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\name{jacobian}
\alias{jacobian}
\alias{jacobian.default}
\title{Gradient of a Vector Valued Function}
\description{
Calculate the m by n numerical approximation of the gradient of a real
m-vector valued function with n-vector argument.
}
\usage{
jacobian(func, x, method="Richardson", method.args=list(), ...)
\method{jacobian}{default}(func, x, method="Richardson",
method.args=list(), ...)
}
\arguments{
\item{func}{a function with a real (vector) result.}
\item{x}{a real or real vector argument to func, indicating the point
at which the gradient is to be calculated.}
\item{method}{one of \code{"Richardson"}, \code{"simple"}, 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{...}{any additional arguments passed to \code{func}.
WARNING: None of these should have names matching other arguments of this function.}
}
\value{A real m by n matrix.}
\details{
For \eqn{f:R^n -> R^m}{f:R^n -> R^m} calculate the \eqn{m x n}{m x n}
Jacobian \eqn{dy/dx}{dy/dx}.
The function \code{jacobian} calculates a numerical approximation of the
first derivative of \code{func} at the point \code{x}. Any additional
arguments in \dots are also passed to \code{func}, but the gradient is not
calculated with respect to these additional arguments.
If method is "Richardson", the calculation is done by
Richardson's extrapolation. See \code{link{grad}} for more details.
For this method \code{methods.args=list(eps=1e-4, d=0.0001,
zero.tol=sqrt(.Machine$double.eps/7e-7), r=4, v=2, show.details=FALSE)}
is set as the default.
If method is "simple", the calculation is done using a simple epsilon
difference.
For method "simple" \code{methods.args=list(eps=1e-4)} is the
default. Only \code{eps} is used by this method.
If method is "complex", the calculation is done using the complex step
derivative approach described in Lyness and Moler. This method requires
that the function be able to handle complex valued arguments and return the
appropriate complex valued result, even though the user may only be
interested in the real case. For cases where it can be used, it is faster than Richardson's extrapolation, and
it also provides gradients that are correct to machine precision (16 digits).
For method "complex", \code{methods.args} is ignored.
The algorithm uses an \code{eps} of \code{.Machine$double.eps} which cannot
(and should not) be modified.
}
\seealso{
\code{\link{grad}},
\code{\link{hessian}},
\code{\link[stats]{numericDeriv}}
}
\examples{
func2 <- function(x) c(sin(x), cos(x))
x <- (0:1)*2*pi
jacobian(func2, x)
jacobian(func2, x, "complex")
}
\keyword{multivariate}
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