1077 lines
45 KiB
Fortran
1077 lines
45 KiB
Fortran
*> \brief \b SGSVJ0 pre-processor for the routine sgesvj.
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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 SGSVJ0 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgsvj0.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/sgsvj0.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/sgsvj0.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 SGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
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* SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
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* REAL EPS, SFMIN, TOL
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* CHARACTER*1 JOBV
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* ..
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* .. Array Arguments ..
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* REAL A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
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* $ WORK( LWORK )
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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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*> SGSVJ0 is called from SGESVJ as a pre-processor and that is its main
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*> purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
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*> it does not check convergence (stopping criterion). Few tuning
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*> parameters (marked by [TP]) are available for the implementer.
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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] JOBV
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*> \verbatim
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*> JOBV is CHARACTER*1
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*> Specifies whether the output from this procedure is used
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*> to compute the matrix V:
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*> = 'V': the product of the Jacobi rotations is accumulated
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*> by postmultiplying the N-by-N array V.
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*> (See the description of V.)
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*> = 'A': the product of the Jacobi rotations is accumulated
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*> by postmultiplying the MV-by-N array V.
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*> (See the descriptions of MV and V.)
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*> = 'N': the Jacobi rotations are not accumulated.
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*> \endverbatim
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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 input 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 input matrix A.
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*> M >= N >= 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 REAL array, dimension (LDA,N)
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*> On entry, M-by-N matrix A, such that A*diag(D) represents
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*> the input matrix.
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*> On exit,
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*> A_onexit * D_onexit represents the input matrix A*diag(D)
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*> post-multiplied by a sequence of Jacobi rotations, where the
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*> rotation threshold and the total number of sweeps are given in
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*> TOL and NSWEEP, respectively.
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*> (See the descriptions of D, TOL and NSWEEP.)
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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[in,out] D
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*> \verbatim
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*> D is REAL array, dimension (N)
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*> The array D accumulates the scaling factors from the fast scaled
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*> Jacobi rotations.
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*> On entry, A*diag(D) represents the input matrix.
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*> On exit, A_onexit*diag(D_onexit) represents the input matrix
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*> post-multiplied by a sequence of Jacobi rotations, where the
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*> rotation threshold and the total number of sweeps are given in
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*> TOL and NSWEEP, respectively.
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*> (See the descriptions of A, TOL and NSWEEP.)
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*> \endverbatim
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*>
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*> \param[in,out] SVA
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*> \verbatim
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*> SVA is REAL array, dimension (N)
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*> On entry, SVA contains the Euclidean norms of the columns of
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*> the matrix A*diag(D).
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*> On exit, SVA contains the Euclidean norms of the columns of
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*> the matrix onexit*diag(D_onexit).
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*> \endverbatim
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*>
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*> \param[in] MV
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*> \verbatim
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*> MV is INTEGER
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*> If JOBV = 'A', then MV rows of V are post-multiplied by a
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*> sequence of Jacobi rotations.
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*> If JOBV = 'N', then MV is not referenced.
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*> \endverbatim
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*>
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*> \param[in,out] V
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*> \verbatim
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*> V is REAL array, dimension (LDV,N)
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*> If JOBV = 'V' then N rows of V are post-multiplied by a
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*> sequence of Jacobi rotations.
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*> If JOBV = 'A' then MV rows of V are post-multiplied by a
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*> sequence of Jacobi rotations.
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*> If JOBV = 'N', then V is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDV
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*> \verbatim
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*> LDV is INTEGER
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*> The leading dimension of the array V, LDV >= 1.
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*> If JOBV = 'V', LDV >= N.
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*> If JOBV = 'A', LDV >= MV.
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*> \endverbatim
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*>
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*> \param[in] EPS
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*> \verbatim
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*> EPS is REAL
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*> EPS = SLAMCH('Epsilon')
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*> \endverbatim
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*>
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*> \param[in] SFMIN
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*> \verbatim
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*> SFMIN is REAL
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*> SFMIN = SLAMCH('Safe Minimum')
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*> \endverbatim
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*>
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*> \param[in] TOL
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*> \verbatim
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*> TOL is REAL
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*> TOL is the threshold for Jacobi rotations. For a pair
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*> A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
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*> applied only if ABS(COS(angle(A(:,p),A(:,q)))) > TOL.
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*> \endverbatim
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*>
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*> \param[in] NSWEEP
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*> \verbatim
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*> NSWEEP is INTEGER
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*> NSWEEP is the number of sweeps of Jacobi rotations to be
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*> performed.
