data = data.frame("x" = c(-1,0,1), "y" = c(0,-1,0)) ) ), structure( list( c("x" = 1, "coef" = 0), c("x" = 2, "coef" = 1), c("coef" = -1) ), class = "mpoly" ) ) expect_equal( as.mpoly( lm( y ~ x + I(x^2), data = data.frame("x" = c(-1,0,1), "y" = c(0,-1,0)) ) ), structure( list( c("x" = 1, "coef" = 0), c("x" = 2, "coef" = 1), c("coef" = -1) ), class = "mpoly" ) ) expect_error( as.mpoly( lm( y ~ poly(x, 2, raw = FALSE), data = data.frame("x" = c(-1,0,1), "y" = c(0,-1,0)) ) ), "poly() statements currently must contain raw = TRUE.", fixed = TRUE ) }) test_that("errors", { expect_error( as.mpoly("x"), "objects of class character not supported by as.mpoly().", fixed = TRUE ) })mpoly/tests/testthat/test-bezier-function.R0000644000176200001440000000205113476534026020627 0ustar liggesuserscontext("bezier_function()") test_that("basic", { t <- seq(0, 1, length.out = 11) points <- data.frame(x = 0:3, y = c(0,1,-1,0)) expect_equal( as.data.frame(bezier_function(points)(t)), structure( list( "x" = c(3, 2.7, 2.4, 2.1, 1.8, 1.5, 1.2, 0.9, 0.6, 0.3, 0), "y" = c(0, -0.216, -0.288, -0.252, -0.144, 0, 0.144, 0.252, 0.288, 0.216, 0) ), class = "data.frame", row.names = c(NA, -11L) ) ) }) test_that("bezierFunction is deprecated", { t <- seq(0, 1, length.out = 11) points <- data.frame(x = 0:3, y = c(0,1,-1,0)) expect_equal( suppressWarnings(as.data.frame(bezierFunction(points)(t))), structure( list( "x" = c(3, 2.7, 2.4, 2.1, 1.8, 1.5, 1.2, 0.9, 0.6, 0.3, 0), "y" = c(0, -0.216, -0.288, -0.252, -0.144, 0, 0.144, 0.252, 0.288, 0.216, 0) ), class = "data.frame", row.names = c(NA, -11L) ) ) expect_warning( bezierFunction(points), "'bezierFunction' is deprecated.", fixed = TRUE ) }) mpoly/tests/testthat/test-eq_mp.R0000644000176200001440000000062613476534026016633 0ustar liggesuserscontext("eq_mp()") test_that("eq_mp() properly deals with spaces", { expect_equal(eq_mp("y = x"), mp("y - x")) expect_equal(eq_mp("y =x"), mp("y - x")) }) test_that("eq_mp() works with = or ==", { expect_equal(eq_mp("y = x"), eq_mp("y == x")) }) test_that("eq_mp() is properly vectorized", { expect_equal( eq_mp(c("y = x", "x^2 + y^2 = 1")), mp(c("y - x", "x^2 + y^2 - 1")) ) }) mpoly/tests/testthat/test-burst.R0000644000176200001440000000063013476534026016664 0ustar liggesuserscontext("burst()") test_that("basic burst works", { expect_equal( burst(2), structure(c(0, 2, 1, 2, 0, 1), .Dim = 3:2) ) }) test_that("bursts of a particular length are computed correctly", { expect_equal( burst(3, 2), structure(c(0, 3, 1, 2, 3, 0, 2, 1), .Dim = c(4L, 2L)) ) }) test_that("bursts of length one is itself", { expect_equal(burst(3, 1), 3L) }) mpoly/tests/testthat/test-deriv.R0000644000176200001440000000135413476534026016642 0ustar liggesuserscontext("deriv()") test_that("basic", { expect_equal( deriv(mp("x y + y z + z^2"), "x"), mp("y") ) expect_equal( deriv(mp("x y + y z + z^2"), "y"), mp("x + z") ) expect_equal( deriv(mp("x y + y z + z^2"), "z"), mp("y + 2 z") ) expect_equal( deriv(mp("x y + y z + z^2"), "t"), mp("0") ) expect_error( deriv(mp("x y + y z + z^2")), "var must be specified, see ?deriv.mpoly", fixed = TRUE ) }) test_that("vector indeterminates", { expect_equal( deriv(mp("x y + y z + z^2"), c("x", "y")), mp(c("y", "x + z")) ) }) test_that("gradient", { expect_equal( gradient(mp("x y + y z + z^2")), mp(c("y", "x + z", "y + 2 z")) ) })mpoly/tests/testthat/test-mpolyArithmetic.R0000644000176200001440000000302013476534026020673 0ustar liggesuserscontext("mpoly arithmetic") test_that("Addition works", { expect_equal(0 + mp("1"), mp("1")) expect_equal(0 + mp("x"), mp("x")) expect_equal(1 + mp("1"), mp("2")) expect_equal(1 + mp("x"), mp("x + 1")) expect_equal(-1 + mp("1"), mp("0")) expect_equal(-1 + mp("x"), mp("-1 + x")) expect_equal(mp("1") + 0, mp("1")) expect_equal(mp("x") + 0, mp("x")) expect_equal(mp("1") + 1, mp("2")) expect_equal(mp("x") + 1, mp("x + 1")) expect_equal(mp("1") + -1, mp("0")) expect_equal(mp("x") + -1, mp("x - 1")) }) test_that("Subtraction works", { expect_equal(0 - mp("1"), mp("-1")) expect_equal(0 - mp("x"), mp("-x")) expect_equal(1 - mp("1"), mp("0")) expect_equal(1 - mp("x"), mp("1 - x")) expect_equal(-1 - mp("1"), mp("-2")) expect_equal(-1 - mp("x"), mp("-1 - x")) expect_equal(mp("1") - 0, mp("1")) expect_equal(mp("x") - 0, mp("x")) expect_equal(mp("1") - 1, mp("0")) expect_equal(mp("x") - 1, mp("x - 1")) expect_equal(mp("1") - -1, mp("2")) expect_equal(mp("x") - -1, mp("x + 1")) }) test_that("Multiplication works", { expect_equal(0 * mp("1"), mp("0")) expect_equal(0 * mp("x"), mp("0")) expect_equal(1 * mp("1"), mp("1")) expect_equal(1 * mp("x"), mp("x")) expect_equal(-1 * mp("1"), mp("-1")) expect_equal(-1 * mp("x"), mp("-x")) expect_equal(mp("1") * 0, mp("0")) expect_equal(mp("x") * 0, mp("0")) expect_equal(mp("1") * 1, mp("1")) expect_equal(mp("x") * 1, mp("x")) expect_equal(mp("1") * -1, mp("-1")) expect_equal(mp("x") * -1, mp("-x")) })mpoly/tests/testthat/test-swap.R0000644000176200001440000000137513476534026016506 0ustar liggesuserscontext("swap()") test_that("swap() works for univariate polynomials", { expect_equal( swap(mp("(x + y)^2"), "x", "t"), mp("(t + y)^2") ) expect_equal( swap(mp("(x + y)^2"), "x", "t_1"), mp("(t_1 + y)^2") ) }) test_that("swap() works for bernstein polynomials", { expect_equal( swap(bernstein(3, 5), "x", "t"), bernstein(3, 5, indeterminate = "t") ) }) test_that("swap() works for chebyshev polynomials", { expect_equal( swap(chebyshev(4), "x", "t"), chebyshev(4, indeterminate = "t") ) }) test_that("swap() correctly errors", { expect_error(swap(mp("x^2 + 1"), "x", "x^2")) expect_error(swap(mp("x^2 + 1"), "x", "x 2")) expect_error(swap(mp("x + y"), "x", "y")) }) mpoly/tests/testthat/test-partitions.R0000644000176200001440000000070213476534026017721 0ustar liggesuserscontext("partitions()") test_that("basic partitions() functionality works", { expect_equal( partitions(3), structure(c(3L, 2L, 1L, 0L, 1L, 1L, 0L, 0L, 1L), .Dim = c(3L, 3L)) ) }) test_that("vectorized", { expect_equal( partitions(1:3), list( structure(1L, .Dim = c(1L, 1L)), structure(c(2L, 1L, 0L, 1L), .Dim = c(2L, 2L)), structure(c(3L, 2L, 1L, 0L, 1L, 1L, 0L, 0L, 1L), .Dim = c(3L, 3L)) ) ) }) mpoly/tests/testthat/test-legendre.R0000644000176200001440000000407513476534026017321 0ustar liggesuserscontext("legendre()") test_that("basic", { expect_equal( legendre(3), structure( list( c("x" = 1, "coef" = -1.5), c("x" = 3, "coef" = 2.5) ), "class" = c("legendre", "mpoly"), "legendre" = list( "degree" = 3, "indeterminate" = "x", "normalized" = FALSE ) ) ) }) test_that("basic", { expect_equal( legendre(3, indeterminate = "t"), structure( list( c("t" = 1, "coef" = -1.5), c("t" = 3, "coef" = 2.5) ), "class" = c("legendre", "mpoly"), "legendre" = list( "degree" = 3, "indeterminate" = "t", "normalized" = FALSE ) ) ) }) test_that("normalized", { expect_equal( legendre(3, normalized = TRUE), structure( list( c("x" = 1, "coef" = -2.806243), c("x" = 3, "coef" = 4.677072) ), "class" = c("legendre", "mpoly"), "legendre" = list( "degree" = 3, "indeterminate" = "x", "normalized" = TRUE ) ), tolerance = 1e-5 ) }) test_that("vectorized", { expect_equal( legendre(0:2), structure( list( structure( list( c("coef" = 1) ), "class" = c("legendre", "mpoly"), "legendre" = list( "degree" = 0, "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("x" = 1, "coef" = 1) ), "class" = c("legendre", "mpoly"), "legendre" = list( "degree" = 1, "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("coef" = -.5), c("x" = 2, "coef" = 1.5) ), "class" = c("legendre", "mpoly"), "legendre" = list( "degree" = 2, "indeterminate" = "x", "normalized" = FALSE ) ) ), class = "mpolyList" ) ) }) mpoly/tests/testthat/test-homogenize.R0000644000176200001440000000250513476534026017674 0ustar liggesuserscontext("homogenizing functions") test_that("is.homogeneous()", { expect_false( is.homogeneous(mp("x^2 + y")) ) expect_true( is.homogeneous(mp("x^2 + y^2")) ) expect_equal( is.homogeneous(mp(c("x + y", "x + y^2"))), c(TRUE, FALSE) ) }) test_that("homogenize()", { expect_equal( homogenize(mp("x^3 + y")), mp("x^3 + y t^2") ) expect_equal( homogenize(mp("x^3 + y"), indeterminate = "z"), mp("x^3 + y z^2") ) expect_equal( homogenize(mp(c("x^3 + y", "x + 1"))), mp(c("x^3 + y t^2", "x + t")) ) expect_error( homogenize("x + x^2"), "homogenize requires mpoly objects.", fixed = TRUE ) }) test_that("dehomogenize()", { expect_equal( dehomogenize(mp("x^3 + y t^2"), indeterminate = "t"), mp("x^3 + y") ) expect_equal( dehomogenize(mp(c("x^2 + y t", "x t^2"))), mp(c("x^2 + y", "x")) ) expect_error( dehomogenize(mp("x^3 + y t"), indeterminate = "t"), "polynomial x^3 + y t is not homogeneous.", fixed = TRUE ) expect_error( dehomogenize("x t + x^2"), "dehomogenize requires mpoly objects.", fixed = TRUE ) }) test_that("homogeneous_components()", { expect_equal( homogeneous_components(mp("1 + x + x^2 + x y + y")), mp(c("1", "x + y", "x^2 + x y")) ) }) mpoly/tests/testthat/test-print-mpolylist.R0000644000176200001440000000235513476534026020721 0ustar liggesuserscontext("print.mpolyList()") test_that("basic print.mpolyList() functionality", { expect_equal( print(mp(c("x + 1", "y^2 - 2")), silent = TRUE), c("x + 1", "y^2 - 2") ) }) test_that("plus_pad works", { expect_equal( print(mp(c("x + 1", "y^2 - 2")), silent = TRUE, plus_pad = 1L), c("x + 1", "y^2 - 2") ) }) test_that("stars works", { expect_equal( print(mp(c("x z + 1", "y^2 - 2")), silent = TRUE, stars = TRUE), c("x * z + 1", "y**2 - 2") ) }) test_that("times_pad works", { expect_equal( print(mp(c("x z + 1", "y^2 - 2")), silent = TRUE, stars = TRUE, times_pad = 0L), c("x*z + 1", "y**2 - 2") ) }) test_that("errors correctly", { expect_error( print(mp(c("x z + 1", "y^2 - 2")), silent = TRUE, varorder = c("x", "y")), "If specified, varorder must contain all computed vars - x, z, y" ) }) test_that("messages correctly", { expect_message( print(mp(c("x z + 1", "y^2 - 2")), silent = TRUE, order = "lex"), "Using variable ordering - x, z, y" ) }) test_that("silent is coded correctly", { expect_equal( capture.output(print(mp(c("x z + 1", "y^2 - 2")), silent = FALSE)), c("x z + 1", "y^2 - 2") ) }) mpoly/tests/testthat/test-mpoly-equal.R0000644000176200001440000000074613476534026020002 0ustar liggesuserscontext("mpoly == mpoly") test_that("basic", { expect_true(mp("x + y^2") == mp("x + y^2")) expect_false(mp("x + y^2") == mp("x")) expect_error( mp("2") == 2, "e1 and e2 must be of class mpoly.", fixed = TRUE ) }) context("mpoly != mpoly") test_that("basic", { expect_false(mp("x + y^2") != mp("x + y^2")) expect_true( mp("x + y^2") != mp("y^2")) expect_error( mp("2") != 2, "e1 and e2 must be of class mpoly.", fixed = TRUE ) }) mpoly/tests/testthat/test-laguerre.R0000644000176200001440000000535113476534026017340 0ustar liggesuserscontext("laguerre()") test_that("basic", { expect_equal( laguerre(2), structure( list( c("coef" = 1), c("x" = 1, "coef" = -2), c("x" = 2, "coef" = .5) ), "class" = c("laguerre", "mpoly"), "laguerre" = list( "degree" = 2, "alpha" = 0, "indeterminate" = "x", "normalized" = FALSE ) ) ) }) test_that("indeterminate", { expect_equal( laguerre(2, indeterminate = "t"), structure( list( c("coef" = 1), c("t" = 1, "coef" = -2), c("t" = 2, "coef" = .5) ), "class" = c("laguerre", "mpoly"), "laguerre" = list( "degree" = 2, "alpha" = 0, "indeterminate" = "t", "normalized" = FALSE ) ) ) }) test_that("alpha", { expect_equal( laguerre(2, alpha = 3), structure( list( c("coef" = 10), c("x" = 1, "coef" = -5), c("x" = 2, "coef" = .5) ), "class" = c("laguerre", "mpoly"), "laguerre" = list( "degree" = 2, "alpha" = 3, "indeterminate" = "x", "normalized" = FALSE ) ) ) }) test_that("normalized", { expect_equal( laguerre(2, alpha = 3, normalized = TRUE), structure( list( c("coef" = 1.290994), c("x" = 1, "coef" = -0.6454972), c("x" = 2, "coef" = 0.06454972) ), "class" = c("laguerre", "mpoly"), "laguerre" = list( "degree" = 2, "alpha" = 3, "indeterminate" = "x", "normalized" = TRUE ) ), tolerance = 1e-5 ) }) test_that("vectorized", { expect_equal( laguerre(0:2), structure( list( structure( list( c("coef" = 1) ), "class" = c("laguerre", "mpoly"), "laguerre" = list( "degree" = 2, "alpha" = 0, "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("coef" = 1), c("x" = 1, "coef" = -1) ), "class" = c("laguerre", "mpoly"), "laguerre" = list( "degree" = 2, "alpha" = 0, "indeterminate" = "x", "normalized" = FALSE ) ), structure( list( c("coef" = 1), c("x" = 1, "coef" = -2), c("x" = 2, "coef" = .5) ), "class" = c("laguerre", "mpoly"), "laguerre" = list( "degree" = 2, "alpha" = 0, "indeterminate" = "x", "normalized" = FALSE ) ) ), "class" = "mpolyList" ) ) })mpoly/tests/testthat/test-solve-unipoly.R0000644000176200001440000000115313476534026020353 0ustar liggesuserscontext("solve_unipoly()") test_that("solve_unipoly() works with real roots", { expect_equal( solve_unipoly(mp("x^2 - 1"), real_only = TRUE), c(1, -1) ) }) test_that("solve_unipoly() works generally", { expect_equal( solve_unipoly(mp("x^2 - 1")), c(1+0i, -1+0i) ) expect_equal( solve_unipoly(mp("x^2 - 2")), c(1.41421356237309-0i, -1.41421356237309+0i) ) expect_equal( solve_unipoly(mp("x^3 - 1")), c(1-0i, -0.5+0.866025403784439i, -0.5-0.866025403784439i) ) expect_equal( solve_unipoly(mp("x^4 - 1")), c(0+1i, -1-0i, 0-1i, 1+0i) ) })mpoly/tests/testthat/test-mpolylist-arithmetic.R0000644000176200001440000000732313476534026021716 0ustar liggesuserscontext("mpolyList arithmetic") test_that("addition", { expect_equal( mp(c("x", "y")) + mp(c("y^2", "x")), structure( list( structure( list( c(x = 1, coef = 1), c(y = 2, coef = 1) ), class = "mpoly" ), structure( list( c(y = 1, coef = 1), c(x = 1, coef = 1) ), class = "mpoly") ), class = "mpolyList" ) ) # expect_equal( # mp("2") + mp(c("x", "y")), # mp(c("x + 2", "y + 2")) # ) expect_equal( 2 + mp(c("x", "y")), mp(c("x + 2", "y + 2")) ) # expect_equal( # mp(c("x", "y")) + mp("2"), # mp(c("x + 2", "y + 2")) # ) expect_equal( mp(c("x", "y")) + 2, mp(c("x + 2", "y + 2")) ) expect_equal( c(2, -3) + mp(c("x", "y")), mp(c("x + 2", "y - 3")) ) expect_equal( mp(c("x", "y")) + c(2, -3), mp(c("x + 2", "y - 3")) ) expect_error( c(2, -3) + mp(c("x", "y", "z")), "e1 and e2 must have equal length.", fixed = TRUE ) expect_error( mp(c("x", "y", "z")) + c(2, -3), "e1 and e2 must have equal length.", fixed = TRUE ) }) test_that("subtraction", { expect_equal( mp(c("x", "y")) - mp(c("x", "x^2")), structure( list( structure(list(c(coef = 0)), class = "mpoly"), structure( list( c(y = 1, coef = 1), c(x = 2, coef = -1) ), class = "mpoly") ), class = "mpolyList" ) ) # expect_equal( # mp("2") - mp(c("x", "y")), # mp(c("-1 x + 2", "-1 y + 2")) # ) expect_equal( 2 - mp(c("x", "y")), mp(c("-1 x + 2", "-1 y + 2")) ) # expect_equal( # mp(c("x", "y")) - mp("2"), # mp(c("x - 2", "y - 2")) # ) expect_equal( mp(c("x", "y")) - 2, mp(c("x - 2", "y - 2")) ) expect_equal( c(2, -3) - mp(c("x", "y")), mp(c("-1 x + 2", "-1 y - 3")) ) expect_equal( mp(c("x", "y")) - c(2, -3), mp(c("x - 2", "y + 3")) ) expect_error( c(2, -3) - mp(c("x", "y", "z")), "e1 and e2 must have equal length.", fixed = TRUE ) expect_error( mp(c("x", "y", "z")) - c(2, -3), "e1 and e2 must have equal length.", fixed = TRUE ) }) test_that("multiplication", { expect_equal( mp(c("x", "y")) * mp(c("x", "x^2")), structure( list( structure(list(c(x = 2, coef = 1)), class = "mpoly"), structure(list(c(y = 1, x = 2, coef = 1)), class = "mpoly") ), class = "mpolyList" ) ) # expect_equal( # mp("2") * mp(c("x", "y")), # mp(c("2 x", "2 y")) # ) expect_equal( 2 * mp(c("x", "y")), mp(c("2 x", "2 y")) ) # expect_equal( # mp(c("x", "y")) * mp("2"), # mp(c("2 x", "2 y")) # ) expect_equal( mp(c("x", "y")) * 2, mp(c("2 x", "2 y")) ) expect_equal( c(2, -3) * mp(c("x", "y")), mp(c("2 x", "-3 y")) ) expect_error( c(2, -3) * mp(c("x", "y", "z")), "e1 and e2 must have equal length.", fixed = TRUE ) }) test_that("exponentiation", { expect_equal( mp(c("x", "y"))^2L, structure( list( structure(list(c(x = 2, coef = 1)), class = "mpoly"), structure(list(c(y = 2, coef = 1)), class = "mpoly") ), class = "mpolyList" ) ) expect_equal( mp(c("x", "y"))^0L, structure( list( structure(list(c(coef = 1)), class = "mpoly"), structure(list(c(coef = 1)), class = "mpoly") ), class = "mpolyList" ) ) expect_error( mp(c("x", "y"))^1.5, "exponent must be a positive integer.", fixed = TRUE ) }) mpoly/tests/testthat.R0000644000176200001440000000006612442331262014540 0ustar liggesuserslibrary(testthat) library(mpoly) test_check("mpoly") mpoly/NEWS0000644000176200001440000001457313476534026012135 0ustar liggesusersVersion 1.2.0 ------------------------------------------------------------------------- CHANGES * grobner is now truly defunct, pointing to m2r::gb. Version 1.1.0 ------------------------------------------------------------------------- CHANGES * Packages are now imported more nicely (@importFrom in all cases). * mp() is now based on R's parser and mpoly arithmetic. The performance boost varies with things like how many parenthetical expressions are used, but in basic cases ranged from one to a few orders of magnitude. More importantly, many hard to catch tokenizing problems should be gone. FIXES * A bug in as.function() that caused it to error on univariate polynomials with indeterminates with numbers in them (e.g. x1) is now fixed. (Thanks @Blaza) * A bug in mpoly/numeric arithmetic. Fixed by @Blaza (thanks!) and tests added. Version 1.0.5 ------------------------------------------------------------------------- NEW FEATURES * Add is.unipoly(), a predicate for univariate polynomials. * Add solve_unipoly(), a mpoly wrapper for polyroot(). * plug() now works on mpolyLists. CHANGES * as.function.mpoly() and as.function.mpolyList() no longer message when the resulting function arguments are clear. Version 1.0.4 ------------------------------------------------------------------------- FIXES * A bug in scientific notation causing mp() to hang has been fixed. (Thanks @sommars) * Move README from figures/ to tools/ * Add https:// for paper reference Version 1.0.3 ------------------------------------------------------------------------- NEW FEATURES * as.mpoly() methods for lm objects using poly(, raw = TRUE) statements now works properly. CHANGES * the print.mpoly and print.mpolyList methods now print with cat() instead of message(). they also accept a silent argument, in which case the output is suppressed; characters are returned silently. the argument order has changed so that stars follows varorder and order. FIXES * a stray browser() statement was removed from as.mpoly(). Version 1.0.2 ------------------------------------------------------------------------- NEW FEATURES * you can now pass * for multiplication in mp, so that mp("x*x") works properly. CHANGES * all java stuff has been removed; see the m2r package for grobner basis computations. Version 1.0.1 ------------------------------------------------------------------------- NEW FEATURES * mpoly now provided functions for extracting the leading term (LT), leading coefficient (LC), leading monomial (LM), and arbitrary term (as in p[3]). You can also compute the multidegree of a polynomial (multideg). * new function exponents extracts the exponents of an mpoly. * new functions homogenize, dehomogenize, homogeneous_components extract those components of a multivariate polynomial. * as.function() now returns vectorized functions when the polynomial(s) are only of one variable. CHANGES * terms.mpoly is now deprecated, use monomials. * reorder.mpoly now defaults to the lex order. Version 1.0.0 ------------------------------------------------------------------------- CHANGES * Versioned directly modulo very minor CRAN changes from 0.1.1 for CRAN consistency. Version 0.1.1 ------------------------------------------------------------------------- NEW FEATURES * A new function called plug allows users to plug values into indeterminates in mpoly objects. * mpoly now imports the polynom and orthopolynom packages, and has as.mpoly methods to parse their objects into mpoly objects. * mpoly now imports reshape2 for some visualization stuff. * A new function called chebyshev allows users to compute Chebyshev polynomials. This is based on orthopolynom. * New functions called bernstein and bernsteinApprox allow users to compute Bernstein polynomials and approximate functions with Bernstein polynomials, see examples. A special as.function method exists for evaluating these in a numerically stable way. * A new function called bezier allows users to compute the polynomials associated with Bezier curves. A special as.function method exists for evaluating these in a numerically stable way via de Casteljau's algorithm, based on a new function called bezierFunction (exported). * Several new predicate functions are exported. * ggplot2 is now imported. CHANGES * The internals of tuples have been changed, but the user shouldn't expect to see them. FIXES * A bug in mp functions where non-parenthetical exponents in parenthetical expressions were mis-parsed has been fixed. * round.mpoly now drops terms rounded to have 0 coefficients. * mpolyList now can be used inside functions and other environments (thanks @gitlongor) Version 0.1.0 ------------------------------------------------------------------------- NEW FEATURES * is.linear function. * round now has a method applicable to mpoly objects that rounds the coefficients. this is particularly nice in combination with as.mpoly.lm. * Hard braces are now allowed for variables, e.g. x[1,1] CHANGES * Vastly enhanced mp parser based on stringr 9.0.0 (now dependent on stringi). * Tests added. * Description file updated for CRAN's standards. Version 0.0.5 ------------------------------------------------------------------------- NEW FEATURES * Parenthetical expressions are now handled. CHANGES * Permutations now returns a better ordered listing of the permutations of a set. It also allows for the more flexible specification of (e.g.) permutations(5) for permutations(1:5). FIXES * Some negative parsing in mp has been fixed. Version 0.0.4 ------------------------------------------------------------------------- CHANGES * Division symbols in mp now return an error. * mpoly now has a GPL-2 license. FIXES * Fixed a glitch in print.mpoly that made mp("-1 + y") print as "-1 1 + y". * mpolyList now works on single elements, either a mpoly object (converting it to a mpolyList) or a call to mp. * mp now properly handles constants raised to powers. Version 0.0.3 ------------------------------------------------------------------------- FIXES * Fixed an issue with as.mpoly (and is.mpoly.lm). Version 0.0.2 ------------------------------------------------------------------------- NEW FEATURES * as.mpoly method now defined for lm objects which makes an mpoly object out of a linear model. Version 0.0.1 ------------------------------------------------------------------------- PACKAGE GENESIS mpoly/R/0000755000176200001440000000000013623306133011613 5ustar liggesusersmpoly/R/attach.r0000644000176200001440000000022113476534026013247 0ustar liggesusers.onAttach <- function(...) { if(!interactive()) return() packageStartupMessage(' Please cite mpoly! See citation("mpoly") for details.') }mpoly/R/LCM.R0000644000176200001440000000077113476534026012370 0ustar liggesusers#' Compute the least common multiple of two numbers. #' #' A simple algorithm to compute the least common multiple of two numbers #' #' @param x an object of class numeric #' @param y an object of class numeric #' @return The least common multiple of x and y. #' @export #' @examples #' LCM(5,7) #' LCM(5,8) #' LCM(5,9) #' LCM(5,10) #' Reduce(LCM, 1:10) # -> 2520 LCM <- function(x, y){ x0 <- x y0 <- y while(x != y){ if(x < y){ x <- x + x0 } else { y <- y + y0 } } x } mpoly/R/bernstein.R0000644000176200001440000000346613476534026013752 0ustar liggesusers#' Bernstein polynomials #' #' Bernstein polynomials #' #' @param k Bernstein polynomial k #' @param n Bernstein polynomial degree #' @param indeterminate indeterminate #' @return a mpoly object #' @author David Kahle #' @name bernstein #' @examples #' #' bernstein(0, 0) #' #' bernstein(0, 1) #' bernstein(1, 1) #' #' bernstein(0, 1, "t") #' #' bernstein(0:2, 2) #' bernstein(0:3, 3) #' bernstein(0:3, 3, "t") #' #' #' bernstein(0:4, 4) #' bernstein(0:10, 10) #' bernstein(0:10, 10, "t") #' bernstein(0:20, 20, "t") #' #' \dontrun{ # visualize the bernstein polynomials #' #' library(ggplot2); theme_set(theme_classic()) #' library(tidyr) #' #' s <- seq(0, 1, length.out = 101) #' N <- 10 # number of bernstein polynomials to plot #' (bernPolys <- bernstein(0:N, N)) #' #' df <- data.frame(s, as.function(bernPolys)(s)) #' names(df) <- c("x", paste0("B_", 0:N)) #' head(df) #' #' mdf <- gather(df, degree, value, -x) #' head(mdf) #' #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' } #' #' #' @rdname bernstein #' @export bernstein <- function(k, n, indeterminate = "x"){ stopifnot(k <= n) ## make it possible for vector k args if(length(k) > 1){ listOPolys <- lapply(k, function(.) bernstein(., n, indeterminate)) class(listOPolys) <- "mpolyList" return(listOPolys) } ## construct coefficients and degrees of terms m <- n - k coefs <- choose(n, k) * (-1)^(0:m) * choose(m, 0:m) degs <- k:n ## construct polynomial as list p <- Map(function(deg, coef) c(x = deg, coef = coef), degs, coefs) ## wipe out zeros p <- lapply(p, function(v) v[v != 0]) ## class list class(p) <- c("bernstein", "mpoly") attr(p, "bernstein") <- list(k = k, n = n, indeterminate = indeterminate) ## swap and return swap(p, "x", indeterminate) } mpoly/R/vars.R0000644000176200001440000000103213476534026012717 0ustar liggesusers#' Determine the variables in a mpoly object. #' #' Determine the variables in a mpoly object. #' #' @param p An mpoly or mpolyList object. #' @return A character vector of the variable names. #' @export #' @examples #' #' p <- mp("x + y^2") #' vars(p) #' #' p <- mp(c("x + y^2", "y - 2 x")) #' vars(p) #' vars <- function(p) { stopifnot(is.mpoly(p) || is.mpolyList(p)) flat_list <- unlist(p) flat_list <- flat_list[names(flat_list) != "coef"] flat_list <- unique(names(flat_list)) unique(gsub('\\^[0-9]?', '', flat_list)) } mpoly/R/bernstein_approx.R0000644000176200001440000000440213476534026015332 0ustar liggesusers#' Bernstein polynomial approximation #' #' Bernstein polynomial approximation #' #' @param f the function to approximate #' @param n Bernstein polynomial degree #' @param lower lower bound for approximation #' @param upper upper bound for approximation #' @param indeterminate indeterminate #' @param ... ... #' @return a mpoly object #' @author David Kahle #' @name bernstein-approx #' @examples #' #' #' #' #' #' \dontrun{ # visualize the bernstein polynomials #' #' library(ggplot2); theme_set(theme_bw()) #' library(reshape2) #' #' #' #' #' f <- function(x) sin(2*pi*x) #' p <- bernstein_approx(f, 20) #' round(p, 3) #' #' x <- seq(0, 1, length.out = 101) #' df <- data.frame( #' x = rep(x, 2), #' y = c(f(x), as.function(p)(x)), #' which = rep(c("actual", "approx"), each = 101) #' ) #' qplot(x, y, data = df, geom = "line", color = which) #' #' #' #' #' #' #' p <- bernstein_approx(sin, 20, pi/2, 1.5*pi) #' round(p, 4) #' #' x <- seq(0, 2*pi, length.out = 101) #' df <- data.frame( #' x = rep(x, 2), #' y = c(sin(x), as.function(p)(x)), #' which = rep(c("actual", "approx"), each = 101) #' ) #' qplot(x, y, data = df, geom = "line", color = which) #' #' #' #' #' #' #' #' #' p <- bernstein_approx(dnorm, 15, -1.25, 1.25) #' round(p, 4) #' #' x <- seq(-3, 3, length.out = 101) #' df <- data.frame( #' x = rep(x, 2), #' y = c(dnorm(x), as.function(p)(x)), #' which = rep(c("actual", "approx"), each = 101) #' ) #' qplot(x, y, data = df, geom = "line", color = which) #' #' #' #' #' #' #' } #' #' @rdname bernstein-approx #' @export bernstein_approx <- function(f, n, lower = 0, upper = 1, indeterminate = "x"){ ## compute support and determine weights s <- (0:n)/n fscaled <- function(.) f( (upper-lower)*. + lower ) weights <- as.list(fscaled(s)) ## convert weights to mpolyList weights <- lapply(weights, function(x) mpoly(list(c(coef = x)))) class(weights) <- "mpolyList" ## multiply weights by basis approxPoly <- Reduce(`+`, weights * bernstein(0:n, n, "temp")) ## compute plugin and plug in pluginPoly <- (upper-lower)^-1 * (mp(indeterminate) + -1*lower) plug(approxPoly, "temp", pluginPoly) } #' @rdname bernstein-approx #' @export bernsteinApprox <- function(...) { .Deprecated("bernstein_approx") bernstein_approx(...) }mpoly/R/print.mpolyList.R0000644000176200001440000000323313476534026015100 0ustar liggesusers#' Pretty printing of a list of multivariate polynomials. #' #' This function iterates print.mpoly on an object of class mpolyList. #' #' @inheritParams print.mpoly #' @return Invisible character vector of the printed objects. #' @export #' @examples #' #' mL <- mp(c("x + 1", "y - 1", "x y^2 z + x^2 z^2 + z^2 + x^3")) #' mL #' print(mL, order = "lex") #' print(mL, order = "glex") #' print(mL, order = "grlex") #' print(mL, order = "glex", varorder = c("z","y","x")) #' print(mL, order = "grlex", varorder = c("z","y","x")) #' print(mL, varorder = c("z","y","x")) #' #' (print(mL, varorder = c("z","y","x"), plus_pad = 1L, silent = TRUE)) #' #' (print(mL, silent = TRUE, stars = TRUE, plus_pad = 1L, times_pad = 0L)) #' print.mpolyList <- function(x, varorder = vars(x), stars = FALSE, order, silent = FALSE, ..., plus_pad = 2L, times_pad = 1L){ stopifnot(is.mpolyList(x)) n <- length(x) vars <- vars(x) if(!missing(varorder) && !all(vars %in% varorder)){ stop( stri_c("If specified, varorder must contain all computed vars - ", stri_c(vars, collapse = ", ")), call. = FALSE ) } if(missing(varorder) && !missing(order)){ message(stri_c('Using variable ordering - ', stri_c(vars, collapse = ', '))) } if(missing(order)){ polys <- vapply(x, print, character(1), varorder = varorder, stars = stars, silent = TRUE, ..., plus_pad = plus_pad, times_pad = times_pad) } else { polys <- vapply(x, print, character(1), varorder = varorder, order = order, stars = stars, silent = TRUE, ..., plus_pad = plus_pad, times_pad = times_pad) } if(!silent) cat(polys, sep = "\n") invisible(polys) } mpoly/R/tuples.R0000644000176200001440000000324713476534026013272 0ustar liggesusers#' Determine all n-tuples using the elements of a set. #' #' Determine all n-tuples using the elements of a set. This is really just a #' simple wrapper for expand.grid, so it is not optimized. #' #' @param set a set #' @param n length of each tuple #' @param repeats if set contains duplicates, should the result? #' @param list tuples as list? #' @return a matrix whose rows are the n-tuples #' @export #' @examples #' #' tuples(1:2, 3) #' tuples(1:2, 3, list = TRUE) #' #' apply(tuples(c("x","y","z"), 3), 1, paste, collapse = "") #' #' # multinomial coefficients #' r <- 2 # number of variables, e.g. x, y #' n <- 2 # power, e.g. (x+y)^2 #' apply(burst(n,r), 1, function(v) factorial(n)/ prod(factorial(v))) # x, y, xy #' mp("x + y")^n #' #' r <- 2 # number of variables, e.g. x, y #' n <- 3 # power, e.g. (x+y)^3 #' apply(burst(n,r), 1, function(v) factorial(n)/ prod(factorial(v))) #' mp("x + y")^n #' #' r <- 3 # number of variables, e.g. x, y, z #' n <- 2 # power, e.g. (x+y+z)^2 #' apply(burst(n,r), 1, function(v) factorial(n)/ prod(factorial(v))) # x, y, z, xy, xz, yz #' mp("x + y + z")^n #' #' tuples <- function(set, n = length(set), repeats = FALSE, list = FALSE){ ## determine how big the output will be r <- length(set) nCombos <- r^n ## expand.grid really does all the work in this function, so ## setup the interior part of expand.grid. ## sets4call looks like "set2, set2, set2" with n = 3 out <- do.call(expand.grid, replicate(n, set, simplify = FALSE)) out <- unname(as.matrix(out)) out <- out[,n:1] ## delete duplicates if(!repeats) out <- unique(out) ## do list if(list) out <- unname(split(out, rep(1:nCombos, n))) ## return out } mpoly/R/as.mpoly.R0000644000176200001440000001014413602203275013500 0ustar liggesusers#' Convert an object to an mpoly #' #' mpoly is the most basic function used to create objects of class mpoly. #' #' @param x an object #' @param ... additional arguments to pass to methods #' @return the object formated as a mpoly object. #' @author David Kahle \email{david@@kahle.io} #' @seealso [mp()] #' @export #' @examples #' #' library(ggplot2); theme_set(theme_classic()) #' library(dplyr) #' #' n <- 101 #' s <- seq(-5, 5, length.out = n) #' #' #' #' # one dimensional case #' #' df <- data.frame(x = seq(-5, 5, length.out = n)) %>% #' mutate(y = -x^2 + 2*x - 3 + rnorm(n, 0, 2)) #' #' (mod <- lm(y ~ x + I(x^2), data = df)) #' (p <- as.mpoly(mod)) #' qplot(x, y, data = df) + #' stat_function(fun = as.function(p), colour = "red", size = 1) #' #' (mod <- lm(y ~ poly(x, 2, raw = TRUE), data = df)) #' (p <- as.mpoly(mod)) #' qplot(x, y, data = df) + #' stat_function(fun = as.function(p), colour = "red", size = 1) #' #' (mod <- lm(y ~ poly(x, 1, raw = TRUE), data = df)) #' (p <- as.mpoly(mod)) #' qplot(x, y, data = df) + #' stat_function(fun = as.function(p), colour = "red", size = 1) #' #' #' #' # two dimensional case with ggplot2 #' #' df <- expand.grid(x = s, y = s) %>% #' mutate(z = x^2 - y^2 + 3*x*y + rnorm(n^2, 0, 3)) #' qplot(x, y, data = df, geom = "raster", fill = z) #' #' (mod <- lm(z ~ x + y + I(x^2) + I(y^2) + I(x*y), data = df)) #' (mod <- lm(z ~ poly(x, y, degree = 2, raw = TRUE), data = df)) #' (p <- as.mpoly(mod)) #' df$fit <- apply(df[,c("x","y")], 1, as.function(p)) #' #' qplot(x, y, data = df, geom = "raster", fill = fit) #' #' qplot(x, y, data = df, geom = "raster", fill = z - fit) # residuals #' #' #' #' #' #' #' as.mpoly <- function(x, ...) UseMethod("as.mpoly") #' @export as.mpoly.default <- function(x, ...) { stop(sprintf("objects of class %s not supported by as.mpoly().", class(x)), call. = FALSE) } #' @export as.mpoly.lm <- function(x, ...){ coefs <- coef(x) coef_names <- names(coefs) coef_names[coef_names == "(Intercept)"] <- 1 # check for "poly(x, 2)1" syntax if(any(str_detect(coef_names, "poly"))) { poly_ndcs <- which(str_detect(coef_names, "poly")) coef_names[poly_ndcs] <- vapply(coef_names[poly_ndcs], parse_model_poly, character(1)) mp_str <- paste(coefs, coef_names, sep = " ", collapse = " + ") } else { # I() notation I_ndcs <- which(str_detect(coef_names, "I([0-9a-zA-Z]*)")) if(length(I_ndcs) > 0){ coef_names[I_ndcs] <- vapply( coef_names[I_ndcs], function(s) str_sub(s, 3, -2), character(1) ) } coef_names <- str_replace_all(coef_names, " \\* ", " ") mp_str <- paste(coefs, coef_names, sep = " ", collapse = " + ") } mp(mp_str) } # s <- "poly(x, y, degree = 2, raw = TRUE)2.0" # parse_model_poly(s) parse_model_poly <- function(s) { # grab vars; check for raw inside <- str_extract_all(s, "poly\\(.