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version 1.19, 2023/08/07 08:39:14
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*> \brief \b DTZRQF |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download DTZRQF + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtzrqf.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtzrqf.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtzrqf.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER INFO, LDA, M, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), TAU( * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> This routine is deprecated and has been replaced by routine DTZRZF. |
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*> |
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*> DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A |
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*> to upper triangular form by means of orthogonal transformations. |
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*> |
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*> The upper trapezoidal matrix A is factored as |
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*> |
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*> A = ( R 0 ) * Z, |
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*> |
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*> where Z is an N-by-N orthogonal matrix and R is an M-by-M upper |
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*> triangular matrix. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The number of rows of the matrix A. M >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The number of columns of the matrix A. N >= M. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimension (LDA,N) |
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*> On entry, the leading M-by-N upper trapezoidal part of the |
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*> array A must contain the matrix to be factorized. |
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*> On exit, the leading M-by-M upper triangular part of A |
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*> contains the upper triangular matrix R, and elements M+1 to |
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*> N of the first M rows of A, with the array TAU, represent the |
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*> orthogonal matrix Z as a product of M elementary reflectors. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,M). |
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*> \endverbatim |
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*> |
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*> \param[out] TAU |
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*> \verbatim |
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*> TAU is DOUBLE PRECISION array, dimension (M) |
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*> The scalar factors of the elementary reflectors. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \ingroup doubleOTHERcomputational |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> The factorization is obtained by Householder's method. The kth |
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*> transformation matrix, Z( k ), which is used to introduce zeros into |
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*> the ( m - k + 1 )th row of A, is given in the form |
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*> |
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*> Z( k ) = ( I 0 ), |
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*> ( 0 T( k ) ) |
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*> |
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*> where |
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*> |
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*> T( k ) = I - tau*u( k )*u( k )**T, u( k ) = ( 1 ), |
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*> ( 0 ) |
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*> ( z( k ) ) |
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*> |
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*> tau is a scalar and z( k ) is an ( n - m ) element vector. |
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*> tau and z( k ) are chosen to annihilate the elements of the kth row |
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*> of X. |
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*> |
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*> The scalar tau is returned in the kth element of TAU and the vector |
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*> u( k ) in the kth row of A, such that the elements of z( k ) are |
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*> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in |
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*> the upper triangular part of A. |
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*> |
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*> Z is given by |
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*> |
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*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO ) |
SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER INFO, LDA, M, N |
INTEGER INFO, LDA, M, N |
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DOUBLE PRECISION A( LDA, * ), TAU( * ) |
DOUBLE PRECISION A( LDA, * ), TAU( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* This routine is deprecated and has been replaced by routine DTZRZF. |
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* |
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* DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A |
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* to upper triangular form by means of orthogonal transformations. |
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* |
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* The upper trapezoidal matrix A is factored as |
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* |
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* A = ( R 0 ) * Z, |
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* |
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* where Z is an N-by-N orthogonal matrix and R is an M-by-M upper |
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* triangular matrix. |
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* |
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* Arguments |
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* ========= |
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* |
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* M (input) INTEGER |
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* The number of rows of the matrix A. M >= 0. |
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* |
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* N (input) INTEGER |
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* The number of columns of the matrix A. N >= M. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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* On entry, the leading M-by-N upper trapezoidal part of the |
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* array A must contain the matrix to be factorized. |
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* On exit, the leading M-by-M upper triangular part of A |
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* contains the upper triangular matrix R, and elements M+1 to |
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* N of the first M rows of A, with the array TAU, represent the |
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* orthogonal matrix Z as a product of M elementary reflectors. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,M). |
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* |
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* TAU (output) DOUBLE PRECISION array, dimension (M) |
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* The scalar factors of the elementary reflectors. |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* |
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* Further Details |
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* =============== |
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* |
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* The factorization is obtained by Householder's method. The kth |
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* transformation matrix, Z( k ), which is used to introduce zeros into |
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* the ( m - k + 1 )th row of A, is given in the form |
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* |
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* Z( k ) = ( I 0 ), |
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* ( 0 T( k ) ) |
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* |
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* where |
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* |
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* T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ), |
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* ( 0 ) |
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* ( z( k ) ) |
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* |
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* tau is a scalar and z( k ) is an ( n - m ) element vector. |
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* tau and z( k ) are chosen to annihilate the elements of the kth row |
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* of X. |
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* |
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* The scalar tau is returned in the kth element of TAU and the vector |
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* u( k ) in the kth row of A, such that the elements of z( k ) are |
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* in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in |
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* the upper triangular part of A. |
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* |
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* Z is given by |
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* |
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* Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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INTRINSIC MAX, MIN |
INTRINSIC MAX, MIN |
* .. |
* .. |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFP, XERBLA |
EXTERNAL DAXPY, DCOPY, DGEMV, DGER, DLARFG, XERBLA |
* .. |
* .. |
* .. Executable Statements .. |
* .. Executable Statements .. |
* |
* |
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* Use a Householder reflection to zero the kth row of A. |
* Use a Householder reflection to zero the kth row of A. |
* First set up the reflection. |
* First set up the reflection. |
* |
* |
CALL DLARFP( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) ) |
CALL DLARFG( N-M+1, A( K, K ), A( K, M1 ), LDA, TAU( K ) ) |
* |
* |
IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN |
IF( ( TAU( K ).NE.ZERO ) .AND. ( K.GT.1 ) ) THEN |
* |
* |
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$ LDA, A( K, M1 ), LDA, ONE, TAU, 1 ) |
$ LDA, A( K, M1 ), LDA, ONE, TAU, 1 ) |
* |
* |
* Now form a( k ) := a( k ) - tau*w |
* Now form a( k ) := a( k ) - tau*w |
* and B := B - tau*w*z( k )'. |
* and B := B - tau*w*z( k )**T. |
* |
* |
CALL DAXPY( K-1, -TAU( K ), TAU, 1, A( 1, K ), 1 ) |
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, |
CALL DGER( K-1, N-M, -TAU( K ), TAU, 1, A( K, M1 ), LDA, |