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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 REAL array, dimension (LWORK)
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> LWORK is the dimension of WORK. LWORK >= M.
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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, then 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 realOTHERcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> SGSVJ0 is used just to enable SGESVJ to call a simplified version of
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*> itself to work on a submatrix of the original matrix.
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*>
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*> \par Contributors:
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* ==================
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*>
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*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
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*>
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*> \par Bugs, Examples and Comments:
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* =================================
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*>
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*> Please report all bugs and send interesting test examples and comments to
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*> drmac@math.hr. Thank you.
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*
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* =====================================================================
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SUBROUTINE SGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
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$ SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
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REAL EPS, SFMIN, TOL
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CHARACTER*1 JOBV
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* ..
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* .. Array Arguments ..
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REAL A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
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$ WORK( LWORK )
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* ..
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*
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* =====================================================================
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*
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* .. Local Parameters ..
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REAL ZERO, HALF, ONE
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PARAMETER ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0)
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* ..
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* .. Local Scalars ..
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REAL AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG,
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$ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS,
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$ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA,
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$ THSIGN
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INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
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$ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL,
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$ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
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LOGICAL APPLV, ROTOK, RSVEC
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* ..
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* .. Local Arrays ..
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REAL FASTR( 5 )
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, FLOAT, MIN, SIGN, SQRT
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* ..
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* .. External Functions ..
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REAL SDOT, SNRM2
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INTEGER ISAMAX
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LOGICAL LSAME
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EXTERNAL ISAMAX, LSAME, SDOT, SNRM2
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* ..
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* .. External Subroutines ..
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EXTERNAL SAXPY, SCOPY, SLASCL, SLASSQ, SROTM, SSWAP,
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$ XERBLA
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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APPLV = LSAME( JOBV, 'A' )
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RSVEC = LSAME( JOBV, 'V' )
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IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( M.LT.0 ) THEN
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INFO = -2
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ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
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INFO = -3
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ELSE IF( LDA.LT.M ) THEN
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INFO = -5
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ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN
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INFO = -8
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ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR.
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$ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN
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INFO = -10
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ELSE IF( TOL.LE.EPS ) THEN
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INFO = -13
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ELSE IF( NSWEEP.LT.0 ) THEN
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INFO = -14
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ELSE IF( LWORK.LT.M ) THEN
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INFO = -16
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ELSE
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INFO = 0
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END IF
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*
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* #:(
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SGSVJ0', -INFO )
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RETURN
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END IF
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*
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IF( RSVEC ) THEN
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MVL = N
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ELSE IF( APPLV ) THEN
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MVL = MV
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END IF
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RSVEC = RSVEC .OR. APPLV
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ROOTEPS = SQRT( EPS )
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ROOTSFMIN = SQRT( SFMIN )
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SMALL = SFMIN / EPS
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BIG = ONE / SFMIN
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ROOTBIG = ONE / ROOTSFMIN
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BIGTHETA = ONE / ROOTEPS
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ROOTTOL = SQRT( TOL )
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*
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* .. Row-cyclic Jacobi SVD algorithm with column pivoting ..
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*
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EMPTSW = ( N*( N-1 ) ) / 2
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NOTROT = 0
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FASTR( 1 ) = ZERO
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*
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* .. Row-cyclic pivot strategy with de Rijk's pivoting ..
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*
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SWBAND = 0
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*[TP] SWBAND is a tuning parameter. It is meaningful and effective
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* if SGESVJ is used as a computational routine in the preconditioned
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* Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
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* ......
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KBL = MIN( 8, N )
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*[TP] KBL is a tuning parameter that defines the tile size in the
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* tiling of the p-q loops of pivot pairs. In general, an optimal
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* value of KBL depends on the matrix dimensions and on the
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* parameters of the computer's memory.
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*
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NBL = N / KBL
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IF( ( NBL*KBL ).NE.N )NBL = NBL + 1
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BLSKIP = ( KBL**2 ) + 1
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*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
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ROWSKIP = MIN( 5, KBL )
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*[TP] ROWSKIP is a tuning parameter.
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LKAHEAD = 1
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*[TP] LKAHEAD is a tuning parameter.
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SWBAND = 0
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PSKIPPED = 0
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*
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DO 1993 i = 1, NSWEEP
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* .. go go go ...