+\\)")[[1]] inside <- str_sub(inside, 6, -2) inside <- str_split(inside, ", ")[[1]] if(!any(inside == "raw = TRUE")) { stop("poly() statements currently must contain raw = TRUE.", call. = FALSE) } vars <- inside[!(str_detect(inside, "=") | !str_detect(inside, "[:alpha:]"))] # parse exponents if(str_sub(s, -1) == ")") { exponents <- 1 } else { exponents <- str_sub(str_extract_all(s, "\\).+")[[1]], 2) if(length(exponents) == 0) exponent <- "1" exponents <- str_split(exponents, "\\.")[[1]] } # put together paste(paste(vars, exponents, sep = "^"), collapse = " ") } #' @export as.mpoly.polynomial <- function(x, indeterminate = "x", ...){ as.mpoly.numeric(unclass(x), indeterminate) } #' @export as.mpoly.numeric <- function(x, indeterminate = "x", ...){ n <- length(x) ## make list and populate p <- list() p[[1]] <- c(coef = x[1]) if(n > 1) for(deg in 1:(n-1)){ v <- c(deg, x[deg+1]) names(v) <- c(indeterminate, "coef") p[[deg+1]] <- v } ## clean out zeros p <- Filter(function(v) v[["coef"]] != 0, p) ## clean in case everything is removed if(length(p) == 0) p <- list(c(coef = 0)) ## class and out class(p) <- "mpoly" p } mpoly/R/components.R0000644000176200001440000000535713623301456014140 0ustar liggesusers#' Polynomial components #' #' Compute quantities/expressions related to a multivariate polynomial. #' #' @param x an object of class mpoly #' @param ndx a subsetting index #' @param varorder the order of the variables #' @param order a total order used to order the terms #' @param reduced if TRUE, don't include zero degrees #' @return An object of class mpoly or mpolyList, depending on the context #' @name components #' @examples #' (p <- mp("x y^2 + x (x+1) (x+2) x z + 3 x^10")) #' p[2] #' p[-2] #' p[2:3] #' #' LT(p) #' LC(p) #' LM(p) #' #' multideg(p) #' totaldeg(p) #' monomials(p) #' #' exponents(p) #' exponents(p, reduce = TRUE) #' lapply(exponents(p), is.integer) #' #' homogeneous_components(p) #' #' @rdname components #' @export `[.mpoly` <- function(x, ndx){ x <- unclass(x)[ndx] class(x) <- "mpoly" x } #' @rdname components #' @export LT <- function(x, varorder = vars(x), order = "lex"){ reorder.mpoly(x, varorder, order)[1] } #' @rdname components #' @export LC <- function(x, varorder = vars(x), order = "lex"){ lt <- LT(x, varorder, order) unname(lt[[1]]["coef"]) } #' @rdname components #' @export LM <- function(x, varorder = vars(x), order = "lex"){ lt <- LT(x, varorder, order) lt[[1]]["coef"] <- 1 lt } #' @rdname components #' @export multideg <- function(x, varorder = vars(x), order = "lex"){ if(is.constant(x)) return(0L) lt <- LT(x, varorder, order) coef_ndx <- which(names(lt[[1]]) == "coef") sparse_multideg <- lt[[1]][-coef_ndx] lt_vars <- names(sparse_multideg) nvars <- length(varorder) mdeg <- rep(0L, length = nvars) names(mdeg) <- varorder for (k in 1:length(lt_vars)) { mdeg[lt_vars[k]] <- mdeg[lt_vars[k]] + sparse_multideg[k] } mdeg } #' @rdname components #' @export totaldeg <- function(x){ if(!is.mpoly(x) && length(x) > 1){ return(vapply(x, totaldeg, integer(1))) } if(!is.mpoly(x)) stop("totaldeg requires an mpoly or mpolyList object.", call. = FALSE) max(vapply(exponents(x), sum, integer(1))) } #' @rdname components #' @export monomials <- function(x){ if(!is.mpoly(x)) stop("monomials requires an mpoly or mpolyList object.", call. = FALSE) xs <- lapply(1:length(x), function(ndx) `[.mpoly`(x,ndx)) class(xs) <- "mpolyList" xs } #' @rdname components #' @export exponents <- function(x, reduced = FALSE){ l <- lapply(x, function(term){ fixed_term <- as.integer(term[-which(names(term) == "coef")]) names(fixed_term) <- names(term[-which(names(term) == "coef")]) fixed_term }) if(reduced) return(l) v <- vars(x) p <- length(v) tmp <- rep.int(0L, p) names(tmp) <- v lapply(l, function(exp){ tmp <- tmp + exp[v] tmp[is.na(tmp)] <- 0 tmp <- as.integer(tmp) names(tmp) <- v tmp }) }mpoly/R/hermite.R0000644000176200001440000000562013527033474013406 0ustar liggesusers#' Hermite polynomials #' #' Hermite polynomials as computed by orthopolynom. #' #' @param degree degree of polynomial #' @param kind "he" (default, probabilists', see Wikipedia article) or "h" #' (physicists) #' @param indeterminate indeterminate #' @param normalized provide normalized coefficients #' @return a mpoly object or mpolyList object #' @author David Kahle calling code from the orthopolynom package #' @seealso [orthopolynom::hermite.h.polynomials()], #' [orthopolynom::hermite.he.polynomials()], #' \url{http://en.wikipedia.org/wiki/Hermite_polynomials} #' @export #' @examples #' #' hermite(0) #' hermite(1) #' hermite(2) #' hermite(3) #' hermite(4) #' hermite(5) #' hermite(6) #' hermite(10) #' #' hermite(0:5) #' hermite(0:5, normalized = TRUE) #' hermite(0:5, indeterminate = "t") #' #' #' #' # visualize the hermite polynomials #' #' library(ggplot2); theme_set(theme_classic()) #' library(tidyr) #' #' s <- seq(-3, 3, length.out = 201) #' N <- 5 # number of hermite polynomials to plot #' (hermPolys <- hermite(0:N)) #' #' # see ?bernstein for a better understanding of #' # how the code below works #' #' df <- data.frame(s, as.function(hermPolys)(s)) #' names(df) <- c("x", paste0("T_", 0:N)) #' mdf <- gather(df, degree, value, -x) #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' #' # hermite polynomials are orthogonal with respect to the gaussian kernel: #' He2 <- as.function(hermite(2)) #' He3 <- as.function(hermite(3)) #' He4 <- as.function(hermite(4)) #' #' w <- dnorm #' integrate(function(x) He2(x) * He3(x) * w(x), lower = -Inf, upper = Inf) #' integrate(function(x) He2(x) * He4(x) * w(x), lower = -Inf, upper = Inf) #' integrate(function(x) He3(x) * He4(x) * w(x), lower = -Inf, upper = Inf) #' #' #' hermite <- function(degree, kind = "he", indeterminate = "x", normalized = FALSE){ stopifnot(all(is.wholenumber(degree))) stopifnot(all(degree >= 0)) ## deal with kind stopifnot(kind %in% c("h","he")) ## make coefs if(kind == "h") coefs <- hermite.h.polynomials(max(degree), normalized) if(kind == "he") coefs <- hermite.he.polynomials(max(degree), normalized) ## if only one degree is wanted, return that if(length(degree) == 1){ coefs <- rev.default(coefs)[[1]] p <- as.mpoly.polynomial(coefs, indeterminate) class(p) <- c("hermite", "mpoly") attr(p, "hermite") <- list( degree = length(polynomial)-1, kind = kind, indeterminate = indeterminate, normalized = normalized ) return(p) } ## if several are wanted, return them coefs <- coefs[degree+1] ps <- lapply(coefs, function(polynomial){ p <- as.mpoly.polynomial(polynomial, indeterminate) class(p) <- c("hermite", "mpoly") attr(p, "hermite") <- list( degree = length(polynomial)-1, kind = kind, indeterminate = indeterminate, normalized = normalized ) p }) class(ps) <- "mpolyList" ps } mpoly/R/predicates.R0000644000176200001440000000443213476534026014076 0ustar liggesusers#' mpoly predicate functions #' #' Various functions to deal with mpoly and mpolyList objects. #' #' @param x object to be tested #' @return Vector of logicals. #' @name predicates #' @examples #' #' p <- mp("5") #' is.mpoly(p) #' is.constant(p) #' #' is.constant(mp(c("x + 1", "7", "y - 2"))) #' #' p <- mp("x + y") #' is.mpoly(p) #' #' is.mpolyList(mp(c("x + 1", "y - 1"))) #' #' #' #' is.linear(mp("0")) #' is.linear(mp("x + 1")) #' is.linear(mp("x + y")) #' is.linear(mp(c("0", "x + y"))) #' #' is.linear(mp("x + x y")) #' is.linear(mp(c("x + x y", "x"))) #' #' #' (p <- bernstein(2, 5)) #' is.mpoly(p) #' is.bernstein(p) #' #' (p <- chebyshev(5)) #' is.mpoly(p) #' is.chebyshev(p) #' str(p) #' @export #' @rdname predicates is.constant <- function(x){ if (is.mpoly(x) && length(x) == 1 && length(x[[1]]) == 1) return(TRUE) if (is.mpolyList(x)) return(vapply(x, is.constant, logical(1))) FALSE } #' @export #' @rdname predicates is.mpoly <- function(x) inherits(x, "mpoly") #' @export #' @rdname predicates is.unipoly <- function(x) { if(is.mpolyList(x)) return(vapply(x, is.unipoly, logical(1))) length(vars(x)) <= 1 } # is.unipoly(mp("1")) # is.unipoly(mp("x+1")) # is.unipoly(mp("x+1+x^2")) # is.unipoly(mp(c("x+1+x^2", "y"))) # is.unipoly(mp(c("x+1+x^2", "y", "x+y"))) #' @export #' @rdname predicates is.bernstein <- function(x) inherits(x, "bernstein") #' @export #' @rdname predicates is.bezier <- function(x) inherits(x, "bezier") #' @export #' @rdname predicates is.chebyshev <- function(x) inherits(x, "chebyshev") #' @export #' @rdname predicates is.mpolyList <- function(x) inherits(x, "mpolyList") #' @export #' @rdname predicates is.linear <- function(x){ stopifnot(is.mpoly(x) || is.mpolyList(x)) if(is.mpolyList(x)) return(vapply(x, is.linear, logical(1))) all( vapply(x, function(term) (length(term) <= 2L) && all(rev(term)[-1] <= 1L), logical(1)) ) } #' Test whether an object is a whole number #' #' Test whether an object is a whole number. #' #' @param x object to be tested #' @param tol tolerance within which a number is said to be whole #' @return Vector of logicals. #' @export #' @examples #' is.wholenumber(seq(-3,3,.5)) is.wholenumber <- function (x, tol = .Machine$double.eps^0.5){ abs(x - round(x)) < tol } mpoly/R/reorder.mpoly.R0000644000176200001440000001144513476534026014556 0ustar liggesusers#' Reorder a multivariate polynomial. #' #' This function is used to set the intrinsic order of a multivariate #' polynomial. It is used for both the in-term variables and the #' terms. #' #' @param x an object of class mpoly #' @param varorder the order of the variables #' @param order a total order used to order the terms, \code{"lex"}, \code{"glex"}, or \code{"grlex"} #' @param ... additional arguments #' @usage \method{reorder}{mpoly}(x, varorder = vars(x), order, ...) #' @return An object of class mpoly. #' @export #' @examples #' #' list <- list( #' c(x = 1, y = 2, z = 1, coef = 1), #' c(x = 2, y = 0, z = 2, coef = 1), #' c(x = 0, y = 0, z = 2, coef = 1), #' c(x = 3, y = 0, z = 0, coef = 1) #' ) #' (p <- mpoly(list)) # -> x y^2 z + x^2 z^2 + z^2 + x^3 #' reorder(p) # -> x y^2 z + x^2 z^2 + z^2 + x^3 #' reorder(p, varorder = c("x","y","z"), order = "lex") #' # -> x^3 + x^2 z^2 + x y^2 z + z^2 #' reorder(p, varorder = c("x","y","z"), order = "glex") #' # -> x^2 z^2 + x y^2 z + x^3 + z^2 #' reorder(p, varorder = c("x","y","z"), order = "grlex") #' # -> x y^2 z + x^2 z^2 + x^3 + z^2 #' #' reorder(mp("x + 1"), varorder = c("y","x","z"), order = "lex") #' reorder(mp("x + y"), varorder = c("y","x","z"), order = "lex") #' reorder(mp("x y + y + 2 x y z^2"), varorder = c("y","x","z")) #' reorder(mp("x^2 + y x + y"), order = "lex") #' #' reorder.mpoly <- function(x, varorder = vars(x), order = "lex", ...){ ## set basic quantities vars <- vars(x) p <- length(vars) n <- length(x) ## argument checking # if(missing(varorder) && missing(order)) return(x) if(!missing(varorder)){ if(!all(vars %in% varorder)){ error <- stri_c( "If specified, varorder must contain all computed vars - ", paste(vars, collapse = ", ") ) stop(error, call. = FALSE) } # reduce varorder to size of vars varorder <- varorder[varorder %in% vars] ## reorder in-term variables x <- lapply(x, function(v){ if(length(v) == 1) return(v) # constant case v <- v[intersect(c(varorder, "coef"), names(v))] }) class(x) <- "mpoly" } if(missing(varorder) && !missing(order)){ message <- stri_c( "Using variable ordering - ", paste(vars, collapse = ", ") ) message(message) } ## argument check - order match.arg(order, c("lex","glex","grlex")) ## order == "lex" if(order == "lex"){ if(n == 1) return(x) # single term polynomial # add zeros l <- lapply(x, function(v){ z <- rep(0, p + 1) names(z) <- c(varorder, "coef") z[names(v)] <- v z }) # construct matrix and sort appropriately m <- matrix(unname(unlist(l)), nrow = length(l), ncol = p + 1, byrow = TRUE) dimnames(m) <- list(1:nrow(m), c(varorder, "coef")) for(k in p:1) m <- m[order(m[,k], decreasing = TRUE),] # split into list and add names list4mpoly <- unname(lapply(split(m, 1:n), function(v){ names(v) <- c(varorder, "coef") v })) # mpoly it and return return( mpoly(list4mpoly, varorder = varorder) ) } ## order == "glex" if(order == "glex"){ if(n == 1) return(x) # single term polynomial # add zeros l <- lapply(x, function(v){ z <- rep(0, p + 1) names(z) <- c(varorder, "coef") z[names(v)] <- v z }) # construct matrix and sort appropriately m <- matrix(unname(unlist(l)), nrow = length(l), ncol = p + 1, byrow = TRUE) dimnames(m) <- list(1:nrow(m), c(varorder, "coef")) for(k in p:1) m <- m[order(m[,k], decreasing = TRUE),] m <- m[order(apply(m[, 1:p, drop = FALSE],1,sum), decreasing = TRUE),] # split into list and add names list4mpoly <- unname(lapply(split(m, 1:n), function(v){ names(v) <- c(varorder, "coef") v })) # mpoly it and return return( mpoly(list4mpoly, varorder = varorder) ) } ## order == "grlex" if(order == "grlex"){ if(n == 1) return(x) # single term polynomial # add zeros l <- lapply(x, function(v){ z <- rep(0, p + 1) names(z) <- c(varorder, "coef") z[names(v)] <- v z }) # construct matrix and sort appropriately m <- matrix(unname(unlist(l)), nrow = length(l), ncol = p + 1, byrow = TRUE) dimnames(m) <- list(1:nrow(m), c(varorder, "coef")) for(k in 1:p) m <- m[order(m[,k]),] m <- m[order(apply(m[, 1:p, drop = FALSE],1,sum), decreasing = TRUE),] # split into list and add names list4mpoly <- unname(lapply(split(m, 1:n), function(v){ names(v) <- c(varorder, "coef") v })) # mpoly it and return return( mpoly(list4mpoly, varorder = varorder) ) } } mpoly/R/permutations.R0000644000176200001440000000366613476534026014515 0ustar liggesusers#' Determine all permutations of a set. #' #' An implementation of the Steinhaus-Johnson-Trotter permutation algorithm. #' #' @param set a set #' @return a matrix whose rows are the permutations of set #' @export #' @examples #' permutations(1:3) #' permutations(c('first','second','third')) #' permutations(c(1,1,3)) #' apply(permutations(letters[1:6]), 1, paste, collapse = '') permutations <- function(set){ r <- length(set) if(r == 1 && is.numeric(set)) return(permutations(1:set)) row2diag <- function(row, direction){ np1 <- length(row) + 1 mat <- matrix(nrow = np1, ncol = np1) if(direction == -1) for(k in 1:np1) mat[k,] <- insert(np1, np1-k+1, row) if(direction == +1) for(k in 1:np1) mat[k,] <- insert(np1, k, row) mat } # row2diag(matrix(c(1,2,2,1),2,2)[,1], -1) # row2diag(matrix(c(1,2,2,1),2,2)[,1], +1) stepUp <- function(mat){ # an r x # matrix c <- ncol(mat) m <- NULL for(k in 1:nrow(mat)) m <- rbind(m, row2diag(mat[k,],(-1)^k)) m } # stepUp(matrix(1)) # stepUp(matrix(c(1,2,2,1),2,2)) # iterate stepUp out <- matrix(1) for(k in 1:(r-1)) out <- stepUp(out) # substitute set values for(k in 1:r) out[out==k] <- paste('_', set[k], sep = '') out <- gsub('_', '', out) # clear PH if(is.numeric(set)){ d <- dim(out) out <- as.numeric(out) dim(out) <- d } # reorder outcomes out <- out[do.call("order", split(out, col(out))),] # return unique(out) } #' Insert an element into a