SYNOPSIS

SUBROUTINE SLATRZ(

M, N, L, A, LDA, TAU, WORK )

    

INTEGER L, LDA, M, N

    

REAL A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal matrix and, R and A1 are M-by-M upper triangular matrices.

ARGUMENTS

M (input) INTEGER

The number of rows of the matrix A. M >= 0.

N (input) INTEGER

The number of columns of the matrix A. N >= 0.

L (input) INTEGER

The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= L >= 0.

A (input/output) REAL array, dimension (LDA,N)

On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading M-by-M upper triangular part of A contains the upper triangular matrix R, and elements N-L+1 to N of the first M rows of A, with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors.

LDA (input) INTEGER

The leading dimension of the array A. LDA >= max(1,M).

TAU (output) REAL array, dimension (M)

The scalar factors of the elementary reflectors.

WORK (workspace) REAL array, dimension (M)

FURTHER DETAILS

Based on contributions by

  A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the ( m - k + 1 )th row of A, is given in the form

   Z( k ) = ( I     0   ),

            ( 0  T( k ) )
where

   T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                               (   0    )
                                               ( z( k ) ) tau is a scalar and z( k ) is an l element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of A2. The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of A2, such that the elements of z( k ) are in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of A1.
Z is given by

   Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).