Provided by: liblapack-doc_3.3.1-1_all

**NAME**

LAPACK-3 - routine i deprecated and has been replaced by routine CTZRZF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This routine is deprecated and has been replaced by routine CTZRZF.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 A = ( R 0 ) * 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 Householder's 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( k ) = ( I 0 ), ( 0 T( k ) ) where T( k ) = I - tau*u( k )*u( k )**H, 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 routine (version 3.3.1) April 2011 CTZRQF(3lapack)