R version 2.9.0 (2009-04-17)
Copyright (C) 2009 The R Foundation for Statistical Computing
ISBN 3-900051-07-0

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>  # c3nwdemo.r         New demo for Chapter 3
>  #                           September 2007, revised 2009
>  #
>  # simpler example for condition number for solving systems
>  #  of linear equations
>  #
>  # here's a matrix
>   A <- matrix(c(4,-2,2,4,-2,2,-2,-2,2,-2,11,5,4,-2,5,6),4,4)
>   A                               # write it out
     [,1] [,2] [,3] [,4]
[1,]    4   -2    2    4
[2,]   -2    2   -2   -2
[3,]    2   -2   11    5
[4,]    4   -2    5    6
>  #
>  # Cholesky factorization
>  #
>   Lt <- chol(A)
>   Lt                              # write it out
     [,1] [,2] [,3] [,4]
[1,]    2   -1    1    2
[2,]    0    1   -1    0
[3,]    0    0    3    1
[4,]    0    0    0    1
>  #
>  # rhs
>  #
>   b <- c(2,0,0,2)
>  #
>  # solve these equations
>  #
>   y <- forwardsolve( t(Lt), b)
>   x <- backsolve( Lt, y )
>   x                               # write it out
[1] 1 1 0 0
>  #
>  # check the solution
>  #
>   A %*% x
     [,1]
[1,]    2
[2,]    0
[3,]    0
[4,]    2
>  #
>  # get infinity norm (from homework exercise)
>  #
>   normA <- max( apply( abs(A), 1, sum) )
>  #
>  # get infinity norm of inverse for condition number
>  #
>   kappa <- normA*max( apply( abs(chol2inv(Lt)),1,sum) )
>   kappa
[1] 73.33333
>  #
>  # small change in matrix A
>  #
>   E <- matrix( 1.e-4, 4, 4)   # recycling
>  #
>  # Cholesky factor of changed matrix
>  #
>   Lte <- chol( A+E )
>   Lte
         [,1]       [,2]       [,3]      [,4]
[1,] 2.000025 -0.9999375  1.0000375 2.0000250
[2,] 0.000000  1.0001125 -0.9998125 0.0000000
[3,] 0.000000  0.0000000  3.0000666 0.9999778
[4,] 0.000000  0.0000000  0.0000000 1.0000222
>  #
>  # solve equations
>  #
>   y <- forwardsolve(t(Lte),b)
>   xe <- backsolve(Lte,y)
>   xe
[1]  9.996668e-01  9.996112e-01 -8.885878e-05  1.332882e-04
>  #
>  # norm of difference
>  # 
>   max( abs(x-xe) ) 
[1] 0.0003887571
>  #
>  #
>  # small change in rhs
>  #
>   be <- b + 1.e-4             # recycling
>  #
>  # solve equations
>  #
>   y <- forwardsolve(t(Lt),be)
>   xe <- backsolve(Lt,y)
>   xe
[1]  1.000167e+00  1.000194e+00  4.444444e-05 -6.666667e-05
>  #
>  # norm of difference
>  # 
>   max( abs(x-xe) ) 
[1] 0.0001944444
>  #
>  # done
>   rm(list=ls())
> 
>