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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: 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

```NAME
LAPACK-3 - 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

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
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*16 array, dimension (M)
The scalar factors of the elementary reflectors.

WORK    (workspace) COMPLEX*16 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 routine (version 3.2.2)   March 2011                       LAPACK-3(3)
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