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*> \brief \b DTZRQF
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DTZRQF + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtzrqf.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtzrqf.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtzrqf.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DTZRZF.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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 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 )**T, 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 ).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), TAU( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, K, M1
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.M ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DTZRQF', -INFO )
RETURN
END IF
*
* Perform the factorization.
*
IF( M.EQ.0 )
$ RETURN
IF( M.EQ.N ) THEN
DO 10 I = 1, N
TAU( I ) = ZERO
10 CONTINUE
ELSE
M1 = MIN( M+1, N )
DO 20 K = M, 1, -1
*
* Use a Householder reflection to zero the kth row of A.
* First set up the reflection.
*
CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) )
*
IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN
*
* We now perform the operation A := A*P( k ).
*
* Use the first ( k - 1 ) elements of TAU to store a( k ),
* where a( k ) consists of the first ( k - 1 ) elements of
* the kth column of A. Also let B denote the first
* ( k - 1 ) rows of the last ( n - m ) columns of A.
*
CALL DCOPY( K-1, A( 1, K ), 1, TAU, 1 )
*
* Form w = a( k ) + B*z( k ) in TAU.
*
CALL DGEMV( 'No transpose', K-1, N-M, ONE, A( 1, M1 ),
$ LDA, A( K, M1 ), LDA, ONE, TAU, 1 )
*
* Now form a( k ) := a( k ) - tau*w
* and B := B - tau*w*z( k )**T.
*
CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 )
CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA,
$ A( 1, M1 ), LDA )
END IF
20 CONTINUE
END IF
*
RETURN
*
* End of DTZRQF
*
END
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