# 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
 # 
 # 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)
   X                #  U12*X
   X <- Givens(X,1,3)
   X                #  U13*U12*X
   X <- Givens(X,2,3)
   X                #  U23*U13*U12*X
   X <- Givens(X,1,4)
   X                #  U14*U23*U13*U12*X
   X <- Givens(X,2,4)
   X                #  U24*U14*U23*U13*U12*X
 # clean and close
 rm(list=ls())
 q()