Computes the numerical Hessian of functions or the symbolic Hessian of characters.

hessian(f, var, params = list(), accuracy = 4, stepsize = NULL, drop = TRUE)

f %hessian% var

## Arguments

f array of characters or a function returning a numeric array. vector giving the variable names with respect to which the derivatives are to be computed and/or the point where the derivatives are to be evaluated. See derivative. list of additional parameters passed to f. degree of accuracy for numerical derivatives. finite differences stepsize for numerical derivatives. It is based on the precision of the machine by default. if TRUE, return the Hessian as a matrix and not as an array for scalar-valued functions.

## Value

Hessian matrix for scalar-valued functions when drop=TRUE, array otherwise.

## Details

In Cartesian coordinates, the Hessian of a scalar-valued function $$F$$ is the square matrix of second-order partial derivatives:

$$(H(F))_{ij} = \partial_{ij}F$$

When the function $$F$$ is a tensor-valued function $$F_{d_1,\dots,d_n}$$, the hessian is computed for each scalar component.

$$(H(F))_{d_1\dots d_n,ij} = \partial_{ij}F_{d_1\dots d_n}$$

It might be tempting to apply the definition of the Hessian as the Jacobian of the gradient to write it in arbitrary orthogonal coordinate systems. However, this results in a Hessian matrix that is not symmetric and ignores the distinction between vector and covectors in tensor analysis. The generalization to arbitrary coordinate system is not currently supported.

## Functions

• %hessian%: binary operator with default parameters.

Guidotti, E. (2020). "calculus: High dimensional numerical and symbolic calculus in R". https://arxiv.org/abs/2101.00086

Other differential operators: curl(), derivative(), divergence(), gradient(), jacobian(), laplacian()

## Examples

### symbolic Hessian
hessian("x*y*z", var = c("x", "y", "z"))
#>      [,1] [,2] [,3]
#> [1,] "0"  "z"  "y"
#> [2,] "z"  "0"  "x"
#> [3,] "y"  "x"  "0"
### numerical Hessian in (x=1, y=2, z=3)
f <- function(x, y, z) x*y*z
hessian(f = f, var = c(x=1, y=2, z=3))
#>              [,1]        [,2]         [,3]
#> [1,] 1.100056e-10 3.00000e+00 2.000000e+00
#> [2,] 3.000000e+00 2.75014e-11 1.000000e+00
#> [3,] 2.000000e+00 1.00000e+00 1.222284e-11
### vectorized interface
f <- function(x) x*x*x
hessian(f = f, var = c(1, 2, 3))
#>              [,1]        [,2]         [,3]
#> [1,] 1.100056e-10 3.00000e+00 2.000000e+00
#> [2,] 3.000000e+00 2.75014e-11 1.000000e+00
#> [3,] 2.000000e+00 1.00000e+00 1.222284e-11
### symbolic vector-valued functions
f <- c("y*sin(x)", "x*cos(y)")
hessian(f = f, var = c("x","y"))
#> , , 1
#>
#>      [,1]            [,2]
#> [1,] "-(y * sin(x))" "cos(x)"
#> [2,] "0"             "-sin(y)"
#>
#> , , 2
#>
#>      [,1]      [,2]
#> [1,] "cos(x)"  "0"
#> [2,] "-sin(y)" "-(x * cos(y))"
#>
### numerical vector-valued functions
f <- function(x) c(sum(x), prod(x))
hessian(f = f, var = c(0,0,0))
#> , , 1
#>
#>              [,1] [,2] [,3]
#> [1,] 1.790455e-13    0    0
#> [2,] 0.000000e+00    0    0
#>
#> , , 2
#>
#>      [,1]         [,2] [,3]
#> [1,]    0 1.790455e-13    0
#> [2,]    0 0.000000e+00    0
#>
#> , , 3
#>
#>      [,1] [,2]         [,3]
#> [1,]    0    0 1.790455e-13
#> [2,]    0    0 0.000000e+00
#>
### binary operator
"x*y^2" %hessian% c(x=1, y=3)
#>      [,1] [,2]
#> [1,]    0    6
#> [2,]    6    2