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*
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MXAAPQ = ZERO
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MXSINJ = ZERO
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ISWROT = 0
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*
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NOTROT = 0
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PSKIPPED = 0
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*
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DO 2000 ibr = 1, NBL
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igl = ( ibr-1 )*KBL + 1
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*
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DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr )
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*
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igl = igl + ir1*KBL
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*
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DO 2001 p = igl, MIN( igl+KBL-1, N-1 )
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* .. de Rijk's pivoting
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q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1
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IF( p.NE.q ) THEN
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CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
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IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1,
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$ V( 1, q ), 1 )
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TEMP1 = SVA( p )
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SVA( p ) = SVA( q )
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SVA( q ) = TEMP1
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TEMP1 = D( p )
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D( p ) = D( q )
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D( q ) = TEMP1
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END IF
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*
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IF( ir1.EQ.0 ) THEN
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*
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* Column norms are periodically updated by explicit
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* norm computation.
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* Caveat:
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* Some BLAS implementations compute SNRM2(M,A(1,p),1)
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* as SQRT(SDOT(M,A(1,p),1,A(1,p),1)), which may result in
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* overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and
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* underflow for ||A(:,p)||_2 < SQRT(underflow_threshold).
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* Hence, SNRM2 cannot be trusted, not even in the case when
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* the true norm is far from the under(over)flow boundaries.
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* If properly implemented SNRM2 is available, the IF-THEN-ELSE
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* below should read "AAPP = SNRM2( M, A(1,p), 1 ) * D(p)".
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*
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IF( ( SVA( p ).LT.ROOTBIG ) .AND.
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$ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
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SVA( p ) = SNRM2( M, A( 1, p ), 1 )*D( p )
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ELSE
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TEMP1 = ZERO
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AAPP = ONE
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CALL SLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
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SVA( p ) = TEMP1*SQRT( AAPP )*D( p )
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END IF
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AAPP = SVA( p )
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ELSE
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AAPP = SVA( p )
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END IF
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*
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IF( AAPP.GT.ZERO ) THEN
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*
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PSKIPPED = 0
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*
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DO 2002 q = p + 1, MIN( igl+KBL-1, N )
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*
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AAQQ = SVA( q )
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IF( AAQQ.GT.ZERO ) THEN
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*
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AAPP0 = AAPP
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IF( AAQQ.GE.ONE ) THEN
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ROTOK = ( SMALL*AAPP ).LE.AAQQ
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IF( AAPP.LT.( BIG / AAQQ ) ) THEN
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AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
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$ q ), 1 )*D( p )*D( q ) / AAQQ )
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$ / AAPP
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ELSE
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CALL SCOPY( M, A( 1, p ), 1, WORK, 1 )
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CALL SLASCL( 'G', 0, 0, AAPP, D( p ),
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$ M, 1, WORK, LDA, IERR )
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AAPQ = SDOT( M, WORK, 1, A( 1, q ),
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$ 1 )*D( q ) / AAQQ
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END IF
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ELSE
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ROTOK = AAPP.LE.( AAQQ / SMALL )
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IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
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AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
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$ q ), 1 )*D( p )*D( q ) / AAQQ )
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$ / AAPP
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ELSE
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CALL SCOPY( M, A( 1, q ), 1, WORK, 1 )
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CALL SLASCL( 'G', 0, 0, AAQQ, D( q ),
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$ M, 1, WORK, LDA, IERR )
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AAPQ = SDOT( M, WORK, 1, A( 1, p ),
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$ 1 )*D( p ) / AAPP
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END IF
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END IF
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*
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MXAAPQ = MAX( MXAAPQ, ABS( AAPQ ) )
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*
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* TO rotate or NOT to rotate, THAT is the question ...
