NAME

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

SYNOPSIS

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

  SUBROUTINE DTZRZF_64( M, N, A, LDA, TAU, WORK, LWORK, INFO)
  INTEGER*8 M, N, LDA, LWORK, INFO
  DOUBLE PRECISION A(LDA,*), TAU(*), WORK(*)

F95 INTERFACE

  SUBROUTINE TZRZF( [M], [N], A, [LDA], TAU, [WORK], [LWORK], [INFO])
  INTEGER :: M, N, LDA, LWORK, INFO
  REAL(8), DIMENSION(:) :: TAU, WORK
  REAL(8), DIMENSION(:,:) :: A

  SUBROUTINE TZRZF_64( [M], [N], A, [LDA], TAU, [WORK], [LWORK], [INFO])
  INTEGER(8) :: M, N, LDA, LWORK, INFO
  REAL(8), DIMENSION(:) :: TAU, WORK
  REAL(8), DIMENSION(:,:) :: A

C INTERFACE

#include <sunperf.h>

void dtzrzf(int m, int n, double *a, int lda, double *tau, int *info);

void dtzrzf_64(long m, long n, double *a, long lda, double *tau, long *info);

PURPOSE

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.

ARGUMENTS

M (input)
The number of rows of the matrix A. M > = 0.
N (input)
The number of columns of the matrix A. N > = 0.
A (input/output)
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)
The leading dimension of the array A. LDA > = max(1,M).
TAU (output)
The scalar factors of the elementary reflectors.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
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)

 = 0:  successful exit

 < 0:  if INFO  = -i, the i-th argument had an illegal 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 ).