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\name{hessian}
\alias{hessian}
\alias{hessian.default}
\title{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(), ...)
\method{hessian}{default}(func, x, method="Richardson",
method.args=list(), ...)
}
\arguments{
\item{func}{a function for which the first (vector) argument
is used as a parameter vector.}
\item{x}{the parameter vector first argument to func.}
\item{method}{one of \code{"Richardson"} or \code{"complex"} indicating
the method to use for the approximation.}
\item{method.args}{arguments passed to method. See \code{\link{grad}}.
(Arguments not specified remain with their default values.)}
\item{...}{an additional arguments passed to \code{func}.
WARNING: None of these should have names matching other arguments of this function.}
}
\value{An n by n matrix of the Hessian of the function calculated at the
point \code{x}.}
\details{
The function \code{hessian} calculates an numerical approximation to
the n x n second derivative of a scalar real valued function with n-vector
argument.
The argument \code{method} can be \code{"Richardson"} or \code{"complex"}.
Method \code{"simple"} is not supported.
For method \code{"complex"} the Hessian matrix is calculated as the Jacobian
of the gradient. The function \code{grad} with method "complex" is used,
and \code{methods.args} is ignored for this (an \code{eps} of
\code{.Machine$double.eps} is used - see \code{\link{grad}} for more details).
However, \code{jacobian} is used in the second step, with method
\code{"Richardson"} and argument \code{methods.args} is used for this.
The default is
\code{methods.args=list(eps=1e-4, d=0.1, zero.tol=sqrt(.Machine$double.eps/7e-7),
r=4, v=2, show.details=FALSE)}. (These are the defaults for \code{hessian}
with method \code{"Richardson"}, which are slightly different from the defaults
for \code{jacobian} with method \code{"Richardson"}.)
Methods \code{"Richardson"} uses \code{\link{genD}} and extracts the
second derivative. For this method
\code{methods.args=list(eps=1e-4, d=0.1, zero.tol=sqrt(.Machine$double.eps/7e-7),
r=4, v=2, show.details=FALSE)} is set as the default.
}
\seealso{
\code{\link{jacobian}},
\code{\link{grad}},
\code{\link{genD}}
}
\examples{
sc2.f <- function(x){
n <- length(x)
sum((1:n) * (exp(x) - x)) / n
}
sc2.g <- function(x){
n <- length(x)
(1:n) * (exp(x) - 1) / n
}
x0 <- rnorm(5)
hess <- hessian(func=sc2.f, x=x0)
hessc <- hessian(func=sc2.f, x=x0, "complex")
all.equal(hess, hessc, tolerance = .Machine$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)
}
\keyword{multivariate}
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