Hessian Matrix — hessian • pracma

Numerically compute the Hessian matrix.

hessian(f, x0, h = .Machine$double.eps^(1/4), ...)

Arguments

f: univariate function of several variables.
x0: point in \(R^n\).
h: step size.
...: variables to be passed to f.

Details

Computes the hessian matrix based on the three-point central difference formula, expanded to two variables.

Assumes that the function has continuous partial derivatives.

Value

An n-by-n matrix with \(\frac{\partial^2 f}{\partial x_i \partial x_j}\) as (i, j) entry.

References

Fausett, L. V. (2007). Applied Numerical Analysis Using Matlab. Second edition, Prentice Hall.

See also

hessdiag, hessvec, laplacian

Examples

f <- function(x) cos(x[1] + x[2])
x0 <- c(0, 0)
hessian(f, x0)
#>      [,1] [,2]
#> [1,]   -1   -1
#> [2,]   -1   -1

f <- function(u) {
    x <- u[1]; y <- u[2]; z <- u[3]
    return(x^3 + y^2 + z^2 +12*x*y + 2*z)
}
x0 <- c(1,1,1)
hessian(f, x0)
#>      [,1] [,2] [,3]
#> [1,]    6   12    0
#> [2,]   12    2    0
#> [3,]    0    0    2