\name{gaussHermite}
\alias{gaussHermite}
\title{
  Gauss-Hermite Quadrature Formula
}
\description{
  Nodes and weights for the n-point Gauss-Hermite quadrature formula.
}
\usage{
gaussHermite(n)
}
\arguments{
  \item{n}{Number of nodes in the interval \code{]-Inf, Inf[}.}
}
\details{
  Gauss-Hermite quadrature is used for integrating functions of the form
  \deqn{\int_{-\infty}^{\infty} f(x) e^{-x^2} dx}
  over the infinite interval \eqn{]-\infty, \infty[}.

  \code{x} and \code{w} are obtained from a tridiagonal eigenvalue problem.
  The value of such an integral is then \code{sum(w*f(x))}.
}
\value{
  List with components \code{x}, the nodes or points in\code{]-Inf, Inf[}, and
  \code{w}, the weights applied at these nodes.
}
\references{
  Gautschi, W. (2004). Orthogonal Polynomials: Computation and Approximation.
  Oxford University Press.

  Trefethen, L. N. (2000). Spectral Methods in Matlab. SIAM, Society for
  Industrial and Applied Mathematics.
}
\author{
  HwB  <hwborchers@googlemail.com>
}
\note{
  The basic quadrature rules are well known and can, e. g., be found in
  Gautschi (2004) --- and explicit Matlab realizations in Trefethen (2000).
  These procedures have also been implemented in Matlab by Geert Van Damme,
  see his entries at MatlabCentral since 2010.

  For other Matlab implementations of many quadrature rules see the web site
  \url{http://people.sc.fsu.edu/~jburkardt/m_src/m_src.html} by J. Burkardt,
  Florida State University.
}
\seealso{
\code{\link{gaussLegendre}}, \code{\link{gaussLaguerre}}
}
\examples{
cc <- gaussHermite(17)
# Integrate  exp(-x^2)  from -Inf to Inf
sum(cc$w)                        #=> 1.77245385090552 == sqrt(pi)
# Integrate  x^2 exp(-x^2)
sum(cc$w * cc$x^2)               #=> 0.88622692545276 == sqrt(pi) /2
# Integrate  cos(x) * exp(-x^2)
sum(cc$w * cos(cc$x))            #=> 1.38038844704314 == sqrt(pi)/exp(1)^0.25
}
\keyword{ math }