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*
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IF( ABS( AAPQ ).GT.TOL ) THEN
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*
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* .. rotate
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* ROTATED = ROTATED + ONE
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*
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IF( ir1.EQ.0 ) THEN
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NOTROT = 0
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PSKIPPED = 0
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ISWROT = ISWROT + 1
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END IF
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*
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IF( ROTOK ) THEN
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*
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AQOAP = AAQQ / AAPP
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APOAQ = AAPP / AAQQ
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THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ
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*
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IF( ABS( THETA ).GT.BIGTHETA ) THEN
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*
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T = HALF / THETA
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FASTR( 3 ) = T*D( p ) / D( q )
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FASTR( 4 ) = -T*D( q ) / D( p )
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CALL SROTM( M, A( 1, p ), 1,
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$ A( 1, q ), 1, FASTR )
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IF( RSVEC )CALL SROTM( MVL,
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$ V( 1, p ), 1,
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$ V( 1, q ), 1,
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$ FASTR )
|
|
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
|
|
$ ONE+T*APOAQ*AAPQ ) )
|
|
AAPP = AAPP*SQRT( MAX( ZERO,
|
|
$ ONE-T*AQOAP*AAPQ ) )
|
|
MXSINJ = MAX( MXSINJ, ABS( T ) )
|
|
*
|
|
ELSE
|
|
*
|
|
* .. choose correct signum for THETA and rotate
|
|
*
|
|
THSIGN = -SIGN( ONE, AAPQ )
|
|
T = ONE / ( THETA+THSIGN*
|
|
$ SQRT( ONE+THETA*THETA ) )
|
|
CS = SQRT( ONE / ( ONE+T*T ) )
|
|
SN = T*CS
|
|
*
|
|
MXSINJ = MAX( MXSINJ, ABS( SN ) )
|
|
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
|
|
$ ONE+T*APOAQ*AAPQ ) )
|
|
AAPP = AAPP*SQRT( MAX( ZERO,
|
|
$ ONE-T*AQOAP*AAPQ ) )
|
|
*
|
|
APOAQ = D( p ) / D( q )
|
|
AQOAP = D( q ) / D( p )
|
|
IF( D( p ).GE.ONE ) THEN
|
|
IF( D( q ).GE.ONE ) THEN
|
|
FASTR( 3 ) = T*APOAQ
|
|
FASTR( 4 ) = -T*AQOAP
|
|
D( p ) = D( p )*CS
|
|
D( q ) = D( q )*CS
|
|
CALL SROTM( M, A( 1, p ), 1,
|
|
$ A( 1, q ), 1,
|
|
$ FASTR )
|
|
IF( RSVEC )CALL SROTM( MVL,
|
|
$ V( 1, p ), 1, V( 1, q ),
|
|
$ 1, FASTR )
|
|
ELSE
|
|
CALL SAXPY( M, -T*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
CALL SAXPY( M, CS*SN*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
D( p ) = D( p )*CS
|
|
D( q ) = D( q ) / CS
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL, -T*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ CS*SN*APOAQ,
|
|
$ V( 1, p ), 1,
|
|
$ V( 1, q ), 1 )
|
|
END IF
|
|
END IF
|
|
ELSE
|
|
IF( D( q ).GE.ONE ) THEN
|
|
CALL SAXPY( M, T*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
CALL SAXPY( M, -CS*SN*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
D( p ) = D( p ) / CS
|
|
D( q ) = D( q )*CS
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL, T*APOAQ,
|
|
$ V( 1, p ), 1,
|
|
$ V( 1, q ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ -CS*SN*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
END IF
|
|
ELSE
|
|
IF( D( p ).GE.D( q ) ) THEN
|
|
CALL SAXPY( M, -T*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
CALL SAXPY( M, CS*SN*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
D( p ) = D( p )*CS
|
|
D( q ) = D( q ) / CS
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL,
|
|
$ -T*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ CS*SN*APOAQ,
|
|
$ V( 1, p ), 1,
|
|
$ V( 1, q ), 1 )
|
|
END IF
|
|
ELSE
|
|
CALL SAXPY( M, T*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
CALL SAXPY( M,
|
|
$ -CS*SN*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
D( p ) = D( p ) / CS
|
|
D( q ) = D( q )*CS
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL,
|
|
$ T*APOAQ, V( 1, p ),
|
|
$ 1, V( 1, q ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ -CS*SN*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
END IF
|
|
END IF
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
ELSE
|
|
* .. have to use modified Gram-Schmidt like transformation
|
|
CALL SCOPY( M, A( 1, p ), 1, WORK, 1 )
|
|
CALL SLASCL( 'G', 0, 0, AAPP, ONE, M,
|
|
$ 1, WORK, LDA, IERR )
|
|
CALL SLASCL( 'G', 0, 0, AAQQ, ONE, M,
|
|
$ 1, A( 1, q ), LDA, IERR )
|
|
TEMP1 = -AAPQ*D( p ) / D( q )
|
|
CALL SAXPY( M, TEMP1, WORK, 1,
|
|
$ A( 1, q ), 1 )
|
|
CALL SLASCL( 'G', 0, 0, ONE, AAQQ, M,
|
|
$ 1, A( 1, q ), LDA, IERR )
|
|
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
|
|
$ ONE-AAPQ*AAPQ ) )
|
|
MXSINJ = MAX( MXSINJ, SFMIN )
|
|
END IF
|
|
* END IF ROTOK THEN ... ELSE
|
|
*
|
|
* In the case of cancellation in updating SVA(q), SVA(p)
|
|
* recompute SVA(q), SVA(p).
