taylor.Rmd

---
title: "Taylor series of multivariate functions"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Taylor series of multivariate functions}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(calculus)
```

The function [`taylor`](https://calculus.eguidotti.com/reference/taylor.html) 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`](https://calculus.eguidotti.com/reference/partitions.html) 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:

```{r}
partitions(n = 2, length = 3, fill = TRUE, perm = TRUE, equal = FALSE)
```

Based on these partitions, the function [`taylor`](https://calculus.eguidotti.com/reference/taylor.html) 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.

```{r}
taylor("exp(x)", var = "x", order = 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`.

```{r}
f <- function(x, y) log(y)*sin(x)
taylor(f, var = c(x = 0, y = 1), order = 2)
```

## Cite as

Guidotti E (2022). “calculus: High-Dimensional Numerical and Symbolic Calculus in R.” _Journal of Statistical Software_, *104*(5), 1-37. [doi:10.18637/jss.v104.i05](https://doi.org/10.18637/jss.v104.i05)

A BibTeX entry for LaTeX users is

```bibtex
@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},
}
```