vector. #' #' Insert an element into a vector. #' #' @param elem element to insert #' @param slot location of insert #' @param v vector to insert into #' @return vector with element inserted #' @export #' @examples #' insert(2, 1, 1) #' insert(2, 2, 1) #' insert('x', 5, letters) insert <- function(elem, slot, v){ n <- length(v) if(slot == 1) return( c(elem, v) ) if(slot == n+1) return( c(v, elem) ) c(v[1:(slot-1)], elem, v[slot:n]) }mpoly/R/chebyshev.R0000644000176200001440000001014713531501672013724 0ustar liggesusers#' Chebyshev polynomials #' #' Chebyshev polynomials as computed by orthopolynom. #' #' @param degree degree of polynomial #' @param kind \code{"t"} or \code{"u"} (Chebyshev polynomials of the first and #' second kinds), or \code{"c"} or \code{"s"} #' @param indeterminate indeterminate #' @param normalized provide normalized coefficients #' @param k,n the k'th root of the n'th chebyshev polynomial #' @return a mpoly object or mpolyList object #' @author David Kahle calling code from the orthopolynom package #' @seealso [orthopolynom::chebyshev.t.polynomials()], #' [orthopolynom::chebyshev.u.polynomials()], #' [orthopolynom::chebyshev.c.polynomials()], #' [orthopolynom::chebyshev.s.polynomials()], #' \url{http://en.wikipedia.org/wiki/Chebyshev_polynomials} #' @name chebyshev #' @examples #' #' chebyshev(0) #' chebyshev(1) #' chebyshev(2) #' chebyshev(3) #' chebyshev(4) #' chebyshev(5) #' chebyshev(6) #' chebyshev(10) #' #' chebyshev(0:5) #' chebyshev(0:5, normalized = TRUE) #' chebyshev(0:5, kind = "u") #' chebyshev(0:5, kind = "c") #' chebyshev(0:5, kind = "s") #' chebyshev(0:5, indeterminate = "t") #' #' #' #' # visualize the chebyshev polynomials #' #' library(ggplot2); theme_set(theme_classic()) #' library(tidyr) #' #' s <- seq(-1, 1, length.out = 201) #' N <- 5 # number of chebyshev polynomials to plot #' (cheb_polys <- chebyshev(0:N)) #' #' # see ?bernstein for a better understanding of #' # how the code below works #' #' df <- data.frame(s, as.function(cheb_polys)(s)) #' names(df) <- c("x", paste0("T_", 0:N)) #' mdf <- gather(df, degree, value, -x) #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' #' #' # roots of chebyshev polynomials #' N <- 5 #' cheb_roots <- chebyshev_roots(1:N, N) #' cheb_fun <- as.function(chebyshev(N)) #' cheb_fun(cheb_roots) #' #' #' #' # chebyshev polynomials are orthogonal in two ways: #' T2 <- as.function(chebyshev(2)) #' T3 <- as.function(chebyshev(3)) #' T4 <- as.function(chebyshev(4)) #' #' w <- function(x) 1 / sqrt(1 - x^2) #' integrate(function(x) T2(x) * T3(x) * w(x), lower = -1, upper = 1) #' integrate(function(x) T2(x) * T4(x) * w(x), lower = -1, upper = 1) #' integrate(function(x) T3(x) * T4(x) * w(x), lower = -1, upper = 1) #' #' (cheb_roots <- chebyshev_roots(1:4, 4)) #' sum(T2(cheb_roots) * T3(cheb_roots) * w(cheb_roots)) #' sum(T2(cheb_roots) * T4(cheb_roots) * w(cheb_roots)) #' sum(T3(cheb_roots) * T4(cheb_roots) * w(cheb_roots)) #' #' sum(T2(cheb_roots) * T3(cheb_roots)) #' sum(T2(cheb_roots) * T4(cheb_roots)) #' sum(T3(cheb_roots) * T4(cheb_roots)) #' #' #' @rdname chebyshev #' @export chebyshev <- function(degree, kind = "t", indeterminate = "x", normalized = FALSE){ stopifnot(all(is.wholenumber(degree))) stopifnot(all(degree >= 0)) ## deal with kind stopifnot(kind %in% c("t","u","c","s")) ## make coefs if(kind == "t") coefs <- chebyshev.t.polynomials(max(degree), normalized) if(kind == "u") coefs <- chebyshev.u.polynomials(max(degree), normalized) if(kind == "c") coefs <- chebyshev.c.polynomials(max(degree), normalized) if(kind == "s") coefs <- chebyshev.s.polynomials(max(degree), normalized) ## if only one degree is wanted, return that if(length(degree) == 1){ coefs <- rev.default(coefs)[[1]] p <- as.mpoly.polynomial(coefs, indeterminate) class(p) <- c("chebyshev", "mpoly") attr(p, "chebyshev") <- list( "degree" = length(polynomial)-1, "kind" = kind, "indeterminate" = indeterminate, "normalized" = normalized ) return(p) } ## if several are wanted, return them coefs <- coefs[degree+1] ps <- lapply(coefs, function(polynomial){ p <- as.mpoly.polynomial(polynomial, indeterminate) class(p) <- c("chebyshev", "mpoly") attr(p, "chebyshev") <- list( "degree" = length(polynomial)-1, "kind" = kind, "indeterminate" = indeterminate, "normalized" = normalized ) p }) class(ps) <- "mpolyList" ps } #' @rdname chebyshev #' @export chebyshev_roots <- function(k, n) { stopifnot( is.wholenumber(k) ) stopifnot( is.wholenumber(n) ) stopifnot( round(k) %in% 1:n ) cos((k - .5) * pi / n) } mpoly/R/terms.mpoly.R0000644000176200001440000000117413476534026014244 0ustar liggesusers#' Extract the terms of a multivariate polynomial. #' #' Compute the terms of an mpoly object as a mpolyList. #' #' @param x an object of class mpoly #' @param ... additional parameters #' @return An object of class mpolyList. #' @usage \method{terms}{mpoly}(x, ...) #' @export #' @examples #' #' \dontrun{ .Deprecated issues a warning #' #' x <- mp("x^2 - y + x y z") #' terms(x) #' monomials(x) #' #' } #' terms.mpoly <- function(x, ...){ .Deprecated("monomials") # mpolyList <- lapply(x, function(v){ # l <- list(v) # class(l) <- 'mpoly' # l # }) # class(mpolyList) <- 'mpolyList' # mpolyList monomials(x) } mpoly/R/laguerre.R0000644000176200001440000000553313530645631013560 0ustar liggesusers#' Generalized Laguerre polynomials #' #' Generalized Laguerre polynomials as computed by orthopolynom. #' #' @param degree degree of polynomial #' @param alpha generalization constant #' @param indeterminate indeterminate #' @param normalized provide normalized coefficients #' @return a mpoly object or mpolyList object #' @author David Kahle calling code from the orthopolynom package #' @seealso [orthopolynom::glaguerre.polynomials()], #' \url{http://en.wikipedia.org/wiki/Laguerre_polynomials} #' @export #' @examples #' #' laguerre(0) #' laguerre(1) #' laguerre(2) #' laguerre(3) #' laguerre(4) #' laguerre(5) #' laguerre(6) #' #' laguerre(2) #' laguerre(2, normalized = TRUE) #' #' laguerre(0:5) #' laguerre(0:5, normalized = TRUE) #' laguerre(0:5, indeterminate = "t") #' #' #' #' # visualize the laguerre polynomials #' #' library(ggplot2); theme_set(theme_classic()) #' library(tidyr) #' #' s <- seq(-5, 20, length.out = 201) #' N <- 5 # number of laguerre polynomials to plot #' (lagPolys <- laguerre(0:N)) #' #' # see ?bernstein for a better understanding of #' # how the code below works #' #' df <- data.frame(s, as.function(lagPolys)(s)) #' names(df) <- c("x", paste0("L_", 0:N)) #' mdf <- gather(df, degree, value, -x) #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' qplot(x, value, data = mdf, geom = "line", color = degree) + #' coord_cartesian(ylim = c(-25, 25)) #' #' #' # laguerre polynomials are orthogonal with respect to the exponential kernel: #' L2 <- as.function(laguerre(2)) #' L3 <- as.function(laguerre(3)) #' L4 <- as.function(laguerre(4)) #' #' w <- dexp #' integrate(function(x) L2(x) * L3(x) * w(x), lower = 0, upper = Inf) #' integrate(function(x) L2(x) * L4(x) * w(x), lower = 0, upper = Inf) #' integrate(function(x) L3(x) * L4(x) * w(x), lower = 0, upper = Inf) #' #' laguerre <- function(degree, alpha = 0, indeterminate = "x", normalized = FALSE){ stopifnot(all(is.wholenumber(degree))) stopifnot(all(degree >= 0)) ## make coefs coefs <- glaguerre.polynomials(max(degree), alpha, normalized = normalized) ## if only one degree is wanted, return that if(length(degree) == 1){ coefs <- rev.default(coefs)[[1]] p <- as.mpoly.polynomial(coefs, indeterminate) class(p) <- c("laguerre", "mpoly") attr(p, "laguerre") <- list( "degree" = length(coefs)-1, "alpha" = alpha, "indeterminate" = indeterminate, "normalized" = normalized ) return(p) } ## if several are wanted, return them coefs <- coefs[degree+1] ps <- lapply(coefs, function(polynomial){ p <- as.mpoly.polynomial(polynomial, indeterminate) class(p) <- c("laguerre", "mpoly") attr(p, "laguerre") <- list( "degree" = length(coefs)-1, "alpha" = alpha, "indeterminate" = indeterminate, "normalized" = normalized ) p }) class(ps) <- "mpolyList" ps } mpoly/R/mpolyArithmetic.R0000644000176200001440000000372513476534026015131 0ustar liggesusers#' Arithmetic with multivariate polynomials #' #' Arithmetic with multivariate polynomials #' #' @param e1 an object of class mpoly #' @param e2 an object of class mpoly #' @return object of class mpoly #' @name mpolyArithmetic #' @examples #' #' p <- mp("x + y") #' p + p #' p - p #' p * p #' p^2 #' p^10 #' #' #' mp("(x+1)^10") #' p + 1 #' 2*p #' #' #' NULL #' @rdname mpolyArithmetic #' @export `+.mpoly` <- function(e1, e2){ # strip to lists e1 <- unclass(e1) e2 <- unclass(e2) # if either is constant, mpoly it if(!is.list(e1)){ stopifnot(is.numeric(e1) && length(e1) == 1) e1 <- list(c(coef = e1)) } if(!is.list(e2)){ stopifnot(is.numeric(e2) && length(e2) == 1) e2 <- list(c(coef = e2)) } # let mpoly do the heavy lifting mpoly(c(e1, e2)) } #' @rdname mpolyArithmetic #' @export `-.mpoly` <- function(e1, e2){ # deal with constants if(is.numeric(e2)) { e2 <- mpoly(list(c("coef" = e2))) } # flip coefficients of each term e2 <- lapply(e2, function(v){ v["coef"] <- -v["coef"] v }) class(e2) <- "mpoly" # add e1 + e2 } #' @rdname mpolyArithmetic #' @export `*.mpoly` <- function(e1, e2){ # allow for multiplication by a constant if(is.numeric(e1) && length(e1) == 1) e1 <- mpoly(list(c(coef = e1))) if(is.numeric(e2) && length(e2) == 1) e2 <- mpoly(list(c(coef = e2))) # argument check stopifnot(is.mpoly(e1)) stopifnot(is.mpoly(e2)) # multiply list <- lapply(e1, function(v){ lapply(e2, function(z){ c(v, z) }) }) # return mpoly( unlist(list, recursive = FALSE) ) } #' @rdname mpolyArithmetic #' @export `^.mpoly` <- function(e1, e2){ if(!is.mpoly(e1)) stop('e1 must be of class mpoly.', call. = FALSE) if(!is.wholenumber(e2) || e2 < 0) stop('exponent must be a positive integer.', call. = FALSE) out <- mpoly(list(c(coef = 1))) if(e2 == 0) return(out) for(k in 1:e2) out <- out * e1 out } mpoly/R/swap.R0000644000176200001440000000345013476534026012724 0ustar liggesusers#' Swap polynomial indeterminates #' #' Swap polynomial indeterminates #' #' @param p polynomial #' @param variable the variable in the polynomial to replace #' @param replacement the replacement variable #' @return a mpoly object #' @author David Kahle #' @export #' @examples #' #' (p <- mp("(x + y)^2")) #' swap(p, "x", "t") #' #' ## the meta data is retained #' (p <- bernstein(3, 5)) #' (p2 <- swap(p, "x", "t")) #' is.bernstein(p2) #' #' (p <- chebyshev(3)) #' (p2 <- swap(p, "x", "t")) #' is.chebyshev(p2) #' #' swap <- function(p, variable, replacement){ # arg check stopifnot(is.mpoly(p)) if (stringi::stri_detect_regex(replacement, "[^\\w.]")) stop("replacement must be a single symbol.", call. = FALSE) # determine variables vars <- vars(p) # if constant, return if(is.constant(p)) return(p) # don't allow replacement of one variable in the presence of many when # the variable is already a part of the polynomial, since # mpoly won't be run again to determine if a variable is duplicated stopifnot(variable %in% vars) if(replacement %in% vars && length(vars) > 1){ stop("the replacement value cannot be a variable in the polynomial, try plug.", call. = FALSE) } # swapping in polynomial p_out <- unclass(p) p_out <- lapply(p, function(v) { names(v)[names(v) == variable] <- replacement v }) class(p_out) <- "mpoly" # custom for special polynomials if (is.bernstein(p)) { class(p_out) <- c("bernstein", "mpoly") attr(p_out, "bernstein") <- attr(p, "bernstein") attr(p_out, "bernstein")["indeterminate"] <- replacement } if (is.chebyshev(p)) { class(p_out) <- c("chebyshev", "mpoly") attr(p_out, "chebyshev") <- attr(p, "chebyshev") attr(p_out, "chebyshev")["indeterminate"] <- replacement } # return p_out } mpoly/R/print.mpoly.R0000644000176200001440000001066413476534026014252 0ustar liggesusers#' Pretty printing of multivariate polynomials. #' #' This is the major function used to view multivariate polynomials. #' #' @param x an object of class mpoly #' @param varorder the order of the variables #' @param order a total order used to order the monomials in the printing #' @param stars print the multivariate polynomial in the more computer-friendly #' asterisk notation (default FALSE) #' @param silent logical; if TRUE, suppresses output #' @param ... additional parameters to go to [base::cat()] #' @param plus_pad number of spaces to the left and right of plus sign #' @param times_pad number of spaces to the left and right of times sign #' @usage \method{print}{mpoly}(x, varorder, order, stars = FALSE, silent = #' FALSE, ..., plus_pad = 2L, times_pad = 1L) #' @return Invisible string of the printed object. #' @export #' @examples #' #' mp("-x^5 - 3 y^2 + x y^3 - 1") #' #' #' (p <- mp("2 x^5 - 3 y^2 + x y^3")) #' print(p) # same #' print(p, silent = TRUE) #' s <- print(p, silent = TRUE) #' s #' #' print(p, order = "lex") # -> 2 x^5 + x y^3 - 3 y^2 #' print(p, order = "lex", varorder = c("y","x")) # -> y^3 x - 3 y^2 + 2 x^5 #' print(p, varorder = c("y","x")) # -> y^3 x - 3 y^2 + 2 x^5 #' #' # this is mostly used internally #' print(p, stars = TRUE) #' print(p, stars = TRUE, times_pad = 0L) #' print(p, stars = TRUE, times_pad = 0L, plus_pad = 1L) #' print(p, stars = TRUE, times_pad = 0L, plus_pad = 0L) #' print(p, plus_pad = 1L) #' print.mpoly <- function(x, varorder, order, stars = FALSE, silent = FALSE, ..., plus_pad = 2L, times_pad = 1L) { # argument checking and basic variable setting stopifnot(is.mpoly(x)) stopifnot(is.logical(stars)) vars <- vars(x) # deal with constant mpolys if (length(vars) == 0) { if (!silent) cat(x[[1]], ...) return(invisible(as.character(x[[1]]))) } # check variable order if (!missing(varorder)) { stopifnot(is.character(varorder)) if(!all(vars %in% varorder)) { stop( "if specified, varorder must contain all computed vars - ", stri_c(vars, collapse = ", "), call. = FALSE ) } } if (!missing(order)) match.arg(order, c("lex", "glex", "grlex")) # reorder if (!missing(order) && !missing(varorder)) { x <- reorder(x, varorder = varorder, order = order) } else if(!missing(varorder)){ x <- reorder(x, varorder = varorder) } else if(!missing(order)){ x <- reorder(x, order = order) } # set delimiters if (!stars) { times <- " " expo <- "^" } else { times_pad <- stringi::stri_dup(" ", times_pad) times <- stri_c(times_pad, "*", times_pad) expo <- "**" } # now ready to start constructing the string. we don't do this by growing # the entire mpoly because growing in R is slow, so we print each of the terms, # merge them together, and take care of edge cases # one of the challenges here is to figure out where edge cases should be handled: # in the term printing or after the merging. we try to take a practical approach. # some things are fixed in term printing (e.g. ^1 or 1 x) and others # (e.g. ?) are done in tidying the final result. # print terms terms <- vapply(x, print_term, character(1L), times = times, exp = expo) # merge terms with +'s and -'s plus_pad <- stringi::stri_dup(" ", plus_pad) plus <- stri_c(plus_pad, "+", plus_pad) s <- stri_c(terms, collapse = plus) s <- stri_replace_all_fixed(s, stri_c("+", plus_pad, "-"), stri_c("-", plus_pad)) # fix subtractions # final cleaning - clean "2 x - 1 y^2" s <- stri_replace_all_fixed(s, stri_c("-", plus_pad, "1", times), stri_c("-", plus_pad)) # print if(!silent) cat(s, ...) ## return invisible(s) } print_term <- function(term, times = " ", expo = "^") { # if a constant term if (length(term) == 1L) return(format(term[["coef"]], scientific=FALSE)) # split vars + exps from coef coef_ndx <- length(term) coef <- term[[ coef_ndx]] mono <- term [-coef_ndx] # build term out <- format(coef, scientific = FALSE) for (k in seq_along(mono)) { if (mono[k] == 1L) { out <- stri_c(out, times, names(mono)[k]) } else { out <- stri_c(out, times, stri_c(names(mono)[k], expo, mono[k])) } } # clean 1 from front if (abs(term[["coef"]] - 1) < sqrt(.Machine$double.eps) ) out <- stri_sub(out, stri_length(times) + 2L) # return out } mpoly/R/burst.R0000644000176200001440000000337213476534026013114 0ustar liggesusers#' Enumerate integer r-vectors summing to n #' #' Determine all r-vectors with nonnegative integer entries summing to n. Note #' that this is not intended to be optimized. #' #' @param n integer to sum to #' @param r number of components #' @return a matrix whose rows are the n-tuples #' @export #' @examples #' burst(4) #' #' burst(4, 4) #' burst(4, 3) #' burst(4, 2) #' #' rowSums(burst(4)) #' rowSums(burst(4, 3)) #' rowSums(burst(4, 2)) #' #' #' burst(10, 