|
|
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
|
|
$ THEN
|
|
IF( ( AAQQ.LT.ROOTBIG ) .AND.
|
|
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
|
|
SVA( q ) = SNRM2( M, A( 1, q ), 1 )*
|
|
$ D( q )
|
|
ELSE
|
|
T = ZERO
|
|
AAQQ = ONE
|
|
CALL SLASSQ( M, A( 1, q ), 1, T,
|
|
$ AAQQ )
|
|
SVA( q ) = T*SQRT( AAQQ )*D( q )
|
|
END IF
|
|
END IF
|
|
IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
|
|
IF( ( AAPP.LT.ROOTBIG ) .AND.
|
|
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
|
|
AAPP = SNRM2( M, A( 1, p ), 1 )*
|
|
$ D( p )
|
|
ELSE
|
|
T = ZERO
|
|
AAPP = ONE
|
|
CALL SLASSQ( M, A( 1, p ), 1, T,
|
|
$ AAPP )
|
|
AAPP = T*SQRT( AAPP )*D( p )
|
|
END IF
|
|
SVA( p ) = AAPP
|
|
END IF
|
|
*
|
|
ELSE
|
|
* A(:,p) and A(:,q) already numerically orthogonal
|
|
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
|
|
PSKIPPED = PSKIPPED + 1
|
|
END IF
|
|
ELSE
|
|
* A(:,q) is zero column
|
|
IF( ir1.EQ.0 )NOTROT = NOTROT + 1
|
|
PSKIPPED = PSKIPPED + 1
|
|
END IF
|
|
*
|
|
IF( ( i.LE.SWBAND ) .AND.
|
|
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
|
|
IF( ir1.EQ.0 )AAPP = -AAPP
|
|
NOTROT = 0
|
|
GO TO 2103
|
|
END IF
|
|
*
|
|
2002 CONTINUE
|
|
* END q-LOOP
|
|
*
|
|
2103 CONTINUE
|
|
* bailed out of q-loop
|
|
|
|
SVA( p ) = AAPP
|
|
|
|
ELSE
|
|
SVA( p ) = AAPP
|
|
IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
|
|
$ NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p
|
|
END IF
|
|
*
|
|
2001 CONTINUE
|
|
* end of the p-loop
|
|
* end of doing the block ( ibr, ibr )
|
|
1002 CONTINUE
|
|
* end of ir1-loop
|
|
*
|
|
*........................................................
|
|
* ... go to the off diagonal blocks
|
|
*
|
|
igl = ( ibr-1 )*KBL + 1
|
|
*
|
|
DO 2010 jbc = ibr + 1, NBL
|
|
*
|
|
jgl = ( jbc-1 )*KBL + 1
|
|
*
|
|
* doing the block at ( ibr, jbc )
|
|
*
|
|
IJBLSK = 0
|
|
DO 2100 p = igl, MIN( igl+KBL-1, N )
|
|
*
|
|
AAPP = SVA( p )
|
|
*
|
|
IF( AAPP.GT.ZERO ) THEN
|
|
*
|
|
PSKIPPED = 0
|
|
*
|
|
DO 2200 q = jgl, MIN( jgl+KBL-1, N )
|
|
*
|
|
AAQQ = SVA( q )
|
|
*
|
|
IF( AAQQ.GT.ZERO ) THEN
|
|
AAPP0 = AAPP
|
|
*
|
|
* .. M x 2 Jacobi SVD ..
|
|
*
|
|
* .. Safe Gram matrix computation ..
