version 1.1, 2010/01/26 15:22:46
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version 1.18, 2018/05/29 07:18:01
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*> \brief \b DLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method. |
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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 DLASYF + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf.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/dlasyf.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/dlasyf.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 DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER UPLO |
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* INTEGER INFO, KB, LDA, LDW, N, NB |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IPIV( * ) |
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* DOUBLE PRECISION A( LDA, * ), W( LDW, * ) |
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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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*> DLASYF computes a partial factorization of a real symmetric matrix A |
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*> using the Bunch-Kaufman diagonal pivoting method. The partial |
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*> factorization has the form: |
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*> |
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*> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: |
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*> ( 0 U22 ) ( 0 D ) ( U12**T U22**T ) |
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*> |
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*> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L' |
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*> ( L21 I ) ( 0 A22 ) ( 0 I ) |
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*> |
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*> where the order of D is at most NB. The actual order is returned in |
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*> the argument KB, and is either NB or NB-1, or N if N <= NB. |
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*> |
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*> DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code |
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*> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or |
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*> A22 (if UPLO = 'L'). |
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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] UPLO |
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*> \verbatim |
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*> UPLO is CHARACTER*1 |
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*> Specifies whether the upper or lower triangular part of the |
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*> symmetric matrix A is stored: |
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*> = 'U': Upper triangular |
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*> = 'L': Lower triangular |
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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 order of the matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] NB |
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*> \verbatim |
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*> NB is INTEGER |
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*> The maximum number of columns of the matrix A that should be |
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*> factored. NB should be at least 2 to allow for 2-by-2 pivot |
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*> blocks. |
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*> \endverbatim |
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*> |
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*> \param[out] KB |
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*> \verbatim |
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*> KB is INTEGER |
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*> The number of columns of A that were actually factored. |
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*> KB is either NB-1 or NB, or N if N <= NB. |
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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 symmetric matrix A. If UPLO = 'U', the leading |
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*> n-by-n upper triangular part of A contains the upper |
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*> triangular part of the matrix A, and the strictly lower |
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*> triangular part of A is not referenced. If UPLO = 'L', the |
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*> leading n-by-n lower triangular part of A contains the lower |
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*> triangular part of the matrix A, and the strictly upper |
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*> triangular part of A is not referenced. |
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*> On exit, A contains details of the partial factorization. |
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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,N). |
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*> \endverbatim |
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*> |
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*> \param[out] IPIV |
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*> \verbatim |
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*> IPIV is INTEGER array, dimension (N) |
