SYNOPSIS

SUBROUTINE DTZRQF(

M, N, A, LDA, TAU, INFO )

    

INTEGER INFO, LDA, M, N

    

DOUBLE PRECISION A( LDA, * ), TAU( * )

PURPOSE

This routine is deprecated and has been replaced by routine DTZRZF.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

   A = ( R  0 ) * Z,
where Z is an N-by-N orthogonal 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) DOUBLE PRECISION 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 orthogonal 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) DOUBLE PRECISION 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 ), 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 ).