|
|
*
|
|
IF( AAQQ.GE.ONE ) THEN
|
|
IF( AAPP.GE.AAQQ ) THEN
|
|
ROTOK = ( SMALL*AAPP ).LE.AAQQ
|
|
ELSE
|
|
ROTOK = ( SMALL*AAQQ ).LE.AAPP
|
|
END IF
|
|
IF( AAPP.LT.( BIG / AAQQ ) ) THEN
|
|
AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
|
|
$ q ), 1 )*D( p )*D( q ) / AAQQ )
|
|
$ / AAPP
|
|
ELSE
|
|
CALL SCOPY( M, A( 1, p ), 1, WORK, 1 )
|
|
CALL SLASCL( 'G', 0, 0, AAPP, D( p ),
|
|
$ M, 1, WORK, LDA, IERR )
|
|
AAPQ = SDOT( M, WORK, 1, A( 1, q ),
|
|
$ 1 )*D( q ) / AAQQ
|
|
END IF
|
|
ELSE
|
|
IF( AAPP.GE.AAQQ ) THEN
|
|
ROTOK = AAPP.LE.( AAQQ / SMALL )
|
|
ELSE
|
|
ROTOK = AAQQ.LE.( AAPP / SMALL )
|
|
END IF
|
|
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
|
|
AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
|
|
$ q ), 1 )*D( p )*D( q ) / AAQQ )
|
|
$ / AAPP
|
|
ELSE
|
|
CALL SCOPY( M, A( 1, q ), 1, WORK, 1 )
|
|
CALL SLASCL( 'G', 0, 0, AAQQ, D( q ),
|
|
$ M, 1, WORK, LDA, IERR )
|
|
AAPQ = SDOT( M, WORK, 1, A( 1, p ),
|
|
$ 1 )*D( p ) / AAPP
|
|
END IF
|
|
END IF
|
|
*
|
|
MXAAPQ = MAX( MXAAPQ, ABS( AAPQ ) )
|
|
*
|
|
* TO rotate or NOT to rotate, THAT is the question ...
|
|
*
|
|
IF( ABS( AAPQ ).GT.TOL ) THEN
|
|
NOTROT = 0
|
|
* ROTATED = ROTATED + 1
|
|
PSKIPPED = 0
|
|
ISWROT = ISWROT + 1
|
|
*
|
|
IF( ROTOK ) THEN
|
|
*
|
|
AQOAP = AAQQ / AAPP
|
|
APOAQ = AAPP / AAQQ
|
|
THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ
|
|
IF( AAQQ.GT.AAPP0 )THETA = -THETA
|
|
*
|
|
IF( ABS( THETA ).GT.BIGTHETA ) THEN
|
|
T = HALF / THETA
|
|
FASTR( 3 ) = T*D( p ) / D( q )
|
|
FASTR( 4 ) = -T*D( q ) / D( p )
|
|
CALL SROTM( M, A( 1, p ), 1,
|
|
$ A( 1, q ), 1, FASTR )
|
|
IF( RSVEC )CALL SROTM( MVL,
|
|
$ V( 1, p ), 1,
|
|
$ V( 1, q ), 1,
|
|
$ FASTR )
|
|
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
|
|
$ ONE+T*APOAQ*AAPQ ) )
|
|
AAPP = AAPP*SQRT( MAX( ZERO,
|
|
$ ONE-T*AQOAP*AAPQ ) )
|
|
MXSINJ = MAX( MXSINJ, ABS( T ) )
|
|
ELSE
|
|
*
|
|
* .. choose correct signum for THETA and rotate
|
|
*
|
|
THSIGN = -SIGN( ONE, AAPQ )
|
|
IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
|
|
T = ONE / ( THETA+THSIGN*
|
|
$ SQRT( ONE+THETA*THETA ) )
|
|
CS = SQRT( ONE / ( ONE+T*T ) )
|
|
SN = T*CS
|
|
MXSINJ = MAX( MXSINJ, ABS( SN ) )
|
|
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
|
|
$ ONE+T*APOAQ*AAPQ ) )
|
|
AAPP = AAPP*SQRT( MAX( ZERO,
|
|
$ ONE-T*AQOAP*AAPQ ) )
|
|
*
|
|
APOAQ = D( p ) / D( q )
|
|
AQOAP = D( q ) / D( p )
|
|
IF( D( p ).GE.ONE ) THEN
|
|
*
|
|
IF( D( q ).GE.ONE ) THEN
|
|
FASTR( 3 ) = T*APOAQ
|
|
FASTR( 4 ) = -T*AQOAP
|
|
D( p ) = D( p )*CS
|
|
D( q ) = D( q )*CS
|
|
CALL SROTM( M, A( 1, p ), 1,
|
|
$ A( 1, q ), 1,
|
|
$ FASTR )
|
|
IF( RSVEC )CALL SROTM( MVL,
|
|
$ V( 1, p ), 1, V( 1, q ),
|
|
$ 1, FASTR )
|
|
ELSE
|
|
CALL SAXPY( M, -T*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
CALL SAXPY( M, CS*SN*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL, -T*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ CS*SN*APOAQ,
|
|
$ V( 1, p ), 1,
|
|
$ V( 1, q ), 1 )
|
|