4) # all possible 2x2 contingency tables with n=10 #' burst(10, 4) / 10 # all possible empirical relative frequencies #' burst <- function(n, r = n){ stopifnot(is.wholenumber(n)) stopifnot(n > 0) stopifnot(is.wholenumber(r)) stopifnot(r > 0) if(r == 1) return(n) ## compute all partitions of the number (order does not matter) parts <- partitions(n) ## convert to list, remove 0's partsWOzeros <- apply(parts, 1, function(row) row[row != 0] ) ## select all those with length less than or equal to r isAtLongest <- function(howLong){ function(v) length(v) <= howLong } #isAtLongest(4)(1:4) #isAtLongest(4)(1:5) partsWOzeros <- Filter(isAtLongest(r), partsWOzeros) ## refill with 0's rvectors <- lapply(partsWOzeros, function(v){ if(length(v) < r) return(c(v, rep(0, r - length(v)))) v }) ## compute permutations of each permsOfEach <- lapply(rvectors, permutations) ## make out matrix m <- sum(vapply(permsOfEach, nrow, integer(1))) n <- r out <- integer(length = m*n) dim(out) <- c(m, n) ## populate out matrix row <- 1 for(i in seq_along(permsOfEach)){ permsInPartition <- nrow(permsOfEach[[i]]) out[row:(row+permsInPartition-1),] <- permsOfEach[[i]] row <- row + permsInPartition } ## return out } mpoly/R/homogenize.R0000644000176200001440000000634713476534026014126 0ustar liggesusers#' Homogenize a polynomial #' #' Homogenize a polynomial. #' #' @param x an mpoly object, see [mpoly()] #' @param indeterminate name of homogenization #' @return a (de/homogenized) mpoly or an mpolyList #' @name homogenize #' @examples #' #' x <- mp("x^4 + y + 2 x y^2 - 3 z") #' is.homogeneous(x) #' (xh <- homogenize(x)) #' is.homogeneous(xh) #' #' homogeneous_components(x) #' #' homogenize(x, "o") #' #' xh <- homogenize(x) #' dehomogenize(xh) # assumes indeterminate = "t" #' plug(xh, "t", 1) # same effect, but dehomogenize is faster #' #' #' #' # the functions are vectorized #' (ps <- mp(c("x + y^2", "x + y^3"))) #' (psh <- homogenize(ps)) #' dehomogenize(psh) #' #' #' # demonstrating a leading property of homogeneous polynomials #' library(magrittr) #' p <- mp("x^2 + 2 x + 3") #' (ph <- homogenize(p, "y")) #' lambda <- 3 #' (d <- totaldeg(p)) #' ph %>% #' plug("x", lambda*mp("x")) %>% #' plug("y", lambda*mp("y")) #' lambda^d * ph #' #' @rdname homogenize #' @export homogenize <- function(x, indeterminate = "t"){ if (is.mpolyList(x)) { hs <- lapply(x, homogenize, indeterminate = indeterminate) class(hs) <- "mpolyList" return(hs) } if(!is.mpoly(x)) stop("homogenize requires mpoly objects.", call. = FALSE) term_exps <- exponents(x) term_degs <- vapply(term_exps, sum, numeric(1)) max_deg <- max(term_degs) vars <- vars(x) p <- length(vars) homo_deg_per_term <- max_deg - term_degs for(k in 1:length(x)){ if(homo_deg_per_term[[k]] == 0) next term <- x[[k]] coef_ndx <- which(names(term) == "coef") homo_piece <- homo_deg_per_term[[k]] names(homo_piece) <- indeterminate x[[k]] <- c(term[-coef_ndx], homo_piece, term[coef_ndx]) } x } #' @rdname homogenize #' @export dehomogenize <- function(x, indeterminate = "t"){ if (is.mpolyList(x)) { hs <- lapply(x, dehomogenize, indeterminate = indeterminate) class(hs) <- "mpolyList" return(hs) } if(!is.mpoly(x)) stop("dehomogenize requires mpoly objects.", call. = FALSE) # if(missing(indeterminate)) stop("dehomogenize requires a variable to dehomogenize.", call. = FALSE) # check for inhomogeneous if (!is.homogeneous(x)) { stop("polynomial ", print.mpoly(x, silent = TRUE), " is not homogeneous.", call. = FALSE) } # dehomogenize # plug(x, indeterminate, 1) # works, but is not optimized, so xdh <- lapply(x, function(term){ var_ndx <- which(names(term) == indeterminate) if (length(var_ndx) == 0) { return(term) # indeterminate not in term } else { return(term[-var_ndx]) } }) class(xdh) <- "mpoly" xdh } #' @rdname homogenize #' @export is.homogeneous <- function(x) { if (is.mpolyList(x)) return(vapply(x, is.homogeneous, logical(1))) term_total_degs <- vapply(exponents(x), sum, numeric(1)) all(term_total_degs == term_total_degs[1]) } #' @rdname homogenize #' @export homogeneous_components <- function(x){ exps <- exponents(x) exps_mat <- t(simplify2array(exps)) term_total_degs <- rowSums(exps_mat) n_terms <- length(x) term_ndcs <- split(1:n_terms, term_total_degs) homo_comps <- lapply(term_ndcs, function(ndcs){ x[ndcs] }) homo_comps <- unname(homo_comps) class(homo_comps) <- "mpolyList" homo_comps } mpoly/R/deriv.mpoly.R0000644000176200001440000000735313623300445014216 0ustar liggesusers#' Compute partial derivatives of a multivariate polynomial. #' #' This is a deriv method for mpoly objects. It does not call the #' [stats::deriv()]. #' #' @param expr an object of class mpoly #' @param var character - the partial derivative desired #' @param ... any additional arguments #' @return An object of class mpoly or mpolyList. #' @export #' @examples #' m <- mp("x y + y z + z^2") #' deriv(m, "x") #' deriv(m, "y") #' deriv(m, "z") #' deriv(m, c("x","y","z")) #' deriv(m, "a") #' is.mpoly(deriv(m, "x")) #' is.mpolyList( deriv(m, c("x","y","z")) ) deriv.mpoly <- function(expr, var, ...){ # argument checks if (missing(var)) stop("var must be specified, see ?deriv.mpoly", call. = FALSE) stopifnot(is.character(var)) # if many vars provided if(length(var) > 1){ mpolyList <- lapply(as.list(var), function(var){ deriv(expr, var = var, ...) }) class(mpolyList) <- "mpolyList" return(mpolyList) } # if the variable is not in the polynomial, return zero if(!(var %in% vars(expr))){ return( mpoly(list(c(coef = 0))) ) } expr <- unclass(expr) # take derivative expr <- lapply(expr, function(v){ if(length(v) == 1) return(c(coef = 0)) p <- length(v) if(!(var %in% names(v[1:p]))) return(c(coef = 0)) v["coef"] <- unname(v[var]) * v["coef"] v[var] <- v[var] - 1 v }) # return mpoly(expr) } #' Compute gradient of a multivariate polynomial. #' #' This is a wrapper for deriv.mpoly. #' #' @param mpoly an object of class mpoly #' @seealso [deriv.mpoly()] #' @return An object of class mpoly or mpolyList. #' @export #' @examples #' m <- mp("x y + y z + z^2") #' gradient(m) #' #' #' # gradient descent illustration using the symbolically #' # computed gradient of the rosenbrock function (shifted) #' rosenbrock <- mp("(1 - x)^2 + 100 (y - x^2)^2") #' fn <- as.function(rosenbrock) #' (rosenbrock_gradient <- gradient(rosenbrock)) #' gr <- as.function(rosenbrock_gradient) #' #' # visualize the function #' library(ggplot2) #' s <- seq(-5, 5, .05) #' df <- expand.grid(x = s, y = s) #' df$z <- apply(df, 1, fn) #' ggplot(df, aes(x = x, y = y)) + #' geom_raster(aes(fill = z + 1e-10)) + #' scale_fill_continuous(trans = "log10") #' #' # run the gradient descent algorithm using line-search #' # step sizes computed with optimize() #' current <- steps <- c(-3,-4) #' change <- 1 #' tol <- 1e-5 #' while(change > tol){ #' last <- current #' delta <- optimize( #' function(delta) fn(current - delta*gr(current)), #' interval = c(1e-10, .1) #' )$minimum #' current <- current - delta*gr(current) #' steps <- unname(rbind(steps, current)) #' change <- abs(fn(current) - fn(last)) #' } #' steps <- as.data.frame(steps) #' names(steps) <- c("x", "y") #' #' # visualize steps, note the optim at c(1,1) #' # the routine took 5748 steps #' ggplot(df, aes(x = x, y = y)) + #' geom_raster(aes(fill = z + 1e-10)) + #' geom_path(data = steps, color = "red") + #' geom_point(data = steps, color = "red", size = .5) + #' scale_fill_continuous(trans = "log10") #' #' # it gets to the general region of space quickly #' # but once it gets on the right arc, it's terrible #' # here's what the end game looks like #' last_steps <- tail(steps, 100) #' rngx <- range(last_steps$x); sx <- seq(rngx[1], rngx[2], length.out = 201) #' rngy <- range(last_steps$y); sy <- seq(rngy[1], rngy[2], length.out = 201) #' df <- expand.grid(x = sx, y = sy) #' df$z <- apply(df, 1, fn) #' ggplot(df, aes(x = x, y = y)) + #' geom_raster(aes(fill = z)) + #' geom_path(data = last_steps, color = "red", size = .25) + #' geom_point(data = last_steps, color = "red", size = 1) + #' scale_fill_continuous(trans = "log10") #' #' gradient <- function(mpoly) deriv(mpoly, var = vars(mpoly)) mpoly/R/solve_unipoly.R0000644000176200001440000000146013476534026014660 0ustar liggesusers#' Solve a univariate mpoly with polyroot #' #' Solve a univariate mpoly with polyroot #' #' @param mpoly an mpoly #' @param real_only return only real solutions? #' @export #' @examples #' #' solve_unipoly(mp("x^2 - 1")) # check x = -1, 1 #' solve_unipoly(mp("x^2 - 1"), real_only = TRUE) #' solve_unipoly <- function(mpoly, real_only = FALSE) { mpoly <- reorder(mpoly) degs <- vapply(mpoly, function(term) { if(length(term) == 1) return(0L) term[[1]] }, double(1)) coefs <- sapply(mpoly, `[[`, "coef") # vapply errors? coef_vec <- rep(0L, degs[1]+1) names(coef_vec) <- 0:degs[1] for(k in 1:length(mpoly)) coef_vec[as.character(degs[k])] <- coefs[k] solns <- polyroot(coef_vec) if (real_only) { is.real <- function(x) Im(x) == 0 solns <- Re(Filter(is.real, solns)) } solns } mpoly/R/mpoly-defunct.R0000644000176200001440000000032413476534026014535 0ustar liggesusers#' Defunct mpoly functions #' #' This is a list of past functions of the mpoly package. #' #' The following are defunct mpoly functions: #' #' \itemize{ #' #' \item grobner #' #' } #' #' @name mpoly-defunct NULL mpoly/R/plug.R0000644000176200001440000000311113522636150012704 0ustar liggesusers#' Switch indeterminates in a polynomial #' #' Switch indeterminates in a polynomial #' #' @param p a polynomial #' @param indeterminate the indeterminate in the polynomial to switch #' @param value the value/indeterminate to substitute #' @return an mpoly object #' @export #' @examples #' #' # on an mpoly #' (p <- mp("(x+y)^3")) #' plug(p, "x", 5) #' plug(p, "x", "t") #' plug(p, "x", "y") #' plug(p, "x", mp("2 y")) #' #' plug(p, "x", mp("x + y")) #' mp("((x+y)+y)^3") #' #' # on an mpolyList #' ps <- mp(c("x+y", "x+1")) #' plug(ps, "x", 1) #' plug <- function(p, indeterminate, value){ # go recursive if an mpolyList if(is.mpolyList(p)) return(structure(lapply(p, plug, indeterminate, value), class = "mpolyList")) # check args stopifnot(length(value) == 1 || is.mpoly(value)) stopifnot(length(indeterminate) == 1) # if plugging in a number if(is.numeric(value)){ pList <- unclass(p) pList <- lapply(pList, function(term){ # term <- pList[[2]] indetNdcs <- which(names(term) == indeterminate) term[indetNdcs] <- value^unname(term[indetNdcs]) names(term)[indetNdcs] <- "coef" term }) return(mpoly(pList)) # this can be optimized } # if plugging in a value if(is.character(value)) value <- str_trim(value) if(is.mpoly(value)) value <- print.mpoly(value, silent = TRUE) if(is.character(value)){ charPoly <- print.mpoly(p, silent = TRUE) charPoly <- str_replace_all(charPoly, indeterminate, paste0("(",value,")")) return(mp(charPoly))# this can be optimized } } mpoly/R/lissajous.R0000644000176200001440000000257513522636150013766 0ustar liggesusers#' Lissajous polynomials #' #' The Lissajous polynomials are the implicit (variety) descriptions of the #' image of the parametric map x = cos(m t + p), y = sin(n t + q). #' #' @param m,n,p,q Trigonometric coefficients, see examples for description #' @param digits The number of digits to round coefficients to, see #' [round.mpoly()]. This is useful for cleaning terms that are numerically #' nonzero, but should be. #' @return a mpoly object #' @seealso [chebyshev()], Merino, J. C (2003). Lissajous figures and Chebyshev #' polynomials. The College Mathematics Journal, 34(2), pp. 122-127. #' @export #' @examples #' #' lissajous(3, 2, -pi/2, 0) #' lissajous(4, 3, -pi/2, 0) #' lissajous <- function(m, n, p, q, digits = 3) { delta <- det(matrix(c(m, n, p, q), nrow = 2)) Tnx <- chebyshev(n, indeterminate = "x") Tmy <- chebyshev(m, indeterminate = "y") is.even <- function(n) (n %% 2L) == 0L is.zero <- function(x, tol = 1e-5) abs(x) <= tol if (is.even(m)) { if (is.zero(sin(delta))) { p <- cos(delta)*Tnx - (-1)^(m/2)*Tmy } else { p <- Tnx^2 + Tmy^2 - 2*(-1)^(m/2)*Tnx*Tmy*cos(delta) - sin(delta)^2 } } else { # m odd if (is.zero(cos(delta))) { p <- sin(delta)*Tnx - (-1)^((m-1)/2)*Tmy } else { p <- Tnx^2 + Tmy^2 - 2*(-1)^((m-1)/2)*Tnx*Tmy*sin(delta) - cos(delta)^2 } } round(p, digits) } mpoly/R/mpoly-package.R0000644000176200001440000000172513522636150014477 0ustar liggesusers#' Multivariate polynomials in R. #' #' A package for symbolic computation and more with multivariate polynomials #' #' @importFrom stringr str_c str_detect str_sub str_replace_all str_replace #' str_count str_dup str_extract str_extract_all str_locate str_locate_all #' str_split str_sub<- str_trim fixed str_remove str_remove_all #' @importFrom stringi stri_c stri_replace_all_fixed stri_sub stri_length #' stri_replace_all_regex stri_sub #' @importFrom partitions parts #' @importFrom plyr ddply #' @importFrom stats deriv reorder coef #' @importFrom ggplot2 ggplot aes #' @importFrom polynom polynomial #' @importFrom orthopolynom chebyshev.c.polynomials chebyshev.s.polynomials #' chebyshev.t.polynomials chebyshev.u.polynomials glaguerre.polynomials #' hermite.h.polynomials hermite.he.polynomials jacobi.g.polynomials #' jacobi.p.polynomials legendre.polynomials #' @importFrom tidyr gather #' @docType package #' @name mpoly #' @aliases mpoly package-mpoly NULLmpoly/R/partitions.R0000644000176200001440000000116613476534026014150 0ustar liggesusers#' Enumerate the partitions of an integer #' #' Determine all unrestricted partitions of an integer. This function is #' equivalent to the function parts in the partitions package. #' #' @param n an integer #' @return a matrix whose rows are the n-tuples #' @author Robin K. S. Hankin, from package partitions #' @export #' @examples #' partitions(5) #' str(partitions(5)) partitions <- function(n){ stopifnot(all(sapply(as.list(n), is.wholenumber))) stopifnot(all(n >= 1)) if(is.vector(n) && length(n) > 1){ return(lapply(as.list(n), function(x){ as.matrix( t(parts(x)) ) })) } as.matrix( t(parts(n)) ) }mpoly/R/as.function.mpolyList.R0000644000176200001440000001140513522636150016164 0ustar liggesusers#' Change a vector of multivariate polynomials into a function. #' #' Transforms an mpolyList object into a function which can be evaluated. #' #' @inheritParams as.function.mpoly #' @usage \method{as.function}{mpolyList}(x, varorder = vars(x), vector = TRUE, #' silent = FALSE, ..., plus_pad = 1L, times_pad = 1L, squeeze = TRUE) #' @seealso [plug()], [as.function.mpolyList()] #' @export #' @examples #' #' # basic examples #' (mpolyList <- mp(c("2 x + 1", "x - z^2"))) #' (f <- as.function(mpolyList)) #' f(c(1,2)) # -> (2*1 + 1, 1-2^2) = 3 -3 #' #' f <- as.function(mpolyList, varorder = c("x","y","z")) #' f(c(1,0,2)) # -> 3 -3 #' f(c(1,4,2)) # -> 3 -3 #' #' f <- as.function(mpolyList, varorder = c("x","y","z"), vector = FALSE) #' f(1, 0, 2) # -> 3 -3 #' f(1, 4, 2) # -> 3 -3 #' #' #' #' # making a gradient function (useful for optim) #' mpoly <- mp("x + y^2 + y z") #' mpolyList <- gradient(mpoly) #' f <- as.function(mpolyList, varorder = vars(mpoly)) #' f(c(0,2,3)) # -> 1 7 2 #' #' #' #' # a univariate mpolyList creates a vectorized function #' ps <- mp(c("x", "x^2", "x^3")) #' f <- as.function(ps) #' f #' s <- seq(-1, 1, length.out = 11) #' f(s) #' #' # another example #' ps <- chebyshev(1:3) #' f <- as.function(ps) #' f(s) #' #' # the binomial pmf as an algebraic (polynomial) map #' # from [0,1] to [0,1]^size #' # p |-> {choose(size, x) p^x (1-p)^(size-x)}_{x = 0, ..., size} #' abinom <- function(size, indet = "p"){ #' chars4mp <- vapply(as.list(0:size), function(x){ #' sprintf("%d %s^%d (1-%s)^%d", choose(size, x), indet, x, indet, size-x) #' }, character(1)) #' mp(chars4mp) #' } #' (ps <- abinom(2, "p")) # = mp(c("(1-p)^2", "2 p (1-p)", "p^2")) #' f <- as.function(ps) #' #' f(.5) # P[X = 0], P[X = 1], and P[X = 2] for X ~ Bin(2, .5) #' dbinom(0:2, 2, .5) #' #' f(.75) # P[X = 0], P[X = 1], and P[X = 2] for X ~ Bin(2, .75) #' dbinom(0:2, 2, .75) #' #' # as the degree gets larger, you'll need to be careful when evaluating #' # the polynomial. as.function() is not currently optimized for #' # stable numerical evaluation of polynomials; it evaluates them in #' # the naive way #' all.equal( #' as.function(abinom(10))(.5), #' dbinom(0:10, 10, .5) #' ) #' #' all.equal( #' as.function(abinom(30))(.5), #' dbinom(0:30, 20, .5) #' ) #' #' #' # the function produced is vectorized: #' number_of_probs <- 11 #' probs <- seq(0, 1, length.out = number_of_probs) #' (mat <- f(probs)) #' colnames(mat) <- sprintf("P[X = %d]", 0:2) #' rownames(mat) <- sprintf("p = %.2f", s) #' mat #' as.function.mpolyList <- function(x, varorder = vars(x), vector = TRUE, silent = FALSE, ..., plus_pad = 1L, times_pad = 1L, squeeze = TRUE){ # argument checking stopifnot(is.character(varorder)) stopifnot(is.logical(vector)) stopifnot(is.mpolyList(x)) if (!missing(varorder) && !all( vars(x) %in% varorder )) stop("varorder must contain all of the variables.", call. = FALSE) # determine the number of variables p <- length(varorder) # print polys with stars mp_str <- print.mpolyList(x, stars = TRUE, silent = TRUE, plus_pad = plus_pad, times_pad = times_pad) # univariate polynomial - vectorize if (p == 1) { mp_str <- stri_c(mp_str, collapse = " , ") mp_str <- stri_replace_all_fixed(mp_str, varorder, ".") if (squeeze) mp_str <- stri_replace_all_fixed(mp_str, " ", "") f <- function(){} formals(f) <- alist(. = ) body(f) <- as.call(c( as.name("{"), expression(if(length(.) > 1) return(t(sapply(., f)))), parse(text = stri_c("c(", mp_str, ")")) )) return(f) } # general polynomials as a vector argument if (vector) { mp_str <- stri_c(" ", mp_str, " ") # pad to make parsing easier for (k in 1L:p) { mp_str <- stri_replace_all_fixed(mp_str, "**", " ^") mp_str <- stri_replace_all_fixed(mp_str, stri_c(" ", varorder[k], " "), stri_c(" .