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685 lines
22 KiB
685 lines
22 KiB
2 years ago
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*> \brief \b ZLALSD uses the singular value decomposition of A to solve the least squares problem.
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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 ZLALSD + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlalsd.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/zlalsd.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/zlalsd.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 ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
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* RANK, WORK, RWORK, IWORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO
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* INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
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* DOUBLE PRECISION RCOND
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* ..
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* .. Array Arguments ..
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* INTEGER IWORK( * )
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* DOUBLE PRECISION D( * ), E( * ), RWORK( * )
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* COMPLEX*16 B( LDB, * ), 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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*> ZLALSD uses the singular value decomposition of A to solve the least
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*> squares problem of finding X to minimize the Euclidean norm of each
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*> column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
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*> are N-by-NRHS. The solution X overwrites B.
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*>
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*> The singular values of A smaller than RCOND times the largest
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*> singular value are treated as zero in solving the least squares
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*> problem; in this case a minimum norm solution is returned.
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*> The actual singular values are returned in D in ascending order.
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*>
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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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*> = 'U': D and E define an upper bidiagonal matrix.
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*> = 'L': D and E define a lower bidiagonal matrix.
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*> \endverbatim
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*>
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*> \param[in] SMLSIZ
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*> \verbatim
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*> SMLSIZ is INTEGER
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*> The maximum size of the subproblems at the bottom of the
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*> computation tree.
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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 dimension of the bidiagonal matrix. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of columns of B. NRHS must be at least 1.
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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 DOUBLE PRECISION array, dimension (N)
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*> On entry D contains the main diagonal of the bidiagonal
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*> matrix. On exit, if INFO = 0, D contains its singular values.
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*> \endverbatim
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*>
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*> \param[in,out] E
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*> \verbatim
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*> E is DOUBLE PRECISION array, dimension (N-1)
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*> Contains the super-diagonal entries of the bidiagonal matrix.
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*> On exit, E has been destroyed.
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is COMPLEX*16 array, dimension (LDB,NRHS)
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*> On input, B contains the right hand sides of the least
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*> squares problem. On output, B contains the solution X.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of B in the calling subprogram.
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*> LDB must be at least max(1,N).
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*> \endverbatim
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*>
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*> \param[in] RCOND
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*> \verbatim
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*> RCOND is DOUBLE PRECISION
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*> The singular values of A less than or equal to RCOND times
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*> the largest singular value are treated as zero in solving
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*> the least squares problem. If RCOND is negative,
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*> machine precision is used instead.
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*> For example, if diag(S)*X=B were the least squares problem,
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*> where diag(S) is a diagonal matrix of singular values, the
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*> solution would be X(i) = B(i) / S(i) if S(i) is greater than
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*> RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
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*> RCOND*max(S).
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*> \endverbatim
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*>
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*> \param[out] RANK
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*> \verbatim
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*> RANK is INTEGER
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*> The number of singular values of A greater than RCOND times
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*> the largest singular value.
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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 COMPLEX*16 array, dimension (N * NRHS)
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*> \endverbatim
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*>
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*> \param[out] RWORK
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*> \verbatim
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*> RWORK is DOUBLE PRECISION array, dimension at least
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*> (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
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*> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ),
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*> where
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*> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension at least
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*> (3*N*NLVL + 11*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 = -i, the i-th argument had an illegal value.
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*> > 0: The algorithm failed to compute a singular value while
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*> working on the submatrix lying in rows and columns
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*> INFO/(N+1) through MOD(INFO,N+1).
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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 complex16OTHERcomputational
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*
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*> \par Contributors:
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* ==================
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*>
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*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
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*> California at Berkeley, USA \n
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*> Osni Marques, LBNL/NERSC, USA \n
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*
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* =====================================================================
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SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
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$ RANK, WORK, RWORK, IWORK, 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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CHARACTER UPLO
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INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
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DOUBLE PRECISION RCOND
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* ..
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* .. Array Arguments ..
