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#
# rmutil : A Library of Special Functions for Repeated Measurements
# Copyright (C) 1998, 1999, 2000, 2001 J.K. Lindsey
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public Licence as published by
# the Free Software Foundation; either version 2 of the Licence, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public Licence for more details.
#
# You should have received a copy of the GNU General Public Licence
# along with this program; if not, write to the Free Software
# Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
#
# SYNOPSIS
#
# gauss.hermite(points, iterlim=10)
#
# DESCRIPTION
#
# Function to compute points and weights for Gauss-Hermite integration
gauss.hermite <- function(points, iterlim=10){
x <- w <- rep(0,points)
m <- (points+1)/2
for(i in 1:m){
z <- if(i==1)sqrt(2*points+1)-2*(2*points+1)^(-1/6)
else if(i==2)z-sqrt(points)/z
else if(i==3||i==4)1.9*z-0.9*x[i-2]
else 2.0*z-x[i-2]
for(j in 1:iterlim){
z1 <- z
p <- hermite(points,z)
z <- z1-p[1]/p[2]
if(abs(z-z1)<=1e-15)break}
if(j==iterlim)warning("iteration limit exceeded")
x[points+1-i] <- -(x[i] <- z)
w[i] <- w[points+1-i] <- 2/p[2]^2}
r <- cbind(x*sqrt(2),w/sum(w))
colnames(r) <- c("Points","Weights")
r}
# orthonormal Hermite polynomials
hermite <- function(points, z){
p1 <- 1/pi^0.4
p2 <- 0
for(j in 1:points){
p3 <- p2
p2 <- p1
p1 <- z*sqrt(2.0/j)*p2-sqrt((j-1)/j)*p3}
pp <- sqrt(2*points)*p2
c(p1,pp)}
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