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*> Details of the interchanges and the block structure of D. |
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*> |
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*> If UPLO = 'U': |
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*> Only the last KB elements of IPIV are set. |
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*> |
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*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
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*> interchanged and D(k,k) is a 1-by-1 diagonal block. |
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*> |
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*> If IPIV(k) = IPIV(k-1) < 0, then rows and columns |
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*> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) |
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*> is a 2-by-2 diagonal block. |
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*> |
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*> If UPLO = 'L': |
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*> Only the first KB elements of IPIV are set. |
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*> |
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*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
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*> interchanged and D(k,k) is a 1-by-1 diagonal block. |
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*> |
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*> If IPIV(k) = IPIV(k+1) < 0, then rows and columns |
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*> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) |
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*> is a 2-by-2 diagonal block. |
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*> \endverbatim |
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*> |
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*> \param[out] W |
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*> \verbatim |
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*> W is DOUBLE PRECISION array, dimension (LDW,NB) |
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*> \endverbatim |
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*> |
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*> \param[in] LDW |
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*> \verbatim |
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*> LDW is INTEGER |
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*> The leading dimension of the array W. LDW >= max(1,N). |
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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 = k, D(k,k) is exactly zero. The factorization |
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*> has been completed, but the block diagonal matrix D is |
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*> exactly singular. |
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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 November 2013 |
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* |
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*> \ingroup doubleSYcomputational |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> \verbatim |
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*> |
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*> November 2013, Igor Kozachenko, |
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*> Computer Science Division, |
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*> University of California, Berkeley |
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*> \endverbatim |
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* |
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* ===================================================================== |
SUBROUTINE DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) |
SUBROUTINE DLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.5.0) -- |
* -- 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 |
* November 2013 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
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DOUBLE PRECISION A( LDA, * ), W( LDW, * ) |
DOUBLE PRECISION A( LDA, * ), W( LDW, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DLASYF computes a partial factorization of a real symmetric matrix A |
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* using the Bunch-Kaufman diagonal pivoting method. The partial |
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* factorization has the form: |
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* |
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* A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: |
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* ( 0 U22 ) ( 0 D ) ( U12' U22' ) |
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* |
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* A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L' |
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* ( L21 I ) ( 0 A22 ) ( 0 I ) |
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* |
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* where the order of D is at most NB. The actual order is returned in |
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* the argument KB, and is either NB or NB-1, or N if N <= NB. |
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* |
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* DLASYF is an auxiliary routine called by DSYTRF. It uses blocked code |
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* (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or |
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* A22 (if UPLO = 'L'). |
