# ctzrqf (l) - Linux Manuals

## NAME

CTZRQF - routine i deprecated and has been replaced by routine CTZRZF

## SYNOPSIS

SUBROUTINE CTZRQF(
M, N, A, LDA, TAU, INFO )

INTEGER INFO, LDA, M, N

COMPLEX A( LDA, * ), TAU( * )

## PURPOSE

This routine is deprecated and has been replaced by routine CTZRZF. CTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A to upper triangular form by means of unitary transformations. The upper trapezoidal matrix A is factored as

Z,
where Z is an N-by-N unitary matrix and R is an M-by-M 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 >= M.
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 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

## FURTHER DETAILS

The factorization is obtained by Householderaqs method. The kth transformation matrix, Z( k ), whose conjugate transpose is used to introduce zeros into the (m - k + 1)th row of A, is given in the form
Z(       ),

T( )
where

T( I - tau*u( )*u( )aq,   u(      ),
)
z( ) 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( ... Z( ).