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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>  # chex57.r           Demonstration of Givens transformations
>  #                     for simple linear regression problem
>  #                     Example 5.7
>  #
>    X <- matrix(c(rep(1,4),(1:4)),4,2)    # design matrix
>    X
     [,1] [,2]
[1,]    1    1
[2,]    1    2
[3,]    1    3
[4,]    1    4
>  # 
>  # function to do Givens on col i, rows i,j
>    Givens <- function(X,i,j) {
+               a <- X[i,i] ;      b <- X[j,i]
+               r <- sqrt(a*a+b*b)                  # length
+               Uij <- matrix( c(a/r,-b/r,b/r,a/r),2,2)  # essential
+               print(Uij)                               # matrix 
+               X[c(i,j),] <- Uij %*% X[c(i,j),]    # main computation
+               U <- diag(4) ;                      # identity
+               U[c(i,j),c(i,j)] <- Uij             # modify row/col i,j
+               print(U)                            # big matrix
+               Givens <- X }                       # return modified X
>  #
>  # go through a sequence of steps, putting zeros in X
>    X <- Givens(X,1,2)
           [,1]      [,2]
[1,]  0.7071068 0.7071068
[2,] -0.7071068 0.7071068
           [,1]      [,2] [,3] [,4]
[1,]  0.7071068 0.7071068    0    0
[2,] -0.7071068 0.7071068    0    0
[3,]  0.0000000 0.0000000    1    0
[4,]  0.0000000 0.0000000    0    1
>    X                #  U12*X
         [,1]      [,2]
[1,] 1.414214 2.1213203
[2,] 0.000000 0.7071068
[3,] 1.000000 3.0000000
[4,] 1.000000 4.0000000
>    X <- Givens(X,1,3)
           [,1]      [,2]
[1,]  0.8164966 0.5773503
[2,] -0.5773503 0.8164966
           [,1] [,2]      [,3] [,4]
[1,]  0.8164966    0 0.5773503    0
[2,]  0.0000000    1 0.0000000    0
[3,] -0.5773503    0 0.8164966    0
[4,]  0.0000000    0 0.0000000    1
>    X                #  U13*U12*X
         [,1]      [,2]
[1,] 1.732051 3.4641016
[2,] 0.000000 0.7071068
[3,] 0.000000 1.2247449
[4,] 1.000000 4.0000000
>    X <- Givens(X,2,3)
           [,1]      [,2]
[1,]  0.5000000 0.8660254
[2,] -0.8660254 0.5000000
     [,1]       [,2]      [,3] [,4]
[1,]    1  0.0000000 0.0000000    0
[2,]    0  0.5000000 0.8660254    0
[3,]    0 -0.8660254 0.5000000    0
[4,]    0  0.0000000 0.0000000    1
>    X                #  U23*U13*U12*X
         [,1]     [,2]
[1,] 1.732051 3.464102
[2,] 0.000000 1.414214
[3,] 0.000000 0.000000
[4,] 1.000000 4.000000
>    X <- Givens(X,1,4)
           [,1]      [,2]
[1,]  0.8660254 0.5000000
[2,] -0.5000000 0.8660254
           [,1] [,2] [,3]      [,4]
[1,]  0.8660254    0    0 0.5000000
[2,]  0.0000000    1    0 0.0000000
[3,]  0.0000000    0    1 0.0000000
[4,] -0.5000000    0    0 0.8660254
>    X                #  U14*U23*U13*U12*X
              [,1]     [,2]
[1,]  2.000000e+00 5.000000
[2,]  0.000000e+00 1.414214
[3,]  0.000000e+00 0.000000
[4,] -1.110223e-16 1.732051
>    X <- Givens(X,2,4)
           [,1]      [,2]
[1,]  0.6324555 0.7745967
[2,] -0.7745967 0.6324555
     [,1]       [,2] [,3]      [,4]
[1,]    1  0.0000000    0 0.0000000
[2,]    0  0.6324555    0 0.7745967
[3,]    0  0.0000000    1 0.0000000
[4,]    0 -0.7745967    0 0.6324555
>    X                #  U24*U14*U23*U13*U12*X
              [,1]     [,2]
[1,]  2.000000e+00 5.000000
[2,] -8.599751e-17 2.236068
[3,]  0.000000e+00 0.000000
[4,] -7.021667e-17 0.000000
>  # clean and close
>  rm(list=ls())
>  q()