dtzrzf(3)

NAME

DTZRZF - reduce the M-by-N ( M<=N ) real upper trapezoidal

matrix A to upper triangular form by means of orthogonal trans

formations

SYNOPSIS

SUBROUTINE DTZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
    INTEGER        INFO, LDA, LWORK, M, N
    DOUBLE         PRECISION A( LDA, * ), TAU( * ),  WORK(
* )

PURPOSE

DTZRZF reduces the M-by-N ( M<=N ) real upper trapezoidal

matrix A to upper triangular form by means of orthogonal trans

formations. 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 >= 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, dimen

sion (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 ille

gal 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

DTZRZF(3)