rootSolve (version 1.7)

hessian: Estimates the hessian matrix

Description

Given a vector of variables (x), and a function (f) that estimates one function value, estimates the hessian matrix by numerical differencing. The hessian matrix is a square matrix of second-order partial derivatives of the function f with respect to x. It contains, on rows i and columns j $$d^2(f(x))/d(x_i)/d(x_j)$$

Usage

hessian(f, x, centered = FALSE, pert = 1e-8, ...)

Arguments

f

function returning one function value, or a vector of function values.

x

either one value or a vector containing the x-value(s) at which the hessian matrix should be estimated.

centered

if TRUE, uses a centered difference approximation, else a forward difference approximation.

pert

numerical perturbation factor; increase depending on precision of model solution.

...

other arguments passed to function f.

Value

The gradient matrix where the number of rows equals the length of f and the number of columns equals the length of x.

the elements on i-th row and j-th column contain: \(d((f(x))_i)/d(x_j)\)

Details

Function hessian(f,x) returns a forward or centered difference approximation of the gradient, which itself is also estimated by differencing. Because of that, it is not very precise.

See Also

gradient, for a full (not necessarily square) gradient matrix

Examples

# NOT RUN {
## =======================================================================
## the banana function
## =======================================================================
   fun <- function(x)  100*(x[2] - x[1]^2)^2 + (1 - x[1])^2
   mm  <- nlm(fun, p = c(0, 0))$estimate
   (Hes <- hessian(fun, mm))
   # can also be estimated by nlm(fun, p=c(0,0), hessian=TRUE)
   solve(Hes)   # estimate of parameter uncertainty
# }