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INTEGER IWORK( * )
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DOUBLE PRECISION D( * ), E( * ), RWORK( * )
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COMPLEX*16 B( LDB, * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE, TWO
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
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COMPLEX*16 CZERO
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PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ) )
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* ..
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* .. Local Scalars ..
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INTEGER BX, BXST, C, DIFL, DIFR, GIVCOL, GIVNUM,
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$ GIVPTR, I, ICMPQ1, ICMPQ2, IRWB, IRWIB, IRWRB,
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$ IRWU, IRWVT, IRWWRK, IWK, J, JCOL, JIMAG,
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$ JREAL, JROW, K, NLVL, NM1, NRWORK, NSIZE, NSUB,
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$ PERM, POLES, S, SIZEI, SMLSZP, SQRE, ST, ST1,
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$ U, VT, Z
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DOUBLE PRECISION CS, EPS, ORGNRM, RCND, R, SN, TOL
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* ..
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* .. External Functions ..
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INTEGER IDAMAX
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DOUBLE PRECISION DLAMCH, DLANST
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EXTERNAL IDAMAX, DLAMCH, DLANST
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEMM, DLARTG, DLASCL, DLASDA, DLASDQ, DLASET,
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$ DLASRT, XERBLA, ZCOPY, ZDROT, ZLACPY, ZLALSA,
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$ ZLASCL, ZLASET
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, LOG, SIGN
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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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INFO = 0
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*
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IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( NRHS.LT.1 ) THEN
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INFO = -4
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ELSE IF( ( LDB.LT.1 ) .OR. ( LDB.LT.N ) ) THEN
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INFO = -8
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZLALSD', -INFO )
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RETURN
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END IF
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*
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EPS = DLAMCH( 'Epsilon' )
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*
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* Set up the tolerance.
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*
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IF( ( RCOND.LE.ZERO ) .OR. ( RCOND.GE.ONE ) ) THEN
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RCND = EPS
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ELSE
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RCND = RCOND
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END IF
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*
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RANK = 0
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*
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* Quick return if possible.
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*
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IF( N.EQ.0 ) THEN
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RETURN
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ELSE IF( N.EQ.1 ) THEN
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IF( D( 1 ).EQ.ZERO ) THEN
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CALL ZLASET( 'A', 1, NRHS, CZERO, CZERO, B, LDB )
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ELSE
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RANK = 1
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CALL ZLASCL( 'G', 0, 0, D( 1 ), ONE, 1, NRHS, B, LDB, INFO )
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D( 1 ) = ABS( D( 1 ) )
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END IF
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RETURN
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END IF
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*
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* Rotate the matrix if it is lower bidiagonal.
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*
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IF( UPLO.EQ.'L' ) THEN
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DO 10 I = 1, N - 1
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CALL DLARTG( D( I ), E( I ), CS, SN, R )
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D( I ) = R
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E( I ) = SN*D( I+1 )
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D( I+1 ) = CS*D( I+1 )
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IF( NRHS.EQ.1 ) THEN
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CALL ZDROT( 1, B( I, 1 ), 1, B( I+1, 1 ), 1, CS, SN )
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ELSE
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RWORK( I*2-1 ) = CS
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RWORK( I*2 ) = SN
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END IF
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10 CONTINUE
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IF( NRHS.GT.1 ) THEN
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DO 30 I = 1, NRHS
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DO 20 J = 1, N - 1
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CS = RWORK( J*2-1 )
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SN = RWORK( J*2 )
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CALL ZDROT( 1, B( J, I ), 1, B( J+1, I ), 1, CS, SN )
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20 CONTINUE
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30 CONTINUE
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END IF
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END IF
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*
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* Scale.
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*
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NM1 = N - 1
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ORGNRM = DLANST( 'M', N, D, E )
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IF( ORGNRM.EQ.ZERO ) THEN
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CALL ZLASET( 'A', N, NRHS, CZERO, CZERO, B, LDB )
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RETURN
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END IF
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*
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CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
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CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, INFO )
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*
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* If N is smaller than the minimum divide size SMLSIZ, then solve
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* the problem with another solver.
