ZLATRZ - factor the M-by-(M+L) complex upper trapezoidal
matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0
) * Z by means of unitary transformations, where Z is an
(M+L)-by-(M+L) unitary matrix and, R and A1 are M-by-M
upper triangular matrices
SYNOPSIS
SUBROUTINE ZLATRZ( M, N, L, A, LDA, TAU, WORK )
INTEGER L, LDA, M, N
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
ZLATRZ factors the M-by-(M+L) complex upper trapezoidal
matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0
) * Z by means of unitary transformations, where Z is an
(M+L)-by-(M+L) unitary 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) COMPLEX*16 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 uni
tary matrix Z as a product of M elementary reflec
tors.
LDA (input) INTEGER
The leading dimension of the array A. LDA >=
max(1,M).
TAU (output) COMPLEX*16 array, dimension (M)
The scalar factors of the elementary reflectors.
WORK (workspace) COMPLEX*16 array, dimension (M)
FURTHER DETAILS
Based on contributions by
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 ele
ments 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 ).
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