https://github.com/cran/pracma
Revision 26e049d70b4a1c237987e260cba68f6a9413736c authored by Hans W. Borchers on 09 April 2019, 04:10:07 UTC, committed by cran-robot on 09 April 2019, 04:10:07 UTC
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Tip revision: 26e049d70b4a1c237987e260cba68f6a9413736c authored by Hans W. Borchers on 09 April 2019, 04:10:07 UTC
version 2.2.5
version 2.2.5
Tip revision: 26e049d
gaussHermite.Rd
\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.
}
\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.
}
\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 }
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