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* |
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* Arguments |
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* ========= |
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* |
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* UPLO (input) CHARACTER*1 |
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* Specifies whether the upper or lower triangular part of the |
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* symmetric matrix A is stored: |
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* = 'U': Upper triangular |
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* = 'L': Lower triangular |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* NB (input) INTEGER |
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* The maximum number of columns of the matrix A that should be |
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* factored. NB should be at least 2 to allow for 2-by-2 pivot |
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* blocks. |
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* |
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* KB (output) INTEGER |
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* The number of columns of A that were actually factored. |
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* KB is either NB-1 or NB, or N if N <= NB. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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* On entry, the symmetric matrix A. If UPLO = 'U', the leading |
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* n-by-n upper triangular part of A contains the upper |
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* triangular part of the matrix A, and the strictly lower |
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* triangular part of A is not referenced. If UPLO = 'L', the |
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* leading n-by-n lower triangular part of A contains the lower |
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* triangular part of the matrix A, and the strictly upper |
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* triangular part of A is not referenced. |
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* On exit, A contains details of the partial factorization. |
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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,N). |
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* |
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* IPIV (output) INTEGER array, dimension (N) |
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* Details of the interchanges and the block structure of D. |
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* If UPLO = 'U', only the last KB elements of IPIV are set; |
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* if UPLO = 'L', only the first KB elements are set. |
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* |
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* If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
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* interchanged and D(k,k) is a 1-by-1 diagonal block. |
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* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and |
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* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) |
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* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = |
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* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were |
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* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. |
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* |
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* W (workspace) DOUBLE PRECISION array, dimension (LDW,NB) |
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* |
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* LDW (input) INTEGER |
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* The leading dimension of the array W. LDW >= max(1,N). |
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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 = k, D(k,k) is exactly zero. The factorization |
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* has been completed, but the block diagonal matrix D is |
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* exactly singular. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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ABSAKK = ABS( W( K, KW ) ) |
ABSAKK = ABS( W( K, KW ) ) |
* |
* |
* IMAX is the row-index of the largest off-diagonal element in |
* IMAX is the row-index of the largest off-diagonal element in |
* column K, and COLMAX is its absolute value |
* column K, and COLMAX is its absolute value. |
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* Determine both COLMAX and IMAX. |
* |
* |
IF( K.GT.1 ) THEN |
IF( K.GT.1 ) THEN |
IMAX = IDAMAX( K-1, W( 1, KW ), 1 ) |
IMAX = IDAMAX( K-1, W( 1, KW ), 1 ) |
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* |
* |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
* |
* |
* Column K is zero: set INFO and continue |
* Column K is zero or underflow: set INFO and continue |
* |
* |
IF( INFO.EQ.0 ) |
IF( INFO.EQ.0 ) |
$ INFO = K |
$ INFO = K |
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* |
* |
KP = IMAX |
KP = IMAX |
* |
* |
* copy column KW-1 of W to column KW |
* copy column KW-1 of W to column KW of W |
* |
* |
CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) |
CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) |
ELSE |
ELSE |
