.TH DLATRZ 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME DLATRZ - 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 .SH SYNOPSIS .TP 19 SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK ) .TP 19 .ti +4 INTEGER L, LDA, M, N .TP 19 .ti +4 DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) .SH 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. .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= 0. .TP 8 L (input) INTEGER The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= L >= 0. .TP 8 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. .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). .TP 8 TAU (output) DOUBLE PRECISION array, dimension (M) The scalar factors of the elementary reflectors. .TP 8 WORK (workspace) DOUBLE PRECISION array, dimension (M) .SH FURTHER DETAILS Based on contributions by .br A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA The factorization is obtained by Householder\(aqs 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 .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )\(aq, 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. .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br