END IF
|
|
D( p ) = D( p )*CS
|
|
D( q ) = D( q ) / CS
|
|
END IF
|
|
ELSE
|
|
IF( D( q ).GE.ONE ) THEN
|
|
CALL SAXPY( M, T*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
CALL SAXPY( M, -CS*SN*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL, T*APOAQ,
|
|
$ V( 1, p ), 1,
|
|
$ V( 1, q ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ -CS*SN*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
END IF
|
|
D( p ) = D( p ) / CS
|
|
D( q ) = D( q )*CS
|
|
ELSE
|
|
IF( D( p ).GE.D( q ) ) THEN
|
|
CALL SAXPY( M, -T*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
CALL SAXPY( M, CS*SN*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
D( p ) = D( p )*CS
|
|
D( q ) = D( q ) / CS
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL,
|
|
$ -T*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ CS*SN*APOAQ,
|
|
$ V( 1, p ), 1,
|
|
$ V( 1, q ), 1 )
|
|
END IF
|
|
ELSE
|
|
CALL SAXPY( M, T*APOAQ,
|
|
$ A( 1, p ), 1,
|
|
$ A( 1, q ), 1 )
|
|
CALL SAXPY( M,
|
|
$ -CS*SN*AQOAP,
|
|
$ A( 1, q ), 1,
|
|
$ A( 1, p ), 1 )
|
|
D( p ) = D( p ) / CS
|
|
D( q ) = D( q )*CS
|
|
IF( RSVEC ) THEN
|
|
CALL SAXPY( MVL,
|
|
$ T*APOAQ, V( 1, p ),
|
|
$ 1, V( 1, q ), 1 )
|
|
CALL SAXPY( MVL,
|
|
$ -CS*SN*AQOAP,
|
|
$ V( 1, q ), 1,
|
|
$ V( 1, p ), 1 )
|
|
END IF
|
|
END IF
|
|
END IF
|
|
END IF
|
|
END IF
|
|
*
|
|
ELSE
|
|
IF( AAPP.GT.AAQQ ) THEN
|
|
CALL SCOPY( M, A( 1, p ), 1, WORK,
|
|
$ 1 )
|
|
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
|
|
$ M, 1, WORK, LDA, IERR )
|
|
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
|
|
$ M, 1, A( 1, q ), LDA,
|
|
$ IERR )
|
|
TEMP1 = -AAPQ*D( p ) / D( q )
|
|
CALL SAXPY( M, TEMP1, WORK, 1,
|
|
$ A( 1, q ), 1 )
|
|
CALL SLASCL( 'G', 0, 0, ONE, AAQQ,
|
|
$ M, 1, A( 1, q ), LDA,
|
|
$ IERR )
|
|
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
|
|
$ ONE-AAPQ*AAPQ ) )
|
|
MXSINJ = MAX( MXSINJ, SFMIN )
|
|
ELSE
|
|
CALL SCOPY( M, A( 1, q ), 1, WORK,
|
|
$ 1 )
|
|
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
|
|
$ M, 1, WORK, LDA, IERR )
|
|
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
|
|
$ M, 1, A( 1, p ), LDA,
|
|
$ IERR )
|
|
TEMP1 = -AAPQ*D( q ) / D( p )
|
|
CALL SAXPY( M, TEMP1, WORK, 1,
|
|
$ A( 1, p ), 1 )
|
|
CALL SLASCL( 'G', 0, 0, ONE, AAPP,
|
|
$ M, 1, A( 1, p ), LDA,
|
|
$ IERR )
|
|
SVA( p ) = AAPP*SQRT( MAX( ZERO,
|
|
$ ONE-AAPQ*AAPQ ) )
|
|
MXSINJ = MAX( MXSINJ, SFMIN )
|
|
END IF
|
|
END IF
|
|
* END IF ROTOK THEN ... ELSE
|
|
*
|
|
* In the case of cancellation in updating SVA(q)
|
|
* .. recompute SVA(q)
|
|
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
|
|
$ THEN
|
|
IF( ( AAQQ.LT.ROOTBIG ) .AND.
|
|
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
|
|
SVA( q ) = SNRM2( M, A( 1, q ), 1 )*
|
|
$ D( q )
|
|
ELSE
|
|
T = ZERO
|
|
AAQQ = ONE
|
|
CALL SLASSQ( M, A( 1, q ), 1, T,
|
|
$ AAQQ )
|
|
SVA( q ) = T*SQRT( AAQQ )*D( q )
|
|
END IF
|
|
END IF
|
|
IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
|
|
IF( ( AAPP.LT.ROOTBIG ) .AND.
