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 -9.767766e-12

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 (2022). “calculus: High-Dimensional Numerical and Symbolic Calculus in R.” Journal of Statistical Software, 104(5), 1-37. doi:10.18637/jss.v104.i05

A BibTeX entry for LaTeX users is

@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},
}