R : Copyright 2005, The R Foundation for Statistical Computing
Version 2.1.1  (2005-06-20), ISBN 3-900051-07-0

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[Previously saved workspace restored]

>  # qgaustb.r                  October 2007
>  #
>  # find abscissas and weights for gauss quadrature
>  # 
>  # define function
>    qgaus <-function(n,type=3,alph=0) {
+  #            default=Hermite, alph parameter in Laguerre
+         nm1 <- 1:(n-1)        # vector one smaller
+  #            main branch
+         if( type == 1 ) {     # Legendre
+           alpha <- nm1/sqrt(4*nm1*nm1-1)
+           beta <- rep(0,n)    }
+         if( type == 2 ) {     # shifted Legendre
+           alpha <- (nm1/2)/sqrt(4*nm1*nm1-1)
+           beta <- rep(1/2,n)  }
+         if( type == 3 ) {     # Hermite
+           alpha <- sqrt(nm1/2)
+           beta <- rep(0,n)    }
+         if( type == 4 ) {     # modified Hermite
+           alpha <- sqrt(nm1)
+           beta <- rep(0,n)    }
+         if( type == 5 ) {     # Laguerre, alph = parameter
+           alpha <- sqrt(nm1*(nm1+alph))
+           beta <- 2*(1:n)+alph-1  }
+  # make matrix
+    A <- rbind(cbind(rep(0,n-1),diag(alpha,n-1)),rep(0,n))
+    A <- A + t(A) + diag(beta,n)
+    e <- eigen(A,symmetric=T)  # eigenvalues and vectors
+    abscissas <- e$values
+    weights <- (e$vectors[1,])*(e$vectors[1,])
+    qgaus <- list(abscissas,weights) 
+                        }      # end of function qgaus
>  # call it to check it out
>    q1 <- qgaus(5,1)      # Legendre
>    q1
[[1]]
[1]  0.9061798  0.5384693  0.0000000 -0.5384693 -0.9061798

[[2]]
[1] 0.1184634 0.2393143 0.2844444 0.2393143 0.1184634

>    q2 <- qgaus(5,2)      # shifted Legendre
>    q2
[[1]]
[1] 0.95308992 0.76923466 0.50000000 0.23076534 0.04691008

[[2]]
[1] 0.1184634 0.2393143 0.2844444 0.2393143 0.1184634

>    sum(q2[[2]])
[1] 1
>    q3 <- qgaus(5,3)      # Hermite
>    q3
[[1]]
[1]  2.020183e+00  9.585725e-01 -4.440892e-16 -9.585725e-01 -2.020183e+00

[[2]]
[1] 0.01125741 0.22207592 0.53333333 0.22207592 0.01125741

>    q4 <- qgaus(5,4)      # modified Hermite
>    q4
[[1]]
[1]  2.856970e+00  1.355626e+00 -4.440892e-16 -1.355626e+00 -2.856970e+00

[[2]]
[1] 0.01125741 0.22207592 0.53333333 0.22207592 0.01125741

>    sum(q4[[2]])
[1] 1
>    q5 <-qgaus(5,5,2)     # Laguerre with parameter 2
>    q5
[[1]]
[1] 15.828474  9.682910  5.620294  2.837213  1.031109

[[2]]
[1] 0.0001313564 0.0142821168 0.1917748618 0.5333529666 0.2604586984

>  # done
>   rm(list=ls())
>   q()