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 x₀ is given by:

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

where f^(k)(x₀) denotes the k-th order derivative evaluated in x₀. 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 = (x₁, …, x_d) is the vector of variables, k = (k₁, …, 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| = k₁ + ⋯ + k_d    k! = k₁!⋯k_d!    x^k = x₁^k₁⋯x_d^k_d

The summation runs for 0 ≤ |k| ≤ n and identifies the set

{(k₁, ⋯, k_d) : k₁ + ⋯k_d ≤ 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 + (1) * x^1 + (0.5) * x^2"
#> 
#> $order
#> [1] 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 x₀ = 0 but the function also supports x₀ ≠ 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
#> [1] "(0.999999999969436) * x^1*(y-1)^1"
#> 
#> $order
#> [1] 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

@Article{calculus,
  title = {{calculus}: High-Dimensional Numerical and Symbolic Calculus in {R}},
  author = {Emanuele Guidotti},
  journal = {Journal of Statistical Software},
  year = {2022},
  volume = {104},
  number = {5},
  pages = {1--37},
  doi = {10.18637/jss.v104.i05},
}