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END IF |
END IF |
END IF |
END IF |
* |
* |
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* ============================================================ |
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* |
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* KK is the column of A where pivoting step stopped |
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* |
KK = K - KSTEP + 1 |
KK = K - KSTEP + 1 |
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* |
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* KKW is the column of W which corresponds to column KK of A |
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* |
KKW = NB + KK - N |
KKW = NB + KK - N |
* |
* |
* Updated column KP is already stored in column KKW of W |
* Interchange rows and columns KP and KK. |
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* Updated column KP is already stored in column KKW of W. |
* |
* |
IF( KP.NE.KK ) THEN |
IF( KP.NE.KK ) THEN |
* |
* |
* Copy non-updated column KK to column KP |
* Copy non-updated column KK to column KP of submatrix A |
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* at step K. No need to copy element into column K |
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* (or K and K-1 for 2-by-2 pivot) of A, since these columns |
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* will be later overwritten. |
* |
* |
A( KP, K ) = A( KK, K ) |
A( KP, KP ) = A( KK, KK ) |
CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), |
CALL DCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), |
$ LDA ) |
$ LDA ) |
CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 ) |
IF( KP.GT.1 ) |
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$ CALL DCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) |
* |
* |
* Interchange rows KK and KP in last KK columns of A and W |
* Interchange rows KK and KP in last K+1 to N columns of A |
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* (columns K (or K and K-1 for 2-by-2 pivot) of A will be |
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* later overwritten). Interchange rows KK and KP |
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* in last KKW to NB columns of W. |
* |
* |
CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA ) |
IF( K.LT.N ) |
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$ CALL DSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), |
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$ LDA ) |
CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), |
CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), |
$ LDW ) |
$ LDW ) |
END IF |
END IF |
* |
* |
IF( KSTEP.EQ.1 ) THEN |
IF( KSTEP.EQ.1 ) THEN |
* |
* |
* 1-by-1 pivot block D(k): column KW of W now holds |
* 1-by-1 pivot block D(k): column kw of W now holds |
* |
* |
* W(k) = U(k)*D(k) |
* W(kw) = U(k)*D(k), |
* |
* |
* where U(k) is the k-th column of U |
* where U(k) is the k-th column of U |
* |
* |
* Store U(k) in column k of A |
* Store subdiag. elements of column U(k) |
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* and 1-by-1 block D(k) in column k of A. |
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* NOTE: Diagonal element U(k,k) is a UNIT element |
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* and not stored. |
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* A(k,k) := D(k,k) = W(k,kw) |
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* A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k) |
* |
* |
CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) |
CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) |
R1 = ONE / A( K, K ) |
R1 = ONE / A( K, K ) |
CALL DSCAL( K-1, R1, A( 1, K ), 1 ) |
CALL DSCAL( K-1, R1, A( 1, K ), 1 ) |
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* |
ELSE |
ELSE |
* |
* |
* 2-by-2 pivot block D(k): columns KW and KW-1 of W now |
* 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold |
* hold |
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* |
* |
* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) |
* ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k) |
* |
* |
* where U(k) and U(k-1) are the k-th and (k-1)-th columns |
* where U(k) and U(k-1) are the k-th and (k-1)-th columns |
* of U |
* of U |
* |
* |
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* Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2 |
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* block D(k-1:k,k-1:k) in columns k-1 and k of A. |
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* NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT |
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* block and not stored. |
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* A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw) |
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* A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) = |
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* = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) ) |
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* |
IF( K.GT.2 ) THEN |
IF( K.GT.2 ) THEN |
* |
* |
* Store U(k) and U(k-1) in columns k and k-1 of A |
* Compose the columns of the inverse of 2-by-2 pivot |
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* block D in the following way to reduce the number |
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* of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by |
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* this inverse |
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* |
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* D**(-1) = ( d11 d21 )**(-1) = |
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* ( d21 d22 ) |
