# Taylor series of multivariate functions

library(calculus)

The function taylor provides a convenient way to compute the Taylor series of arbitrary unidimensional or multidimensional functions. The mathematical function can be specified both as a character string or as a function. Symbolic or numerical methods are applied accordingly. For univariate functions, the $$n$$-th order Taylor approximation centered in $$x_0$$ is given by:

$f(x) \simeq \sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$

where $$f^{(k)}(x_0)$$ denotes the $$k$$-th order derivative evaluated in $$x_0$$. By using multi-index notation, the Taylor series is generalized to multidimensional functions with an arbitrary number of variables:

$f(x) \simeq \sum_{|k|=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$

where now $$x=(x_1,\dots,x_d)$$ is the vector of variables, $$k=(k_1,\dots,k_d)$$ gives the order of differentiation with respect to each variable $$f^{(k)}=\frac{\partial^{(|k|)}f}{\partial^{(k_1)}_{x_1}\cdots \partial^{(k_d)}_{x_d}}$$, and:

$|k| = k_1+\cdots+k_d \quad\quad k!=k_1!\cdots k_d! \quad\quad x^k=x_1^{k_1}\cdots x_d^{k_d}$

The summation runs for $$0\leq |k|\leq n$$ and identifies the set

$\{(k_1,\cdots,k_d):k_1+\cdots k_d \leq n\}$

that corresponds to the partitions of the integer $$n$$. These partitions can be computed with the function partitions that is included in the package and optimized in C++ for speed and flexibility. For example, the following call generates the partitions needed for the $$2$$-nd order Taylor expansion for a function of $$3$$ variables:

partitions(n = 2, length = 3, fill = TRUE, perm = TRUE, equal = FALSE)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,]    0    0    0    1    0    0    2    0    1     1
#> [2,]    0    0    1    0    0    2    0    1    0     1
#> [3,]    0    1    0    0    2    0    0    1    1     0

Based on these partitions, the function taylor computes the corresponding derivatives and builds the Taylor series. The output is a list containing the Taylor series, the order of the expansion, and a data.frame containing the variables, coefficients and degrees of each term in the Taylor series.

taylor("exp(x)", var = "x", order = 2)
#> $f #>  "(1) * 1 + (1) * x^1 + (0.5) * x^2" #> #>$order
#>  2
#>
#> $terms #> var coef degree #> 0 1 1.0 0 #> 1 x^1 1.0 1 #> 2 x^2 0.5 2 By default, the series is centered in $$x_0=0$$ but the function also supports $$x_0\neq 0$$, the multivariable case, and the approximation of user defined R functions. f <- function(x, y) log(y)*sin(x) taylor(f, var = c(x = 0, y = 1), order = 2) #>$f
#>  "(0.999999999969436) * x^1*(y-1)^1"
#>
#> $order #>  2 #> #>$terms
#>             var coef degree
#> 0,0           1    0      0
#> 0,1     (y-1)^1    0      1
#> 1,0         x^1    0      1
#> 0,2     (y-1)^2    0      2
#> 2,0         x^2    0      2
#> 1,1 x^1*(y-1)^1    1      2

## Cite as

Guidotti, E. (2020). “calculus: High dimensional numerical and symbolic calculus in R”. https://arxiv.org/abs/2101.00086

A BibTeX entry for LaTeX users is

@Misc{,
title = {calculus: High Dimensional Numerical and Symbolic Calculus in R},
author = {Emanuele Guidotti},
year = {2020},
eprint = {2101.00086},
archiveprefix = {arXiv},
primaryclass = {cs.MS},
url = {https://arxiv.org/abs/2101.00086}
}