CLATRZ(l) ) CLATRZ(l)NAME
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 unitary 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) COMPLEX array, dimension (M)
The scalar factors of the elementary reflectors.
WORK (workspace) COMPLEX 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 trans‐
formation 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 ).
LAPACK version 3.0 15 June 2000 CLATRZ(l)