[", k, "] ")) mp_str <- stri_replace_all_fixed(mp_str, " ^", "**") } if (squeeze) mp_str <- stri_replace_all_fixed(mp_str, " ", "") v <- stri_c("(", stri_c(varorder, collapse = ", "), ")") if((silent == FALSE) && (missing(vector) || missing(varorder))) message("f(.) with . = ", v) mp_str <- stri_c("function(.){ c(", stri_c(mp_str, collapse = ", "), ") }") return(eval(parse(text = mp_str))) } # general polynomials as a bunch of arguments if (!vector) { if (squeeze) mp_str <- stri_replace_all_fixed(mp_str, " ", "") mp_str <- stri_c(mp_str, collapse = ", ") if((silent == FALSE) && (missing(vector) || missing(varorder))) message("f(", paste(varorder, collapse = ", "), ")", sep = "") mp_str <- stri_c( "function(", stri_c(varorder, collapse = ", "), ") { c(", mp_str, ") }" ) return(eval(parse(text = mp_str))) } } mpoly/R/as.function.mpoly.R0000644000176200001440000001137213476535265015350 0ustar liggesusers#' Change a multivariate polynomial into a function. #' #' Transforms an mpoly object into a function which can be evaluated. #' #' @inheritParams print.mpoly #' @param vector whether the function should take a vector argument (TRUE) or a #' series of arguments (FALSE) #' @param ... any additional arguments #' @param squeeze minify code in the created function #' @usage \method{as.function}{mpoly}(x, varorder = vars(x), vector = TRUE, #' silent = FALSE, ..., plus_pad = 1L, times_pad = 1L, squeeze = TRUE) #' @seealso [plug()], [as.function.mpolyList()] #' @export #' @examples #' #' p <- mp("(x - 1)^2") #' (f <- as.function(p)) #' f(1) #' f(seq(0, 2, .1)) #' #' p <- mp("x + 3 x y + z^2 x") #' (f <- as.function(p)) #' f(1:3) # -> 16 #' f(c(1,1,1)) # -> 5 #' #' f <- as.function(p, vector = FALSE) #' f(1, 2, 3) # -> 16 #' f(1, 1, 1) # -> 5 #' #' f <- as.function(p, varorder = c("z","y","x"), vector = FALSE) #' f(3, 2, 1) # -> 16 #' f(1, 1, 1) # -> 5 #' #' # for univariate mpolys, as.function() returns a vectorized function #' # that can even apply to arrays #' p <- mp("x^2") #' f <- as.function(p) #' f(1:10) #' (mat <- matrix(1:4, 2)) #' f(mat) #' #' #' p <- mp("1 2 3 4") #' f <- as.function(p) #' f(10) # -> 24 #' #' bernstein(1, 2) #' s <- seq(0, 1, .01) #' as.function(bernstein(1, 2))(s) #' plot( #' s, #' as.function(bernstein(1, 2))(s) #' ) #' #' #' as.function(mp("x + xx")) #' as.function(mp("x + xx"), squeeze = FALSE) #' #' as.function.mpoly <- function(x, varorder = vars(x), vector = TRUE, silent = FALSE, ..., plus_pad = 1L, times_pad = 1L, squeeze = TRUE){ # argument checking stopifnot(is.character(varorder)) stopifnot(is.logical(vector)) stopifnot(is.mpoly(x)) if (!setequal(varorder, vars(x))) stop("varorder must contain all of the variables of x.", call. = FALSE) # determine the number of variables p <- length(vars(x)) # deal with constant polynomials if (is.constant(x)) return( function(.) 0*. + unlist(x)[["coef"]] ) # print poly with stars mp_str <- print.mpoly(x, stars = TRUE, silent = TRUE, plus_pad = plus_pad, times_pad = times_pad) # univariate polynomial if (p == 1L) { mp_str <- stri_replace_all_fixed(mp_str, vars(x), ".") if (squeeze) mp_str <- stri_replace_all_fixed(mp_str, " ", "") if (!silent) message("f(.) with . = ", vars(x)) f <- function(){} formals(f) <- alist(. = ) body(f) <- as.call(c( as.name("{"), expression(if(length(.) > 1){ .[] <- sapply(., f) return(.) }), parse(text = mp_str) )) return(f) } # general polynomials as a vector argument if (vector) { mp_str <- stri_c(" ", mp_str, " ") # pad to make parsing easier for (k in 1L:p) { mp_str <- stri_replace_all_fixed(mp_str, "**", " ^") mp_str <- stri_replace_all_fixed(mp_str, stri_c(" ", varorder[k], " "), stri_c(" .[", k, "] ")) mp_str <- stri_replace_all_fixed(mp_str, " ^", "**") } if (squeeze) mp_str <- stri_replace_all_fixed(mp_str, " ", "") v <- stri_c("(", stri_c(varorder, collapse = ", "), ")") if(!silent) message("f(.) with . = ", v) mp_str <- stri_c("function(.) {", mp_str, "}") return(eval(parse(text = mp_str))) } # general polynomials as a bunch of arguments if (!vector) { if(!silent) message("f(", stri_c(varorder, collapse = ", "), ")", sep = "") mp_str <- stri_c( "function(", stri_c(varorder, collapse = ", "), ") {", mp_str, "}", sep = "" ) return(eval(parse(text = mp_str))) } } as.function.bernstein <- function(x, ...){ # grab bernstein values k <- attr(x, "bernstein")$k n <- attr(x, "bernstein")$n # return exp'd log function function(.) { out <- vector("numeric", length = length(.)) non_pos_ndcs <- (. <= 0) sup_one_ndcs <- (. >= 1) if (any(non_pos_ndcs)) out[non_pos_ndcs] <- choose(n, k) * .[non_pos_ndcs]^k * (1 - .[non_pos_ndcs])^(n-k) if (any(sup_one_ndcs)) out[sup_one_ndcs] <- choose(n, k) * .[sup_one_ndcs]^k * (1 - .[sup_one_ndcs])^(n-k) if (!all(sup_one_ndcs | sup_one_ndcs)) out[0 < . & . < 1] <- exp( lchoose(n, k) + k*log(.[0 < . & . < 1]) + (n-k)*log(1-.[0 < . & . < 1]) ) out } } # as.function.jacobi <- function(x, ...){ # return(as.function.mpoly(x)) ## below is broken. # # # grab bernstein values # d <- attr(x, "jacobi")$degree # k <- attr(x, "jacobi")$kind # i <- attr(x, "jacobi")$indeterminate # n <- attr(x, "jacobi")$normalized # a <- attr(x, "jacobi")$alpha # b <- attr(x, "jacobi")$beta # # # return exp'd log function # # #http://en.wikipedia.org/wiki/Jacobi_polynomials function(.) # #pochhammer(a+1, d) / factorial(d) * hypergeo(-d, 1+a+b+d, a+1, # #(1-.)/2) # # } mpoly/R/bezier.R0000644000176200001440000001657413476534026013245 0ustar liggesusers#' Bezier polynomials #' #' Compute the Bezier polynomials of a given collection of points. Note that #' using [mpoly::as.function.mpoly()] on the resulting Bezier polynomials is #' made numerically stable by taking advantage of de Casteljau's algorithm; it #' does not use the polynomial that is printed to the screen. See #' [bezier_function()] for details. #' #' @param ... either a sequence of points or a matrix/data frame of points, see #' examples #' @param indeterminate the indeterminate of the resulting polynomial #' @return a mpoly object #' @author David Kahle #' @seealso [bezier_function()] #' @export #' @examples #' #' p1 <- c(0, 0) #' p2 <- c(1, 1) #' p3 <- c(2, -1) #' p4 <- c(3, 0) #' bezier(p1, p2, p3, p4) #' #' #' points <- data.frame(x = 0:3, y = c(0,1,-1,0)) #' bezier(points) #' #' #' points <- data.frame(x = 0:2, y = c(0,1,0)) #' bezier(points) #' #' #' #' #' #' #' #' # visualize the bernstein polynomials #' #' library(ggplot2); theme_set(theme_bw()) #' #' s <- seq(0, 1, length.out = 101) #' #' #' #' ## example 1 #' points <- data.frame(x = 0:3, y = c(0,1,-1,0)) #' (bezPolys <- bezier(points)) #' #' f <- as.function(bezPolys) #' df <- as.data.frame(f(s)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red") + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' ## example 1 with weights #' f <- as.function(bezPolys, weights = c(1,5,5,1)) #' df <- as.data.frame(f(s)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red") + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' #' ## example 2 #' points <- data.frame(x = 0:2, y = c(0,1,0)) #' (bezPolys <- bezier(points)) #' f <- as.function(bezPolys) #' df <- as.data.frame(f(s)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red") + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' ## example 3 #' points <- data.frame(x = c(-1,-2,2,1), y = c(0,1,1,0)) #' (bezPolys <- bezier(points)) #' f <- as.function(bezPolys) #' df <- as.data.frame(f(s)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red") + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' ## example 4 #' points <- data.frame(x = c(-1,2,-2,1), y = c(0,1,1,0)) #' (bezPolys <- bezier(points)) #' f <- as.function(bezPolys) #' df <- as.data.frame(f(s)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red") + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' ## example 5 #' qplot(speed, dist, data = cars) #' #' s <- seq(0, 1, length.out = 201) #' p <- bezier(cars) #' f <- as.function(p) #' df <- as.data.frame(f(s)) #' qplot(speed, dist, data = cars) + #' geom_path(data = df, color = "red") #' #' # the curve is not invariant to permutations of the points #' # but it always goes through the first and last points #' permute_rows <- function(df) df[sample(nrow(df)),] #' p <- bezier(permute_rows(cars)) #' f <- as.function(p) #' df <- as.data.frame(f(s)) #' qplot(speed, dist, data = cars) + #' geom_path(data = df, color = "red") #' #' #' bezier <- function(..., indeterminate = "t"){ ## grab input dots <- as.list(match.call(expand.dots = FALSE))$"..." names(dots) <- as.character(match.call(expand.dots = FALSE)$"...") dots <- lapply(dots, eval) ## parse input into a data frame if(length(dots) == 1 && (is.data.frame(dots[[1]]) || is.matrix(dots[[1]]))){ points <- dots[[1]] if(is.matrix(points)) points <- as.data.frame(points) if(is.null(names(points))) names(points) <- paste0("x", 1:ncol(points)) } else if( all(vapply(dots, is.numeric, logical(1))) && all(vapply(dots, length, numeric(1)) == length(dots[[1]])) ){ points <- as.data.frame(do.call(rbind, dots)) if(is.null(names(dots[[1]]))){ names(points) <- paste0("x", 1:ncol(points)) } else { names(points) <- names(dots[[1]]) } row.names(points) <- NULL } ## make polynomial n <- nrow(points) bernPolys <- bernstein(0:(n-1), n-1, indeterminate) ## initialize bezPoly d <- ncol(points) bezPoly <- mp(rep("0", d)) for(k in 1:n){ ## convert vector of numerics into mpolyList polyWeights <- lapply(unname(points[k,]), function(.) mpoly(list(c(coef = .)))) class(polyWeights) <- "mpolyList" ## duplicate the bern basis function to the same length basis <- replicate(d, bernPolys[[k]], simplify = FALSE) class(basis) <- "mpolyList" ## accumulate onto bezPoly bezPoly <- bezPoly + polyWeights * basis } ## class with meta-data class(bezPoly) <- c("bezier", "mpolyList") attr(bezPoly, "bezier") <- list(points = points) ## return bezPoly } #' Bezier function #' #' Compute the Bezier function of a collection of polynomials. By Bezier #' function we mean the Bezier curve function, a parametric map running from t = #' 0, the first point, to t = 1, the last point, where the coordinate mappings #' are linear combinations of Bernstein polynomials. #' #' The function returned is vectorized and evaluates the Bezier curve in a #' numerically stable way with de Castlejau's algorithm (implemented in R). #' #' @param points a matrix or data frame of numerics. the rows represent points. #' @param weights the weights in a weighted Bezier curve #' @param ... ...; used internally #' @return function of a single parameter #' @author David Kahle #' @references \url{http://en.wikipedia.org/wiki/Bezier_curve}, #' \url{http://en.wikipedia.org/wiki/De_Casteljau's_algorithm} #' @seealso [bezier()] #' @name bezier_function #' @examples #' #' library(ggplot2); theme_set(theme_bw()) #' #' #' t <- seq(0, 1, length.out = 201) #' points <- data.frame(x = 0:3, y = c(0,1,-1,0)) #' #' #' f <- bezier_function(points) #' df <- as.data.frame(f(t)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red", size = 8) + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' f <- bezier_function(points, weights = c(1,5,5,1)) #' df <- as.data.frame(f(t)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red", size = 8) + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' f <- bezier_function(points, weights = c(1,10,10,1)) #' df <- as.data.frame(f(t)) #' #' ggplot(aes(x = x, y = y), data = df) + #' geom_point(data = points, color = "red", size = 8) + #' geom_path(data = points, color = "red") + #' geom_path() #' #' #' #' #' #' #' #' #' #' #' @rdname bezier_function #' @export bezier_function <- function(points, weights = rep(1L, nrow(points))){ n <- nrow(points) degree <- n-1 points <- cbind(1, as.matrix(points)) points <- weights * points combineTwo <- function(vec, t) t*vec[-length(vec)] + (1-t)*vec[-1] single_point_bezier <- function(.){ for(i in 1:degree) points <- apply(points, 2, combineTwo, t = .) points[-1] / points[1] } function(.){ if(length(.) > 1) return(t(sapply(., single_point_bezier))) single_point_bezier(.) } } #' @rdname bezier_function #' @export bezierFunction <- function(...) { .Deprecated("bezier_function") bezier_function(...) } #' @export #' @rdname as.function.mpolyList as.function.bezier <- function(x, ...) bezier_function(attr(x, "bezier")$points, ...) mpoly/R/mp.R0000644000176200001440000004643413522636150012370 0ustar liggesusers#' Define a multivariate polynomial. #' #' mp is a smart function which attempts to create a formal mpoly object from a #' character string containing the usual representation of a multivariate #' polynomial. #' #' @param string a character string containing a polynomial, see examples #' @param varorder (optional) order of variables in string #' @param stars_only if you format your multiplications using asterisks, setting #' this to \code{TRUE} will reduce preprocessing time #' @param vars a character vector of indeterminates #' @return An object of class mpoly. #' @author David Kahle \email{david@@kahle.io} #' @seealso [mpoly()] #' @name mp #' @examples #' #' ( m <- mp("x + y + x y") ) #' is.mpoly( m ) #' unclass(m) #' #' #' mp("x + 2 y + x^2 y + x y z") #' mp("x + 2 y + x^2 y + x y z", varorder = c("y", "z", "x")) #' #' ( ms <- mp(c("x + y", "2 x")) ) #' is.mpolyList(ms) #' #' #' gradient( mp("x + 2 y + x^2 y + x y z") ) #' gradient( mp("(x + y)^10") ) #' #' # mp and the print methods are kinds of inverses of each other #' ( polys <- mp(c("x + y", "x - y")) ) #' strings <- print(polys, silent = TRUE) #' strings #' mp(strings) #' #' @export #' @rdname mp make_indeterminate_list <- function (vars) { make_indeterminate <- function(var, ...) { v <- c(1, 1) names(v) <- c(var, "coef") structure(list(v), class = "mpoly") } uvars <- unique(vars) l <- lapply(uvars, make_indeterminate) names(l) <- uvars l } # make_indeterminate_list("a") # make_indeterminate_list(letters) #' @export #' @rdname mp mp <- function (string, varorder, stars_only = FALSE) { # deal with mpolyLists if (length(string) > 1) { # do basic mpoly parsing ps <- structure( lapply(string, mp, stars_only = stars_only), class = "mpolyList" ) # enforce varorder if present if (!missing(varorder)) { for (k in seq_along(ps)) { ps[[k]] <- structure( lapply(ps[[k]], function(term) { vars_in_term <- intersect(varorder, names(term)) term[c(vars_in_term, "coef")] }), class = "mpoly" ) } } # return early return(ps) } # clean spaces if needed if(!stars_only) { # put *s in for spaces, twice for situations like "x y z" while (str_detect(string, "([\\w\\^.*\\(\\)]+) +([\\w\\^.*\\(\\)]+)")) { string <- str_replace_all(string, "([\\w\\^.*\\(\\)]+) +([\\w\\^.*\\(\\)]+)", "\\1*\\2") } # fix )('s and situations like 2(x+1) # string <- str_replace_all(string, pattern = "\\)\\(", replacement = ")*(") # string <- str_replace_all(string, "([\\w\\^.*\\(\\)]+)(\\(\\))", "\\1*\\2") # fix things like "-x" string <- str_replace_all(string, "^-([\\w\\^.*\\(]+)", "-1*\\1") } # parse using R's parser and mpoly arithmetic expr <- parse(text = string)[[1]] vars <- stringr::str_extract_all( deparse(expr), "(? 