dtzrqf(l) [redhat man page]

DTZRQF(l)								 )								 DTZRQF(l)

NAME

       DTZRQF - routine is deprecated and has been replaced by routine DTZRZF

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       This  routine is deprecated and has been replaced by routine DTZRZF.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 ).

LAPACK version 3.0						   15 June 2000 							 DTZRQF(l)

DTZRZF(l)								 )								 DTZRZF(l)

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	  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 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 >= 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

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(l)