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 + (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 \(x_0=0\) 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
#> [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
```

*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