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version 1.15, 2014/01/27 09:28:23
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*> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations. |
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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 DLATRZ + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrz.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/dlatrz.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/dlatrz.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 DLATRZ( M, N, L, A, LDA, TAU, WORK ) |
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* |
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* .. Scalar Arguments .. |
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* INTEGER L, LDA, M, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) |
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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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*> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix |
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*> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means |
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*> of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal |
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*> matrix and, R and A1 are M-by-M upper triangular matrices. |
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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 >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] L |
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*> \verbatim |
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*> L is INTEGER |
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*> The number of columns of the matrix A containing the |
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*> meaningful part of the Householder vectors. N-M >= L >= 0. |
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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 N-L+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] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (M) |
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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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*> \date September 2012 |
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* |
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*> \ingroup doubleOTHERcomputational |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA |
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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 l element vector. tau and z( k ) |
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*> are chosen to annihilate the elements of the kth row of A2. |
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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 A2, such that the elements of z( k ) are |
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*> in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in |
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*> the upper triangular part of A1. |
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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 DLATRZ( M, N, L, A, LDA, TAU, WORK ) |
SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.2) -- |
* -- 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 |
* September 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
INTEGER L, LDA, M, N |
INTEGER L, LDA, M, N |
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DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) |
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix |
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* [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means |
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* of orthogonal transformations. Z is an (M+L)-by-(M+L) orthogonal |
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* matrix and, R and A1 are M-by-M upper triangular matrices. |
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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 >= 0. |
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* |
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* L (input) INTEGER |
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* The number of columns of the matrix A containing the |
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* meaningful part of the Householder vectors. N-M >= L >= 0. |
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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 N-L+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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* WORK (workspace) DOUBLE PRECISION array, dimension (M) |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA |
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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 l element vector. tau and z( k ) |
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* are chosen to annihilate the elements of the kth row of A2. |
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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 A2, such that the elements of z( k ) are |
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* in a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in |
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* the upper triangular part of A1. |
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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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INTEGER I |
INTEGER I |
* .. |
* .. |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL DLARFP, DLARZ |
EXTERNAL DLARFG, DLARZ |
* .. |
* .. |
* .. Executable Statements .. |
* .. Executable Statements .. |
* |
* |
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* Generate elementary reflector H(i) to annihilate |
* Generate elementary reflector H(i) to annihilate |
* [ A(i,i) A(i,n-l+1:n) ] |
* [ A(i,i) A(i,n-l+1:n) ] |
* |
* |
CALL DLARFP( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) ) |
CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) ) |
* |
* |
* Apply H(i) to A(1:i-1,i:n) from the right |
* Apply H(i) to A(1:i-1,i:n) from the right |
* |
* |