clatrz(3)

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 matri

ces.

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 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 ).

LAPACK version 3.0 15 June 2000

CLATRZ(3)