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>  # chlsoi.r                         J F Monahan, June 2008
>  #                                 
>  # Cholesky factorization and inverse of a positive definite matrix
>  A <- matrix(c(4,2,2,4, 2,5,7,0, 2,7,19,11, 4,0,11,25), 4, 4)
>  "matrix"
[1] "matrix"
>  A
     [,1] [,2] [,3] [,4]
[1,]    4    2    2    4
[2,]    2    5    7    0
[3,]    2    7   19   11
[4,]    4    0   11   25
>  Lt <- chol(A)
>  "Cholesky factor -- transposed to be upper triangular"
[1] "Cholesky factor -- transposed to be upper triangular"
>  Lt
     [,1] [,2] [,3] [,4]
[1,]    2    1    1    2
[2,]    0    2    3   -1
[3,]    0    0    3    4
[4,]    0    0    0    2
>  AI <- chol2inv(Lt)
>  "inverse"
[1] "inverse"
>  AI
           [,1]     [,2]       [,3]       [,4]
[1,]  0.9461806 -1.15625  0.5555556 -0.3958333
[2,] -1.1562500  2.06250 -1.0000000  0.6250000
[3,]  0.5555556 -1.00000  0.5555556 -0.3333333
[4,] -0.3958333  0.62500 -0.3333333  0.2500000
>  "multiply to get identity"
[1] "multiply to get identity"
>  A %*% AI  
              [,1] [,2]          [,3]         [,4]
[1,]  1.000000e+00    0 -4.440892e-16 1.110223e-16
[2,] -9.992007e-16    1 -9.992007e-16 2.775558e-16
[3,] -2.164935e-15    0  1.000000e+00 4.440892e-16
[4,] -7.216450e-16    0 -7.216450e-16 1.000000e+00
>  "solve equations"              
[1] "solve equations"
>  b <-c( -1, 1, 2.5, .25)
>  "Right hand side"
[1] "Right hand side"
>  b
[1] -1.00  1.00  2.50  0.25
>  x <- AI %*% b
>  "Solution"
[1] "Solution"
>  x
        [,1]
[1,] -0.8125
[2,]  0.8750
[3,] -0.2500
[4,]  0.2500
>  rm(list=ls())
>   q()