# High order derivatives of multivariate functions

    library(calculus)

The function `derivative` performs high-order symbolic and numerical differentiation for generic tensors with respect to an arbitrary number of variables. The function behaves differently depending on the arguments `order`, the order of differentiation, and `var`, the variable names with respect to which the derivatives are computed.

When multiple variables are provided and `order` is a single integer \(n\), then the \(n\)-th order derivative is computed for each element of the tensor with respect to each variable:

\[D = \partial^{(n)} \otimes F\]

that is:

\[D_{i,\dots,j,k} = \partial^{(n)}_{k} F_{i,\dots,j}\]

where \(F\) is the tensor of functions and \(\partial_k^{(n)}\) denotes the \(n\)-th order partial derivative with respect to the \(k\)-th variable.

When `order` matches the length of `var`, it is assumed that the differentiation order is provided for each variable. In this case, each element is derived \(n_k\) times with respect to the \(k\)-th variable, for each of the \(m\) variables.

\[D_{i,\dots,j} = \partial^{(n_1)}_1\cdots\partial^{(n_m)}_m F_{i,\dots,j}\]

The same applies when `order` is a named vector giving the differentiation order for each variable. For example, `order = c(x=1, y=2)` differentiates once with respect to \(x\) and twice with respect to \(y\). A call with `order = c(x=1, y=0)` is equivalent to `order = c(x=1)`.

To compute numerical derivatives or to evaluate symbolic derivatives at a point, the function accepts a named vector for the argument `var`; e.g. `var = c(x=1, y=2)` evaluates the derivatives in \(x=1\) and \(y=2\). For `functions` where the first argument is used as a parameter vector, `var` should be a `numeric` vector indicating the point at which the derivatives are to be calculated.

## Examples

Symbolic derivatives of univariate functions: \(\partial_x sin(x)\).

    derivative(f = "sin(x)", var = "x")
    #> [1] "cos(x)"

Evaluation of symbolic and numerical derivatives: \(\partial_x sin(x)|_{x=0}\).

    sym <- derivative(f = "sin(x)", var = c(x = 0))
    num <- derivative(f = function(x) sin(x), var = c(x = 0))

    #> Symbolic  Numeric
    #>        1        1

High order symbolic and numerical derivatives: \(\partial^{(4)}_x sin(x)|_{x=0}\).

    sym <- derivative(f = "sin(x)", var = c(x = 0), order = 4)
    num <- derivative(f = function(x) sin(x), var = c(x = 0), order = 4)

    #>      Symbolic       Numeric
    #>  0.000000e+00 -2.922023e-11

Symbolic derivatives of multivariate functions: \(\partial_x^{(1)}\partial_y^{(2)} y^2sin(x)\).

    derivative(f = "y^2*sin(x)", var = c("x", "y"), order = c(1, 2))
    #> [1] "2 * cos(x)"

Numerical derivatives of multivariate functions: \(\partial_x^{(1)}\partial_y^{(2)} y^2sin(x)|_{x=0,y=0}\) with degree of accuracy \(O(h^6)\).

    f <- function(x, y) y^2*sin(x)
    derivative(f, var = c(x=0, y=0), order = c(1, 2), accuracy = 6)
    #> [1] 2

Symbolic gradient of multivariate functions: \(\partial_{x,y}x^2y^2\).

    derivative("x^2*y^2", var = c("x", "y"))
    #>      [,1]          [,2]
    #> [1,] "2 * x * y^2" "x^2 * (2 * y)"

High order derivatives of multivariate functions: \(\partial^{(6)}_{x,y}x^6y^6\).

    derivative("x^6*y^6", var = c("x", "y"), order = 6)
    #>      [,1]                            [,2]
    #> [1,] "6 * (5 * (4 * (3 * 2))) * y^6" "x^6 * (6 * (5 * (4 * (3 * 2))))"

Numerical gradient of multivariate functions: \(\partial_{x,y}x^2y^2|_{x = 1, y = 2}\).

    f <- function(x, y) x^2*y^2
    derivative(f, var = c(x=1, y=2))
    #>      [,1] [,2]
    #> [1,]    8    4

Numerical Jacobian of vector valued functions: \(\partial_{x,y}[xy,x^2y^2]|_{x = 1, y = 2}\).

    f <- function(x, y) c(x*y, x^2*y^2)
    derivative(f, var = c(x=1, y=2))
    #>      [,1] [,2]
    #> [1,]    2    1
    #> [2,]    8    4

Numerical Jacobian of vector valued where the first argument is used as a parameter vector: \(\partial_{X}[\sum_ix_i, \prod_ix_i]|_{X = 0}\).

    f <- function(x) c(sum(x), prod(x))
    derivative(f, var = c(0, 0, 0))
    #>      [,1] [,2] [,3]
    #> [1,]    1    1    1
    #> [2,]    0    0    0

## 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}
    }