1){ # if(missing(varorder)){ # mpolyList <- lapply(string, mp) # } else { # mpolyList <- lapply(string, mp, varorder = varorder) # } # class(mpolyList) <- "mpolyList" # return(mpolyList) # } # # # switch *'s for " "'s # string <- str_replace_all(string, fixed("*"), " ") # # # fix leading - # string <- str_replace(string, "^(\\s*-)", "0 + -1 ") # # # clean whitespace # string <- str_replace_all(string, " {2,}", " ") # string <- str_replace_all(string, "[\\n\\t]", " ") # # # clean -(x + y) -> -1 (x + y) # string <- str_replace_all(string, "^ *- *\\(", "-1 (") # # # clean double parens # string <- str_replace_all(string, fixed(")("), ") (") # # # check for bad things # unmatched_parentheses_stop(string) # empty_parenthetical_stop(string) # # # compute # out <- parse_parenthetical_polynomial(string) # # # check varorder argument # if(!missing(varorder)){ # # vars <- vars(out) # # if(!all(vars %in% varorder)){ # error <- sprintf( # "If specified, varorder must contain all computed vars - %s", # str_c(vars, collapse = ", ") # ) # stop(error, call. = FALSE) # } # # # order vars appropriately # vars <- intersect(varorder, vars) # out <- reorder.mpoly(out, varorder = vars) # } # # # return # out # # } # # # # string <- "x ((x+y)+2)" # parse_parenthetical_polynomial(string) # parse_parenthetical_polynomial("x ((x+y) + 2)") # parse_parenthetical_polynomial("-(x + y) + 2 x (x + y)^2 + 3 y") # parse_parenthetical_polynomial <- function(string){ # # # fix term joins # terms <- extract_polynomial_terms(string) # # # parse into mpolys # mpolys <- lapply(terms, parse_parenthetical_term) # # # add and return # Reduce(`+.mpoly`, mpolys) # # } # parse_parenthetical_term("3 y") # parse_parenthetical_term(" 3 (x + y) 4 (x - y) ") # parse_parenthetical_term("(x + y) (x - y)") # parse_parenthetical_term("(x + y)") # parse_parenthetical_term("(x + y)^2") # parse_parenthetical_term("(x + y)(x-y)") # parse_parenthetical_term("-2 (x + y)^2 (x - y)^0 4 (1+1)^3") # # # more complex usage # parse_parenthetical_term("((x^2))") # parse_parenthetical_term("((5^2))") # string <- "(1+1) (2^3 z (x+y)^2)^2" # parse_parenthetical_term(string) # parse_parenthetical_term("6 (x)") # parse_parenthetical_term("6.18033988749895 (x)") # parse_parenthetical_term <- function(string){ # # # short circuit if simpler # if(!contains_parenthetical_expression(string)) # return(parse_nonparenthetical_term(string)) # # # break into parenthetical pieces ("bubbles") # pieces <- term_parentheticals(string) # pieces <- pieces[pieces != ""] # # # mpoly pieces # mpolys <- lapply(pieces, function(piece){ # # # identify expression and exponent components # expr <- str_extract(piece, "\\([-()\\w+ \\^.]+\\)") # expr <- str_sub(expr, 2, -2) # take off parens # # # check for exponent on the outer parenthetical # last_paren_ndx <- nchar(piece) - str_locate(str_rev(piece), fixed(")"))[[1]] + 1L # string_after_paren <- str_sub(piece, last_paren_ndx + 1L) # "" or "^3" # # # if "^3", extract, otherwise 1 # if(str_detect(string_after_paren, fixed("^"))){ # exponent <- as.numeric(str_rev(str_extract(str_rev(string_after_paren), "[0-9]+"))) # gets first # } else { # exponent <- 1 # } # # # parse # if(contains_nested_parenthetical_expression(piece)){ # parse_parenthetical_polynomial(expr)^exponent # } else { # parse_nonparenthetical_polynomial(expr)^exponent # } # # }) # # # product and return # Reduce(`*.mpoly`, mpolys) # # } # parse_nonparenthetical_polynomial(" -1") # parse_nonparenthetical_polynomial("x-1") # parse_nonparenthetical_polynomial("5-2x") # parse_nonparenthetical_polynomial("5 - 2 x") # parse_nonparenthetical_polynomial("5 + -2x") # parse_nonparenthetical_polynomial("1--1") # parse_nonparenthetical_polynomial("1 - - 1") # parse_nonparenthetical_polynomial("5^2x") # parse_nonparenthetical_polynomial("5^2-x") # parse_nonparenthetical_polynomial("-x") # parse_nonparenthetical_polynomial("-1") # parse_nonparenthetical_polynomial("1+-x-x") # parse_nonparenthetical_polynomial("1 - -3") # # parse_nonparenthetical_polynomial("-x + 2y - 4x - -4") # # string <- "-4 + 2+2 x + 1 x y^4 -3 prq^3 -y - 3 x 2 - 3 y -2" # parse_nonparenthetical_polynomial(string) # parse_nonparenthetical_polynomial("x + y") # parse_nonparenthetical_polynomial("x - y+-xy") # parse_nonparenthetical_polynomial("1e-2 x") # parse_nonparenthetical_polynomial("1e+2 x") # parse_nonparenthetical_polynomial <- function(string){ # # # check to see if it's a single term # if (!str_detect(string, "[+]") && !str_detect(str_sub(string, 2), "[-]")) { # return(parse_nonparenthetical_term(string)) # } # # # regularize term joins (deal with minuses) # string <- fix_term_joins(string) # # # split polynomial # terms <- str_split(string, fixed(" + "))[[1]] # # # parse terms # mpolyTerms <- lapply(terms, parse_nonparenthetical_term) # # # combine and return # Reduce(`+.mpoly`, mpolyTerms) # # } # parse_parenthetical_term("t1a") # parse_nonparenthetical_term("12var 2 y 2x") # parse_nonparenthetical_term("-2 7") # parse_nonparenthetical_term("2 x y^2 3 2 3^2") # parse_nonparenthetical_term("2 x -2") # -> warn # parse_nonparenthetical_term("x") # parse_nonparenthetical_term("-x") # error # parse_nonparenthetical_term("+x") # correctly wrong # parse_nonparenthetical_term("-5x") # parse_nonparenthetical_term("-0x") # parse_nonparenthetical_term("1.5x") # parse_nonparenthetical_term("1.5^2x") # parse_nonparenthetical_term("1e-2 x") # correctly error # parse_nonparenthetical_term <- function(string){ # # # fix spaces around exponents "x ^ 2" -> "x^2" # string <- str_replace_all(string, " *\\^ *", "^") # # # fix spaces around minuses "x - 2" -> "x-2" # string <- str_replace_all(string, " *- *", "-") # # # split based on spaces # parts <- str_split(string, " ")[[1]] # parts <- parts[nchar(parts) > 0] # for "2 -2" # # # if more than one negative provided error # if(str_detect(str_sub(string, 2), fixed("-"))) # stop("Negative signs are only allowed at the beginning of terms.", call. = FALSE) # # # fix, e.g. "2x" # smashed_var_bool <- str_detect(parts, "^[-. ^0-9]+[a-zA-Z]") # if(any(smashed_var_bool)){ # places_to_break <- str_locate(parts[smashed_var_bool], "[a-zA-Z]")[,1] # for(k in seq_along(places_to_break)){ # parts[smashed_var_bool][k] <- str_c( # str_sub(parts[smashed_var_bool][k], 1, places_to_break[k]-1), # "|", # str_sub(parts[smashed_var_bool][k], places_to_break[k]) # ) # } # parts <- unlist(str_split(parts, fixed("|"))) # } # # # fix, e.g. "-y" # minus_var_bool <- str_detect(parts, "\\-[a-zA-Z]") # if(any(minus_var_bool)){ # parts[minus_var_bool] <- str_c("-1 ", str_sub(parts[minus_var_bool], 2)) # parts <- unlist(str_split(parts, " ")) # } # # # collect numeric elements # parts_with_vars <- str_detect(parts, "[a-zA-Z]") # if(all(parts_with_vars)){ # coef <- 1L # } else { # coef <- prod( # vapply( # as.list(parts[which(!parts_with_vars)]), # function(.) eval(parse(text = .)), # double(1) # ) # ) # this multiplies even, e.g., 5^2 # } # # # if only coefs are given, return # if(all(parts_with_vars == FALSE)) return(mpoly(list(c(coef = coef)))) # # # parse variable exponents # var_parts <- parts[parts_with_vars] # var_parts_with_exps_bool <- str_detect(var_parts, fixed("^")) # var_parts[!var_parts_with_exps_bool] <- str_c(var_parts[!var_parts_with_exps_bool], "^1") # var_parts <- str_split(var_parts, fixed("^")) # vars <- vapply(var_parts, `[`, character(1), 1L) # exps <- as.integer(vapply(var_parts, `[`, character(1), 2L)) # names(exps) <- vars # # # mpoly and return # mpoly(list(c(coef = coef, exps))) # } # fix_term_joins("-2 - -2x + y - -3 y - 2") # fix_term_joins("1e2x - 1e-2x + 1e+2x") # fix_term_joins("1-1") # fix_term_joins("x[1]") # fix_term_joins("x[1,1]") # fix_term_joins("1--1") # fix_term_joins("1 - - 1") # fix_term_joins("5 - 2 x") # fix_term_joins("5^2x - 1") # fix_term_joins("1+-xx-x") # fix_term_joins("-1-1") # fix_term_joins("1e-2 x") # fix_term_joins("1e+2 x") # fix_term_joins("1e2 x") # fix_term_joins("-1-1-") # error # fix_term_joins("-1-1+") # error # fix_term_joins <- function(string){ # # # make sure last char is not a sign # if(str_detect(string, "[+-]$")) stop(sprintf("Term %s does not terminate.", string), call. = FALSE) # # # zero trick for leading symbol, e.g. "-1 + x" -> "0 + -1 + x" # if (str_detect(string, "^[+-]")) { # if (str_detect(string, "^[+-]{2,}")) stop( # sprintf("%s cannot start an expression.", str_extract(string, "^[+-]+")), # call. = FALSE # ) # string <- str_c("0 + ", string) # } # # # fix scientific notation # sciRegex <- "[0-9.]+e[+-]?[0-9]+" # while(str_detect(string, sciRegex)){ # stringToReplace <- str_extract(string, sciRegex) # replacement <- format(as.numeric(stringToReplace)) # string <- str_replace(string, sciRegex, replacement) # } # # # break string into pieces of terms and joins # terms <- str_extract_all(string, "[\\w^.,|\\[\\]]+")[[1]] # joins <- str_split(string, "[\\w^.,|\\[\\]]+")[[1]] # if(joins[1] == "") joins <- joins[-1] # if(joins[length(joins)] == "") joins <- joins[-length(joins)] # if(length(joins) == 0L) return(string) # # # fix joins # pureJoins <- str_replace_all(joins, "\\s", "") # pureJoins[pureJoins == ""] <- "|" # if(any(nchar(pureJoins) > 3)) stop("Arithmetic sign sequence of more than two detected.", call. = FALSE) # cleanJoinMap <- c( # "-" = " + -1 ", "+" = " + ", "--" = " + ", # "++" = " + ", "+-" = " + -1 ", "-+" = " + -1 ", "|" = " " # ) # cleanedJoins <- unname(cleanJoinMap[pureJoins]) # cbind(joins, cleanedJoins) # # # reconstruct # n <- length(terms) + length(joins) # n always odd, first term always a \\w # temp <- character(n) # temp[seq.int(1L, n, 2L)] <- terms # temp[seq.int(2L, n-1L, 2L)] <- cleanedJoins # string <- str_c(temp, collapse = "") # # # strip leading "0 + " if needed # if(str_sub(string, 1L, 4L) == "0 + ") string <- str_sub(string, 5L) # # # return # string # } # string <- "-1 (x + y)+ 2 x (x + y) + 3 y" # string <- "2 (1 + x + (x - y))+ 2 x (x + y) + 3 y" # extract_polynomial_terms(string) # extract_polynomial_terms <- function(string){ # # # str_split(string, " *(? 0){ # for(k in 1:nrow(string_ndcs)){ # str_sub(piped_string, piped_ndcs[k,1], piped_ndcs[k,2]) <- # str_sub(string, string_ndcs[k,1], string_ndcs[k,2]) # } # } # # # split # str_split(piped_string, fixed("*"))[[1]] # } # an inner parenthetical is one that does not contain parentheticals # extract_leftmost_inner_parenthetical("(x + 5)") # extract_leftmost_inner_parenthetical("(x + 5)", contents_only = TRUE) # # extract_leftmost_inner_parenthetical("(x + 5)^10") # extract_leftmost_inner_parenthetical("(x + 5)^10", contents_only = TRUE) # extract_leftmost_inner_parenthetical("((x + 5)^10+2)^2") # extract_leftmost_inner_parenthetical("((x + 5)^10+2)", contents_only = TRUE) # extract_leftmost_inner_parenthetical("(1 + (x + 5)^10+2)^2") # extract_leftmost_inner_parenthetical <- function(string, contents_only = FALSE){ # string <- str_extract(string, "\\([^()]*\\)(?:\\^[0-9]+)?") # if(!contents_only) return(string) # str_extract(string, "\\(.*\\)") ->.; str_sub(., 2L, -2L) # } # blank_parentheticals(" -1 1 x (3 x + -1 (7 + -1 2 x))^2 7 (x + 1) -3 ") # blank_parentheticals(" -1 1 x (3 x + -1 (7 + -1 2 x))^2 7 (x + 1) -3 ", "*") # blank_parentheticals(" -1 1 x (3 x + -1 (7 + -1 2 x))^2 7 (x + 1) -3 ", "_") # blank_parentheticals <- function(string, char = "-"){ # # " -1 1 x (3 x + -1 (7 + -1 2 x))^2 7 (x + 1) -3 " -> # # " -1 1 x ------------------------- 7 ------- -3 " # # this blanks parentheticals from the inside out # # inside parentheticals are done first # # while(contains_parenthetical_expression(string)){ # bad <- extract_leftmost_inner_parenthetical(string) # string <- str_replace( # string, # "\\([^()]*\\)(?:\\^[0-9]+)?", # str_dup(char, nchar(bad)) # ) # } # string # } # string <- " -3 -1 (x + y)^2 4 (x - y)x 4 " # extract_nonparenthetical_elements(string) # string <- " (x + y)^2 (x - y)(x) " # extract_nonparenthetical_elements(string) # extract_nonparenthetical_elements <- function(string){ # # # remove parenthetical stuff # parenthetical_regex <- "\\([-+*a-zA-Z0-9.^ ()]+\\)(\\^\\d+)?" # nonparem_elts <- str_remove_all(string, parenthetical_regex) # nonparem_elts <- str_replace_all(nonparem_elts, "\\s+", " ") # nonparem_elts <- str_trim(nonparem_elts) # # # parenthesize and return # if (nonparem_elts == "") { # "" # } else { # str_c("(", str_trim(nonparem_elts), ")") # } # } # string <- " -3 (x + y)^2 4 (x - y)x 4 " # delete_nonparenthetical_elements(string) # string <- " (x + y)^2 (x - y) (x) (y) " # delete_nonparenthetical_elements(string) # string <- ".2 (x)" # delete_nonparenthetical_elements(string) # extract_parenthetical_elements <- function(string){ # # parenthetical_regex <- "\\([-+*a-zA-Z0-9.^ ()]+\\)(\\^\\d+)?" # str_extract_all(string, parenthetical_regex)[[1]] # # } # string <- " -3 (x + y)^2 4 (x - y)(x)x 4 " # term_parentheticals(string) # string <- " -(x + y)^2 3(x - y)(x) " # term_parentheticals(string) # string <- ".2 (x)" # term_parentheticals(string) # term_parentheticals <- function(string){ # # nonparens <- extract_nonparenthetical_elements(string) # parens <- extract_parenthetical_elements(string) # c(nonparens, parens) # # } # contains_parenthetical_expression <- function(string){ # any(str_detect(string, fixed("("))) # } # contains_nested_parenthetical_expression("5+5") # contains_nested_parenthetical_expression("(5+5)") # contains_nested_parenthetical_expression("((5+5))") # contains_nested_parenthetical_expression("x + (5 y) + 2") # contains_nested_parenthetical_expression("x + ((5 y) + 2)") # contains_nested_parenthetical_expression <- function(string){ # only_parentheses <- str_replace_all(string, "[^()]", "") # str_detect(only_parentheses, fixed("((")) # } # unmatched_parentheses_stop <- function(string){ # if(contains_parenthetical_expression(string)){ # open_paren_count <- str_count(string, fixed("(")) # closed_paren_count <- str_count(string, fixed(")")) # if (open_paren_count > closed_paren_count){ # stop("Parenthetical error: excess ('s detected.", call. = FALSE) # } else if(open_paren_count < closed_paren_count) { # stop("Parenthetical error: excess )'s detected.", call. = FALSE) # } # } # invisible() # } # empty_parenthetical_stop <- function(string) { # if (str_detect(string, "\\( *\\)")) { # stop("Expression contains empty parenthetical.", call. = FALSE) # } # } # str_rev <- function(string) str_c(rev.default(str_split(string, "")[[1]]), collapse = "") mpoly/R/mpoly.R0000644000176200001440000001261713476534026013117 0ustar liggesusers#' Define an mpoly object. #' #' mpoly is the most basic function used to create objects of class mpoly. #' However, it is not a general purpose function; for that see [mp()]. #' #' @param list a list from which to construct an mpoly object #' @param varorder (optional) a character vector setting the intrinsic variable #' order of the polynomial #' @return Object of class mpoly. #' @author David Kahle \email{david@@kahle.io} #' @seealso [mp()] #' @export #' @examples #' list <- list( #' c(x = 1, coef = 1, y = 0), #' c(x = 0, y = 1, coef = 2), #' c(y = 1, coef = -6), #' c(z = 1, coef = -3, x = 2), #' c(x = 1, coef = 0, x = 3), #' c(t = 1, coef = 4, t = 2, y = 4), #' c(x = 1), #' c(x = 1), #' c(coef = 5), #' c(coef = 5), #' c(coef = -5) #' ) #' #' mpoly(list) # 3 x - 4 y - 3 x^2 z + 4 y^4 t^3 + 5 #' mpoly(list, varorder = c("y", "z", "t", "x")) #' #' list <- list( c(x = 5, x = 2, coef = 5, coef = 6, y = 0) ) #' mpoly(list) #' #' mpoly <- function(list, varorder){ ## argument checking if(!is.list(list)) stop("input to mpoly must be a list.", call. = FALSE) if(!all(vapply(list, is.numeric, logical(1)))){ stop("each element of list must be of type numeric.", call. = FALSE) } if(any( unlist(lapply(list, function(v) nchar(names(v))), use.names = FALSE) == 0 )){ stop("each element of list must be named for every element.", call. = FALSE) } flatList <- unlist(list) flatList <- flatList[names(flatList) != "coef"] if(!all(is.wholenumber(flatList)) || any(flatList < 0)){ stop("degrees must be nonnegative integers.", call. = FALSE) } # give terms without a coefficient a coef of 1 addCoefIfMissing <- function(v){ if(any("coef" == names(v))) return(v) c(v, coef = 1) } list <- lapply(list, addCoefIfMissing) ## organize # remove terms with coef 0 list <- filterOutZeroTerms(list) # remove 0 degrees, combine like degrees, single coef as rightmost element list <- lapply(list, function(v){ # separate vardegs from coefs coef_ndx <- which(names(v) == "coef") coefs <- v[coef_ndx] v <- v[-coef_ndx] # combine like degrees (sum) if(length(names(v)) != length(unique(names(v)))) v <- fastNamedVecTapply(v, sum) # combine like coefficients (product) coefs <- c(coef = prod(coefs)) # remove zero degree elements, combine and return c(v[v != 0], coefs) }) ## set