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* |
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* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) = |
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* ( (-d21 ) ( d11 ) ) |
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* |
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* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) * |
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* |
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* * ( ( d22/d21 ) ( -1 ) ) = |
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* ( ( -1 ) ( d11/d21 ) ) |
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* |
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* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) = |
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* ( ( -1 ) ( D22 ) ) |
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* |
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* = 1/d21 * T * ( ( D11 ) ( -1 ) ) |
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* ( ( -1 ) ( D22 ) ) |
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* |
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* = D21 * ( ( D11 ) ( -1 ) ) |
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* ( ( -1 ) ( D22 ) ) |
* |
* |
D21 = W( K-1, KW ) |
D21 = W( K-1, KW ) |
D11 = W( K, KW ) / D21 |
D11 = W( K, KW ) / D21 |
D22 = W( K-1, KW-1 ) / D21 |
D22 = W( K-1, KW-1 ) / D21 |
T = ONE / ( D11*D22-ONE ) |
T = ONE / ( D11*D22-ONE ) |
D21 = T / D21 |
D21 = T / D21 |
|
* |
|
* Update elements in columns A(k-1) and A(k) as |
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* dot products of rows of ( W(kw-1) W(kw) ) and columns |
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* of D**(-1) |
|
* |
DO 20 J = 1, K - 2 |
DO 20 J = 1, K - 2 |
A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) |
A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) |
A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) ) |
A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) ) |
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A( K-1, K-1 ) = W( K-1, KW-1 ) |
A( K-1, K-1 ) = W( K-1, KW-1 ) |
A( K-1, K ) = W( K-1, KW ) |
A( K-1, K ) = W( K-1, KW ) |
A( K, K ) = W( K, KW ) |
A( K, K ) = W( K, KW ) |
|
* |
END IF |
END IF |
|
* |
END IF |
END IF |
* |
* |
* Store details of the interchanges in IPIV |
* Store details of the interchanges in IPIV |
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* |
* |
* Update the upper triangle of A11 (= A(1:k,1:k)) as |
* Update the upper triangle of A11 (= A(1:k,1:k)) as |
* |
* |
* A11 := A11 - U12*D*U12' = A11 - U12*W' |
* A11 := A11 - U12*D*U12**T = A11 - U12*W**T |
* |
* |
* computing blocks of NB columns at a time |
* computing blocks of NB columns at a time |
* |
* |
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50 CONTINUE |
50 CONTINUE |
* |
* |
* Put U12 in standard form by partially undoing the interchanges |
* Put U12 in standard form by partially undoing the interchanges |
* in columns k+1:n |
* in columns k+1:n looping backwards from k+1 to n |
* |
* |
J = K + 1 |
J = K + 1 |
60 CONTINUE |
60 CONTINUE |
JJ = J |
* |
JP = IPIV( J ) |
* Undo the interchanges (if any) of rows JJ and JP at each |
IF( JP.LT.0 ) THEN |
* step J |
JP = -JP |
* |
|
* (Here, J is a diagonal index) |
|
JJ = J |
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JP = IPIV( J ) |
|
IF( JP.LT.0 ) THEN |
|
JP = -JP |
|
* (Here, J is a diagonal index) |
|
J = J + 1 |
|
END IF |
|
* (NOTE: Here, J is used to determine row length. Length N-J+1 |
|
* of the rows to swap back doesn't include diagonal element) |
J = J + 1 |
J = J + 1 |
END IF |
IF( JP.NE.JJ .AND. J.LE.N ) |
J = J + 1 |
$ CALL DSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) |
IF( JP.NE.JJ .AND. J.LE.N ) |
IF( J.LT.N ) |
$ CALL DSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) |
|
IF( J.LE.N ) |
|
$ GO TO 60 |
$ GO TO 60 |
* |
* |
* Set KB to the number of columns factorized |
* Set KB to the number of columns factorized |
Line 386
|
Line 553
|
ABSAKK = ABS( W( K, K ) ) |
ABSAKK = ABS( W( K, K ) ) |
* |
* |
* IMAX is the row-index of the largest off-diagonal element in |
* IMAX is the row-index of the largest off-diagonal element in |
* column K, and COLMAX is its absolute value |
* column K, and COLMAX is its absolute value. |
|
* Determine both COLMAX and IMAX. |
* |
* |
IF( K.LT.N ) THEN |
IF( K.LT.N ) THEN |
IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 ) |
IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 ) |
Line 397
|
Line 565
|
* |
* |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
* |
* |
* Column K is zero: set INFO and continue |
* Column K is zero or underflow: set INFO and continue |
* |
* |
IF( INFO.EQ.0 ) |
IF( INFO.EQ.0 ) |
$ INFO = K |
$ INFO = K |
Line 440
|
Line 608
|
* |
* |
KP = IMAX |
KP = IMAX |
* |
* |
* copy column K+1 of W to column K |
* copy column K+1 of W to column K of W |
* |
* |
CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) |
CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) |
ELSE |
ELSE |
Line 453
|
Line 621
|
END IF |
END IF |
END IF |
END IF |
* |
* |
|
* ============================================================ |
|
* |
|
* KK is the column of A where pivoting step stopped |
|
* |
KK = K + KSTEP - 1 |
KK = K + KSTEP - 1 |
* |
* |
* Updated column KP is already stored in column KK of W |
* Interchange rows and columns KP and KK. |
|
* Updated column KP is already stored in column KK of W. |
* |
* |
IF( KP.NE.KK ) THEN |
IF( KP.NE.KK ) THEN |
* |
* |
* Copy non-updated column KK to column KP |
* Copy non-updated column KK to column KP of submatrix A |
|
* at step K. No need to copy element into column K |
|
* (or K and K+1 for 2-by-2 pivot) of A, since these columns |
|
