
CTZRZF(l) ) CTZRZF(l)
NAME
CTZRZF  reduce the MbyN ( M<=N ) complex upper trapezoidal matrix A to upper triangular
form by means of unitary transformations
SYNOPSIS
SUBROUTINE CTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
CTZRZF reduces the MbyN ( M<=N ) complex upper trapezoidal matrix A to upper triangular
form by means of unitary transformations. The upper trapezoidal matrix A is factored as
A = ( R 0 ) * Z,
where Z is an NbyN unitary matrix and R is an MbyM upper triangular matrix.
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.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the leading MbyN upper trapezoidal part of the array A must contain
the matrix to be factorized. On exit, the leading MbyM upper triangular part of
A contains the upper triangular matrix R, and elements M+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/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,M). For optimum performance
LWORK >= M*NB, where NB is the optimal blocksize.
If LWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
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 ( n  m ) element vector. tau and z( k ) are chosen to
annihilate the elements of the kth row of X.
The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row
of A, such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements
of R are returned in the upper triangular part of A.
Z is given by
Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
LAPACK version 3.0 15 June 2000 CTZRZF(l) 
