DTZRZF - reduce the M-by-N ( M<=N ) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations

SUBROUTINE DTZRZF(

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

INTEGER INFO, LDA, LWORK, M, N

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

DTZRZF 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.

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) 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.

WORK (workspace/output) DOUBLE PRECISION 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 i-th argument had an illegal value

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 ).