* will be later overwritten. |
* |
* |
A( KP, K ) = A( KK, K ) |
A( KP, KP ) = A( KK, KK ) |
CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA ) |
CALL DCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), |
CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 ) |
$ LDA ) |
|
IF( KP.LT.N ) |
|
$ CALL DCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) |
* |
* |
* Interchange rows KK and KP in first KK columns of A and W |
* Interchange rows KK and KP in first K-1 columns of A |
|
* (columns K (or K and K+1 for 2-by-2 pivot) of A will be |
|
* later overwritten). Interchange rows KK and KP |
|
* in first KK columns of W. |
* |
* |
CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) |
IF( K.GT.1 ) |
|
$ CALL DSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) |
CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) |
CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) |
END IF |
END IF |
* |
* |
Line 475
|
Line 657
|
* |
* |
* 1-by-1 pivot block D(k): column k of W now holds |
* 1-by-1 pivot block D(k): column k of W now holds |
* |
* |
* W(k) = L(k)*D(k) |
* W(k) = L(k)*D(k), |
* |
* |
* where L(k) is the k-th column of L |
* where L(k) is the k-th column of L |
* |
* |
* Store L(k) in column k of A |
* Store subdiag. elements of column L(k) |
|
* and 1-by-1 block D(k) in column k of A. |
|
* (NOTE: Diagonal element L(k,k) is a UNIT element |
|
* and not stored) |
|
* A(k,k) := D(k,k) = W(k,k) |
|
* A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k) |
* |
* |
CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) |
CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) |
IF( K.LT.N ) THEN |
IF( K.LT.N ) THEN |
R1 = ONE / A( K, K ) |
R1 = ONE / A( K, K ) |
CALL DSCAL( N-K, R1, A( K+1, K ), 1 ) |
CALL DSCAL( N-K, R1, A( K+1, K ), 1 ) |
END IF |
END IF |
|
* |
ELSE |
ELSE |
* |
* |
* 2-by-2 pivot block D(k): columns k and k+1 of W now hold |
* 2-by-2 pivot block D(k): columns k and k+1 of W now hold |
Line 495
|
Line 683
|
* where L(k) and L(k+1) are the k-th and (k+1)-th columns |
* where L(k) and L(k+1) are the k-th and (k+1)-th columns |
* of L |
* of L |
* |
* |
|
* Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2 |
|
* block D(k:k+1,k:k+1) in columns k and k+1 of A. |
|
* (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT |
|
* block and not stored) |
|
* A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1) |
|
* A(k+2:N,k:k+1) := L(k+2:N,k:k+1) = |
|
* = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) ) |
|
* |
IF( K.LT.N-1 ) THEN |
IF( K.LT.N-1 ) THEN |
* |
* |
* Store L(k) and L(k+1) in columns k and k+1 of A |
* Compose the columns of the inverse of 2-by-2 pivot |
|
* block D in the following way to reduce the number |
|
* of FLOPS when we myltiply panel ( W(k) W(k+1) ) by |
|
* this inverse |
|
* |
|
* D**(-1) = ( d11 d21 )**(-1) = |
|
* ( d21 d22 ) |
|
* |
|
* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) = |
|
* ( (-d21 ) ( d11 ) ) |
|
* |
|
* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) * |
|
* |
|
* * ( ( d22/d21 ) ( -1 ) ) = |
|
* ( ( -1 ) ( d11/d21 ) ) |
|
* |
|
* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) = |
|
* ( ( -1 ) ( D22 ) ) |
|
* |
|
* = 1/d21 * T * ( ( D11 ) ( -1 ) ) |
|
* ( ( -1 ) ( D22 ) ) |
|
* |
|
* = D21 * ( ( D11 ) ( -1 ) ) |
|
* ( ( -1 ) ( D22 ) ) |
* |
* |
D21 = W( K+1, K ) |
D21 = W( K+1, K ) |
D11 = W( K+1, K+1 ) / D21 |
D11 = W( K+1, K+1 ) / D21 |
D22 = W( K, K ) / D21 |
D22 = W( K, K ) / D21 |
T = ONE / ( D11*D22-ONE ) |
T = ONE / ( D11*D22-ONE ) |
D21 = T / D21 |
D21 = T / D21 |
|
* |
|
* Update elements in columns A(k) and A(k+1) as |
|
* dot products of rows of ( W(k) W(k+1) ) and columns |
|
* of D**(-1) |
|
* |
DO 80 J = K + 2, N |
DO 80 J = K + 2, N |
A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) ) |
A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) ) |
A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) ) |
A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) ) |
Line 515
|
Line 739
|
A( K, K ) = W( K, K ) |
A( K, K ) = W( K, K ) |
A( K+1, K ) = W( K+1, K ) |
A( K+1, K ) = W( K+1, K ) |
A( K+1, K+1 ) = W( K+1, K+1 ) |
A( K+1, K+1 ) = W( K+1, K+1 ) |
|
* |
END IF |
END IF |
|
* |
END IF |
END IF |
* |
* |
* Store details of the interchanges in IPIV |
* Store details of the interchanges in IPIV |
Line 536
|
Line 762
|
* |
* |
* Update the lower triangle of A22 (= A(k:n,k:n)) as |
* Update the lower triangle of A22 (= A(k:n,k:n)) as |
* |
* |
* A22 := A22 - L21*D*L21' = A22 - L21*W' |
* A22 := A22 - L21*D*L21**T = A22 - L21*W**T |
* |
* |
* computing blocks of NB columns at a time |
* computing blocks of NB columns at a time |
* |
* |
Line 560
|
Line 786
|
110 CONTINUE |
110 CONTINUE |
* |
* |
* Put L21 in standard form by partially undoing the interchanges |
* Put L21 in standard form by partially undoing the interchanges |
* in columns 1:k-1 |
* of rows in columns 1:k-1 looping backwards from k-1 to 1 |
* |
* |
J = K - 1 |
J = K - 1 |
120 CONTINUE |
120 CONTINUE |
JJ = J |
* |
JP = IPIV( J ) |
* Undo the interchanges (if any) of rows JJ and JP at each |
IF( JP.LT.0 ) THEN |
* step J |
JP = -JP |
* |
|
* (Here, J is a diagonal index) |
|
JJ = J |
|
JP = IPIV( J ) |
|
IF( JP.LT.0 ) THEN |
|
JP = -JP |
|
* (Here, J is a diagonal index) |
|
J = J - 1 |
|
END IF |
|
* (NOTE: Here, J is used to determine row length. Length J |
|
* of the rows to swap back doesn't include diagonal element) |
J = J - 1 |
J = J - 1 |
END IF |
IF( JP.NE.JJ .AND. J.GE.1 ) |
J = J - 1 |
$ CALL DSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) |
IF( JP.NE.JJ .AND. J.GE.1 ) |
IF( J.GT.1 ) |
$ CALL DSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) |
|
IF( J.GE.1 ) |
|
$ GO TO 120 |
$ GO TO 120 |
* |
* |
* Set KB to the number of columns factorized |
* Set KB to the number of columns factorized |