R Graphical Manual

Browse All

Last data update: 2014.03.03

R: Calculate Hessian Matrix

hessian

R Documentation

Calculate Hessian Matrix

Description

Calculate a numerical approximation to the Hessian matrix of a function at a parameter value.

Usage

    hessian(func, x, method="Richardson", method.args=list(), ...)

    ## Default S3 method:
hessian(func, x, method="Richardson",
        method.args=list(), ...)

Arguments

`func`	a function for which the first (vector) argument is used as a parameter vector.
`x`	the parameter vector first argument to func.
`method`	one of `"Richardson"` or `"complex"` indicating the method to use for the approximation.
`method.args`	arguments passed to method. See `grad`. (Arguments not specified remain with their default values.)
`...`	an additional arguments passed to `func`. WARNING: None of these should have names matching other arguments of this function.

Details

The function hessian calculates an numerical approximation to the n x n second derivative of a scalar real valued function with n-vector argument.

The argument method can be "Richardson" or "complex". Method "simple" is not supported.

For method "complex" the Hessian matrix is calculated as the Jacobian of the gradient. The function grad with method "complex" is used, and method.args is ignored for this (an eps of .Machine$double.eps is used). However, jacobian is used in the second step, with method "Richardson" and argument method.args is used for this. The default is method.args=list(eps=1e-4, d=0.1, zero.tol=sqrt(.Machine$double.eps/7e-7), r=4, v=2, show.details=FALSE). (These are the defaults for hessian with method "Richardson", which are slightly different from the defaults for jacobian with method "Richardson".) See addition comments in grad before choosing method "complex".

Methods "Richardson" uses genD and extracts the second derivative. For this method method.args=list(eps=1e-4, d=0.1, zero.tol=sqrt(.Machine$double.eps/7e-7), r=4, v=2, show.details=FALSE) is set as the default. hessian does one evaluation of func in order to do some error checking before calling genD, so the number of function evaluations will be one more than indicated for genD.

Value

An n by n matrix of the Hessian of the function calculated at the point x.

Examples

  sc2.f <- function(x){
    n <- length(x)
    sum((1:n) * (exp(x) - x)) / n
    }

  sc2.g <- function(x){
    n <- length(x)
    (1:n) * (exp(x) - 1) / n
    }

  x0 <- rnorm(5)
  hess <- hessian(func=sc2.f, x=x0)
  hessc <- hessian(func=sc2.f, x=x0, "complex")
  all.equal(hess, hessc, tolerance = .Machine$double.eps)
  
#  Hessian = Jacobian of the gradient
  jac  <- jacobian(func=sc2.g, x=x0)
  jacc <- jacobian(func=sc2.g, x=x0, "complex")
  all.equal(hess, jac, tolerance = .Machine$double.eps)
  all.equal(hessc, jacc, tolerance = .Machine$double.eps)