## SYNOPSIS

- SUBROUTINE DLATRZ(
- M, N, L, A, LDA, TAU, WORK )

- INTEGER L, LDA, M, N

- DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )

## PURPOSE

DLATRZ 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (M)
- The scalar factors of the elementary reflectors.
- WORK (workspace) DOUBLE PRECISION array, dimension (M)

## FURTHER DETAILS

Based on contributions byA. 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 ).