intrinsic varorder - done again after 0 degrees are removed vars <- unique(names(flatList)) # deal with varorder argument if(!missing(varorder)){ if( !setequal(vars, varorder) ){ stop("If specified varorder must be a permutation of ", stri_c(vars, collapse = ", "), call. = FALSE) } vars <- varorder } # sort variables in terms list <- lapply(list, function(v){ p <- length(v) - 1L if(p == 0L) return(v) c( (v[1:p])[intersect(vars, names(v[1:p]))], v["coef"] ) }) ## prepare to check if like terms are present monomials <- vapply(list, function(v){ p <- length(v) - 1 # remove coef on monomials paste(names(v[1:p]), v[1:p], sep = "", collapse = "") }, character(1)) # e.g. c("x1", "y1", "y1", "x2z1", "y4t3", "x1", "x1", "coef5") unique_monomials <- unique(monomials) ## check if like terms are present and, if so, correct if(length(monomials) != length(unique_monomials)){ matchedMonomials <- match(monomials, unique_monomials) matchedMonomials <- factor(matchedMonomials, levels = 1:max(matchedMonomials)) ndcs2combine <- split.default(1:length(list), matchedMonomials) list <- lapply(ndcs2combine, function(v){ if(length(v) == 1) return(list[[v]]) coef <- sum(vapply(list[v], `[`, double(1), length(list[[v[1]]]))) v <- list[[v[1]]] v["coef"] <- coef v }) names(list) <- NULL # i.e. list <- unname(list) } ## combine constant terms ## mpoly(list(c(x = 1, coef = 1), c(coef = 1), c(coef = 2))) nonConstantTerms <- fastFilter(isNotLengthOne, list) constantTerms <- fastFilter(isLengthOne, list) if(length(constantTerms) > 0){ list <- c( nonConstantTerms, list(Reduce(`+`, constantTerms)) ) } ## re-organize after like-terms combined list <- filterOutZeroTerms(list) ## return classed list class(list) <- "mpoly" list } filterOutZeroTerms <- function(list){ # a term is zero if any of its coefficients are 0 hasNonZeroCoef <- function(v) all(v["coef"] != 0) # remove terms with coef 0 list <- fastFilter(hasNonZeroCoef, list) # if all terms have been eliminated, recreate it with a 0 coef if(length(list) == 0) list <- list(c(coef = 0)) # return list } isLengthOne <- function(x) length(x) == 1L isNotLengthOne <- function(x) length(x) > 1L fastFilter <- function(f, x) x[vapply(x, f, logical(1))] fastNamedVecTapply <- function(x, f, type = double(1)){ uniqueNames <- unique(names(x)) matchedNames <- match(names(x), uniqueNames) # indices matchedNames <- factor(matchedNames, levels = 1:max(matchedNames)) groupIndices <- split.default(1:length(x), matchedNames) out <- vapply(groupIndices, function(ndcs) f(x[ndcs]), type) names(out) <- uniqueNames out } # x <- 1:10 # names(x) <- sample(letters[1:3], 10, replace = TRUE) # fastNamedVecTapply(x, sum) # tapply(x, names(x), sum) mpoly/R/mpolyListArithmetic.R0000644000176200001440000000656413476534026015771 0ustar liggesusers#' Element-wise arithmetic with vectors of multivariate polynomials. #' #' Element-wise arithmetic with vectors of multivariate polynomials. #' #' @param e1 an object of class mpolyList #' @param e2 an object of class mpolyList #' @return An object of class mpolyList. #' @name mpolyListArithmetic #' @examples #' #' ( ms1 <- mp( c("x", 'y') ) ) #' ( ms2 <- mp( c("y", '2 x^2') ) ) #' ms1 + ms2 #' ms1 - ms2 #' ms1 * ms2 #' ms1^3 #' NULL #' @rdname mpolyListArithmetic #' @export `+.mpolyList` <- function(e1, e2){ # argument check if (is.numeric(e1)) { e1 <- structure( lapply(e1, function(.) structure(list(c("coef" = .)), class = "mpoly")), class = "mpolyList" ) } # if (is.mpoly(e1)) e1 <- mpolyList(e1) if (is.numeric(e2)) { e2 <- structure( lapply(e2, function(.) structure(list(c("coef" = .)), class = "mpoly")), class = "mpolyList" ) } # if (is.mpoly(e2)) e2 <- mpolyList(e2) stopifnot(is.mpolyList(e1)) stopifnot(is.mpolyList(e2)) # fix lengths if (length(e1) == 1L && length(e2) != 1L) { e1 <- structure( replicate(length(e2), e1[[1L]], simplify = FALSE), class = "mpolyList" ) } if (length(e1) != 1L && length(e2) == 1L) { e2 <- structure( replicate(length(e1), e2[[1L]], simplify = FALSE), class = "mpolyList" ) } if(length(e1) != length(e2)) stop("e1 and e2 must have equal length.", call. = FALSE) # template outcome out <- vector(length = length(e1), mode = "list") # compute sums for (k in seq_along(out)) out[[k]] <- e1[[k]] + e2[[k]] # return structure(out, class = "mpolyList") } #' @rdname mpolyListArithmetic #' @export `-.mpolyList` <- function(e1, e2) e1 + -1*e2 #' @rdname mpolyListArithmetic #' @export `*.mpolyList` <- function(e1, e2){ # argument check if (is.numeric(e1)) { e1 <- structure( lapply(e1, function(.) structure(list(c("coef" = .)), class = "mpoly")), class = "mpolyList" ) } # if (is.mpoly(e1)) e1 <- mpolyList(e1) if (is.numeric(e2)) { e2 <- structure( lapply(e2, function(.) structure(list(c("coef" = .)), class = "mpoly")), class = "mpolyList" ) } # if (is.mpoly(e2)) e2 <- mpolyList(e2) stopifnot(is.mpolyList(e1)) stopifnot(is.mpolyList(e2)) # fix lengths if (length(e1) == 1L && length(e2) != 1L) { e1 <- structure( replicate(length(e2), e1[[1L]], simplify = FALSE), class = "mpolyList" ) } if (length(e1) != 1L && length(e2) == 1L) { e2 <- structure( replicate(length(e1), e2[[1L]], simplify = FALSE), class = "mpolyList" ) } if(length(e1) != length(e2)) stop("e1 and e2 must have equal length.", call. = FALSE) # template outcome out <- vector(length = length(e1), mode = "list") # compute products for (k in seq_along(out)) out[[k]] <- e1[[k]] * e2[[k]] # return structure(out, class = "mpolyList") } #' @rdname mpolyArithmetic #' @export `^.mpolyList` <- function(e1, e2){ stopifnot(is.mpolyList(e1)) if(!is.wholenumber(e2) || e2 < 0) stop("exponent must be a positive integer.", call. = FALSE) out <- structure( replicate( length(e1), structure(list(c("coef" = 1L)), class = "mpoly"), simplify = FALSE ), class = "mpolyList" ) if(e2 == 0) return(out) for(k in 1:e2) out <- out * e1 out } mpoly/R/mpoly-equal.R0000644000176200001440000000215013476534026014213 0ustar liggesusers#' Determine whether two multivariate polynomials are equal. #' #' Determine whether two multivariate polynomials are equal. #' #' @param e1 an object of class mpoly #' @param e2 an object of class mpoly #' @method == mpoly #' @aliases ==.mpoly == #' @return A logical value. #' @name mpoly-equal #' @examples #' #' p1 <- mp("x + y + 2 z") #' p1 == p1 #' #' p2 <- reorder(p1, order = "lex", varorder = c("z","y","x")) #' p1 == p2 #' p2 <- reorder(p1, order = "lex", varorder = c("z","w","y","x")) #' p1 == p2 #' p1 == ( 2 * p2 ) #' #' p1 <- mp("x + 1") #' p2 <- mp("x + 1") #' identical(p1, p2) #' p1 == p2 #' #' mp("x + 1") == mp("y + 1") #' mp("2") == mp("1") #' mp("1") == mp("1") #' mp("0") == mp("-0") #' #' #' @rdname mpoly-equal #' @export `==.mpoly` <- function(e1, e2) { if(!is.mpoly(e1) || !is.mpoly(e2)) stop("e1 and e2 must be of class mpoly.", call. = FALSE) diff <- e1 - e2 if( length(diff) == 1 && length(diff[[1]]) == 1 && diff[[1]] == 0 ){ return(TRUE) } else { return(FALSE) } } #' @rdname mpoly-equal #' @export `!=.mpoly` <- function(e1, e2) !(e1 == e2) mpoly/R/round.mpoly.R0000644000176200001440000000235213476534026014240 0ustar liggesusers#' Round the coefficients of a polynomial #' #' Round the coefficients of an mpoly object. #' #' @param x an mpoly object #' @param digits number of digits to round to #' @return the rounded mpoly object #' @author David Kahle \email{david@@kahle.io} #' @seealso [mp()] #' @export #' @examples #' #' p <- mp("x + 3.14159265")^4 #' p #' round(p) #' round(p, 0) #' #' \dontrun{ #' library(plyr) #' library(ggplot2) #' library(stringr) #' #' n <- 101 #' s <- seq(-5, 5, length.out = n) #' #' # one dimensional case #' df <- data.frame(x = s) #' df <- mutate(df, y = -x^2 + 2*x - 3 + rnorm(n, 0, 2)) #' qplot(x, y, data = df) #' mod <- lm(y ~ x + I(x^2), data = df) #' p <- as.mpoly(mod) #' qplot(x, y, data = df) + #' stat_function(fun = as.function(p), colour = 'red') #' #' p #' round(p, 1) #' qplot(x, y, data = df) + #' stat_function(fun = as.function(p), colour = 'red') + #' stat_function(fun = as.function(round(p,1)), colour = 'blue') #' #' #' } #' round.mpoly <- function(x, digits = 3){ ## round coefficients p <- lapply(x, function(term){ term["coef"] <- round(term["coef"], digits = digits) term }) ## drop zero terms p <- Filter(function(v) v[["coef"]] != 0, p) ## class class(p) <- "mpoly" ## out p } mpoly/R/legendre.R0000644000176200001440000000637313527034713013541 0ustar liggesusers#' Legendre polynomials #' #' Legendre polynomials as computed by orthopolynom. #' #' @param degree degree of polynomial #' @param indeterminate indeterminate #' @param normalized provide normalized coefficients #' @return a mpoly object or mpolyList object #' @author David Kahle calling code from the orthopolynom package #' @seealso [orthopolynom::legendre.polynomials()], #' \url{http://en.wikipedia.org/wiki/Legendre_polynomials} #' @export #' @examples #' #' legendre(0) #' legendre(1) #' legendre(2) #' legendre(3) #' legendre(4) #' legendre(5) #' legendre(6) #' #' legendre(2) #' legendre(2, normalized = TRUE) #' #' legendre(0:5) #' legendre(0:5, normalized = TRUE) #' legendre(0:5, indeterminate = "t") #' #' #' #' # visualize the legendre polynomials #' #' library(ggplot2); theme_set(theme_classic()) #' library(tidyr) #' #' s <- seq(-1, 1, length.out = 201) #' N <- 5 # number of legendre polynomials to plot #' (legPolys <- legendre(0:N)) #' #' # see ?bernstein for a better understanding of #' # how the code below works #' #' df <- data.frame(s, as.function(legPolys)(s)) #' names(df) <- c("x", paste0("P_", 0:N)) #' mdf <- gather(df, degree, value, -x) #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' #' # legendre polynomials and the QR decomposition #' n <- 201 #' x <- seq(-1, 1, length.out = n) #' mat <- cbind(1, x, x^2, x^3, x^4, x^5) #' Q <- qr.Q(qr(mat)) #' df <- as.data.frame(cbind(x, Q)) #' names(df) <- c("x", 0:5) #' mdf <- gather(df, degree, value, -x) #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' Q <- apply(Q, 2, function(x) x / x[n]) #' df <- as.data.frame(cbind(x, Q)) #' names(df) <- c("x", paste0("P_", 0:5)) #' mdf <- gather(df, degree, value, -x) #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' #' # chebyshev polynomials are orthogonal in two ways: #' P2 <- as.function(legendre(2)) #' P3 <- as.function(legendre(3)) #' P4 <- as.function(legendre(4)) #' #' integrate(function(x) P2(x) * P3(x), lower = -1, upper = 1) #' integrate(function(x) P2(x) * P4(x), lower = -1, upper = 1) #' integrate(function(x) P3(x) * P4(x), lower = -1, upper = 1) #' #' n <- 10000L #' u <- runif(n, -1, 1) #' 2 * mean(P2(u) * P3(u)) #' 2 * mean(P2(u) * P4(u)) #' 2 * mean(P3(u) * P4(u)) #' #' (2/n) * sum(P2(u) * P3(u)) #' (2/n) * sum(P2(u) * P4(u)) #' (2/n) * sum(P3(u) * P4(u)) #' legendre <- function(degree, indeterminate = "x", normalized = FALSE){ stopifnot(all(is.wholenumber(degree))) stopifnot(all(degree >= 0)) ## make coefs coefs <- legendre.polynomials(max(degree), normalized = normalized) ## if only one degree is wanted, return that if(length(degree) == 1){ coefs <- rev.default(coefs)[[1]] p <- as.mpoly.polynomial(coefs, indeterminate) class(p) <- c("legendre", "mpoly") attr(p, "legendre") <- list(degree = length(coefs)-1, indeterminate = indeterminate, normalized = normalized) return(p) } ## if several are wanted, return them coefs <- coefs[degree+1] ps <- lapply(coefs, function(polynomial){ p <- as.mpoly.polynomial(polynomial, indeterminate) class(p) <- c("legendre", "mpoly") attr(p, "legendre") <- list(degree = length(polynomial)-1, indeterminate = indeterminate, normalized = normalized) p }) class(ps) <- "mpolyList" ps } mpoly/R/eq_mp.R0000644000176200001440000000135413476534026013054 0ustar liggesusers#' Convert an equation to a polynomial #' #' Convert characters of the form \code{"p1 = p2"} (or similar) to an mpoly #' object representing \code{p1 - p2}. #' #' @param string a character string containing a polynomial, see examples #' @param ... arguments to pass to [mpoly()] #' @return An object of class mpoly or mpolyList. #' @author David Kahle \email{david@@kahle.io} #' @seealso [mpoly()] #' @export #' @examples #' #' eq_mp(c("y = x", "y == (x + 2)")) #' #' #' eq_mp <- function (string, ...) { # "y = x" -> "(y) - (x)" strings <- vapply( stringi::stri_split_regex(string, "\\s*==?\\s*"), function(x) stringi::stri_c("(", x[1], ") - (", x[2], ")"), character(1) ) # parse and output mp(strings, ...) } mpoly/R/mpolyList.R0000644000176200001440000000156013476534026013746 0ustar liggesusers#' Define a collection of multivariate polynomials. #' #' Combine a series of mpoly objects into a mpolyList. #' #' @param ... a series of mpoly objects. #' @return An object of class mpolyList. #' @export #' @examples #' ( p1 <- mp("t^4 - x") ) #' ( p2 <- mp("t^3 - y") ) #' ( p3 <- mp("t^2 - z") ) #' ( ms <- mpolyList(p1, p2, p3) ) #' is.mpolyList( ms ) #' #' mpolyList(mp("x + 1")) #' p <- mp("x + 1") #' mpolyList(p) #' #' ps <- mp(c("x + 1", "y + 2")) #' is.mpolyList(ps) #' #' #' f <- function(){ #' a <- mp("1") #' mpolyList(a) #' } #' f() #' #' mpolyList <- function(...){ arguments <- as.list(match.call()[-1]) out <- lapply(arguments, eval, parent.frame(1)) # if(is.mpoly(out)) out <- list(out) if(!all(vapply(out, is.mpoly, logical(1)))){ stop("Each argument must be of class mpoly.", call. = FALSE) } class(out) <- "mpolyList" out } mpoly/R/jacobi.R0000644000176200001440000000527513476534026013210 0ustar liggesusers#' Jacobi polynomials #' #' Jacobi polynomials as computed by orthopolynom. #' #' @param degree degree of polynomial #' @param alpha the first parameter, also called p #' @param beta the second parameter, also called q #' @param kind "g" or "p" #' @param indeterminate indeterminate #' @param normalized provide normalized coefficients #' @return a mpoly object or mpolyList object #' @author David Kahle calling code from the orthopolynom package #' @seealso [orthopolynom::jacobi.g.polynomials()], #' [orthopolynom::jacobi.p.polynomials()], #' \url{http://en.wikipedia.org/wiki/Jacobi_polynomials} #' @export #' @examples #' #' jacobi(0) #' jacobi(1) #' jacobi(2) #' jacobi(3) #' jacobi(4) #' jacobi(5) #' jacobi(6) #' jacobi(10, 2, 2, normalized = TRUE) #' #' jacobi(0:5) #' jacobi(0:5, normalized = TRUE) #' jacobi(0:5, kind = "g") #' jacobi(0:5, indeterminate = "t") #' #' #' #' # visualize the jacobi polynomials #' #' library(ggplot2); theme_set(theme_classic()) #' library(tidyr) #' #' s <- seq(-1, 1, length.out = 201) #' N <- 5 # number of jacobi polynomials to plot #' (jacPolys <- jacobi(0:N, 2, 2)) #' #' df <- data.frame(s, as.function(jacPolys)(s)) #' names(df) <- c("x", paste0("P_", 0:N)) #' mdf <- gather(df, degree, value, -x) #' qplot(x, value, data = mdf, geom = "line", color = degree) #' #' qplot(x, value, data = mdf, geom = "line", color = degree) + #' coord_cartesian(ylim = c(-30, 30)) #' #' #' jacobi <- function(degree, alpha = 1, beta = 1, kind = "p", indeterminate = "x", normalized = FALSE){ stopifnot(all(is.wholenumber(degree))) stopifnot(all(degree >= 0)) ## deal with kind stopifnot(kind %in% c("g","p")) ## make coefs if(kind == "g") coefs <- jacobi.g.polynomials(max(degree), p = alpha, q = beta, normalized) if(kind == "p") coefs <- jacobi.p.polynomials(max(degree), alpha = alpha, beta = beta, normalized) ## if only one degree is wanted, return that if(length(degree) == 1){ coefs <- rev.default(coefs)[[1]] p <- as.mpoly.polynomial(coefs, indeterminate) class(p) <- c("jacobi", "mpoly") attr(p, "jacobi") <- list( degree = length(polynomial)-1, kind = kind, indeterminate = indeterminate, normalized = normalized, alpha = alpha, beta = beta ) return(p) } ## if several are wanted, return them coefs <- coefs[degree+1] ps <- lapply(coefs, function(polynomial){ p <- 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*tools/README-laguerre-1.png 8ca5ced1ce2197ccb56ae6e9de8c487f *tools/README-legendre-1.png 8ae72dd511f3004106aad700a91ebf97 *tools/README-lm-1.png mpoly/inst/0000755000176200001440000000000013230676064012376 5ustar liggesusersmpoly/inst/CITATION0000644000176200001440000000075413216242052013527 0ustar liggesuserscitHeader("To cite mpoly in publications, please use:") citEntry(entry = "article", author = "David Kahle", title = "mpoly: Multivariate Polynomials in {R}", journal = "The R Journal", year = "2013", volume = "5", number = "1", pages = "162--170", url = "https://journal.r-project.org/archive/2013-1/kahle.pdf", textVersion = "D. Kahle (2013). mpoly: Multivariate Polynomials in R. The R Journal, 5(1), 162-170. URL https://journal.r-project.org/archive/2013-1/kahle.pdf" )