CLATRZ - 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 CLATRZ( M, N, L, A, LDA, TAU, WORK )
INTEGER L, LDA, M, N
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
CLATRZ 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 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 array, dimension (M)
The scalar factors of the elementary reflectors.
WORK (workspace) COMPLEX 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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