|
|
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
|
|
AAPP = SNRM2( M, A( 1, p ), 1 )*
|
|
$ D( p )
|
|
ELSE
|
|
T = ZERO
|
|
AAPP = ONE
|
|
CALL SLASSQ( M, A( 1, p ), 1, T,
|
|
$ AAPP )
|
|
AAPP = T*SQRT( AAPP )*D( p )
|
|
END IF
|
|
SVA( p ) = AAPP
|
|
END IF
|
|
* end of OK rotation
|
|
ELSE
|
|
NOTROT = NOTROT + 1
|
|
PSKIPPED = PSKIPPED + 1
|
|
IJBLSK = IJBLSK + 1
|
|
END IF
|
|
ELSE
|
|
NOTROT = NOTROT + 1
|
|
PSKIPPED = PSKIPPED + 1
|
|
IJBLSK = IJBLSK + 1
|
|
END IF
|
|
*
|
|
IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) )
|
|
$ THEN
|
|
SVA( p ) = AAPP
|
|
NOTROT = 0
|
|
GO TO 2011
|
|
END IF
|
|
IF( ( i.LE.SWBAND ) .AND.
|
|
$ ( PSKIPPED.GT.ROWSKIP ) ) THEN
|
|
AAPP = -AAPP
|
|
NOTROT = 0
|
|
GO TO 2203
|
|
END IF
|
|
*
|
|
2200 CONTINUE
|
|
* end of the q-loop
|
|
2203 CONTINUE
|
|
*
|
|
SVA( p ) = AAPP
|
|
*
|
|
ELSE
|
|
IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
|
|
$ MIN( jgl+KBL-1, N ) - jgl + 1
|
|
IF( AAPP.LT.ZERO )NOTROT = 0
|
|
END IF
|
|
|
|
2100 CONTINUE
|
|
* end of the p-loop
|
|
2010 CONTINUE
|
|
* end of the jbc-loop
|
|
2011 CONTINUE
|
|
*2011 bailed out of the jbc-loop
|
|
DO 2012 p = igl, MIN( igl+KBL-1, N )
|
|
SVA( p ) = ABS( SVA( p ) )
|
|
2012 CONTINUE
|
|
*
|
|
2000 CONTINUE
|
|
*2000 :: end of the ibr-loop
|
|
*
|
|
* .. update SVA(N)
|
|
IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) )
|
|
$ THEN
|
|
SVA( N ) = SNRM2( M, A( 1, N ), 1 )*D( N )
|
|
ELSE
|
|
T = ZERO
|
|
AAPP = ONE
|
|
CALL SLASSQ( M, A( 1, N ), 1, T, AAPP )
|
|
SVA( N ) = T*SQRT( AAPP )*D( N )
|
|
END IF
|
|
*
|
|
* Additional steering devices
|
|
*
|
|
IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR.
|
|
$ ( ISWROT.LE.N ) ) )SWBAND = i
|
|
*
|
|
IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.FLOAT( N )*TOL ) .AND.
|
|
$ ( FLOAT( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN
|
|
GO TO 1994
|
|
END IF
|
|
*
|
|
IF( NOTROT.GE.EMPTSW )GO TO 1994
|
|
|
|
1993 CONTINUE
|
|
* end i=1:NSWEEP loop
|
|
* #:) Reaching this point means that the procedure has completed the given
|
|
* number of iterations.
|
|
INFO = NSWEEP - 1
|
|
GO TO 1995
|
|
1994 CONTINUE
|
|
* #:) Reaching this point means that during the i-th sweep all pivots were
|
|
* below the given tolerance, causing early exit.
|
|
*
|
|
INFO = 0
|
|
* #:) INFO = 0 confirms successful iterations.
|
|
1995 CONTINUE
|
|
*
|
|
* Sort the vector D.
|
|
DO 5991 p = 1, N - 1
|
|
q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1
|
|
IF( p.NE.q ) THEN
|
|
TEMP1 = SVA( p )
|
|
SVA( p ) = SVA( q )
|
|
SVA( q ) = TEMP1
|
|
TEMP1 = D( p )
|
|
D( p ) = D( q )
|
|
D( q ) = TEMP1
|
|
CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
|
|
IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
|
|
END IF
|
|
5991 CONTINUE
|
|
*
|
|
RETURN
|
|
* ..
|
|
* .. END OF SGSVJ0
|
|
* ..
|
|
END
|