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*
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IF( N.LE.SMLSIZ ) THEN
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IRWU = 1
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IRWVT = IRWU + N*N
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IRWWRK = IRWVT + N*N
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IRWRB = IRWWRK
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IRWIB = IRWRB + N*NRHS
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IRWB = IRWIB + N*NRHS
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CALL DLASET( 'A', N, N, ZERO, ONE, RWORK( IRWU ), N )
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CALL DLASET( 'A', N, N, ZERO, ONE, RWORK( IRWVT ), N )
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CALL DLASDQ( 'U', 0, N, N, N, 0, D, E, RWORK( IRWVT ), N,
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$ RWORK( IRWU ), N, RWORK( IRWWRK ), 1,
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$ RWORK( IRWWRK ), INFO )
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IF( INFO.NE.0 ) THEN
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RETURN
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END IF
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*
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* In the real version, B is passed to DLASDQ and multiplied
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* internally by Q**H. Here B is complex and that product is
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* computed below in two steps (real and imaginary parts).
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*
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J = IRWB - 1
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DO 50 JCOL = 1, NRHS
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DO 40 JROW = 1, N
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J = J + 1
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RWORK( J ) = DBLE( B( JROW, JCOL ) )
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40 CONTINUE
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50 CONTINUE
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CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, RWORK( IRWU ), N,
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$ RWORK( IRWB ), N, ZERO, RWORK( IRWRB ), N )
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J = IRWB - 1
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DO 70 JCOL = 1, NRHS
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DO 60 JROW = 1, N
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J = J + 1
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RWORK( J ) = DIMAG( B( JROW, JCOL ) )
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60 CONTINUE
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70 CONTINUE
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CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, RWORK( IRWU ), N,
|
||
|
|
$ RWORK( IRWB ), N, ZERO, RWORK( IRWIB ), N )
|
||
|
|
JREAL = IRWRB - 1
|
||
|
|
JIMAG = IRWIB - 1
|
||
|
|
DO 90 JCOL = 1, NRHS
|
||
|
|
DO 80 JROW = 1, N
|
||
|
|
JREAL = JREAL + 1
|
||
|
|
JIMAG = JIMAG + 1
|
||
|
|
B( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
|
||
|
|
$ RWORK( JIMAG ) )
|
||
|
|
80 CONTINUE
|
||
|
|
90 CONTINUE
|
||
|
|
*
|
||
|
|
TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
|
||
|
|
DO 100 I = 1, N
|
||
|
|
IF( D( I ).LE.TOL ) THEN
|
||
|
|
CALL ZLASET( 'A', 1, NRHS, CZERO, CZERO, B( I, 1 ), LDB )
|
||
|
|
ELSE
|
||
|
|
CALL ZLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, B( I, 1 ),
|
||
|
|
$ LDB, INFO )
|
||
|
|
RANK = RANK + 1
|
||
|
|
END IF
|
||
|
|
100 CONTINUE
|
||
|
|
*
|
||
|
|
* Since B is complex, the following call to DGEMM is performed
|
||
|
|
* in two steps (real and imaginary parts). That is for V * B
|
||
|
|
* (in the real version of the code V**H is stored in WORK).
|
||
|
|
*
|
||
|
|
* CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO,
|
||
|
|
* $ WORK( NWORK ), N )
|
||
|
|
*
|
||
|
|
J = IRWB - 1
|
||
|
|
DO 120 JCOL = 1, NRHS
|
||
|
|
DO 110 JROW = 1, N
|
||
|
|
J = J + 1
|
||
|
|
RWORK( J ) = DBLE( B( JROW, JCOL ) )
|
||
|
|
110 CONTINUE
|
||
|
|
120 CONTINUE
|
||
|
|
CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, RWORK( IRWVT ), N,
|
||
|
|
$ RWORK( IRWB ), N, ZERO, RWORK( IRWRB ), N )
|
||
|
|
J = IRWB - 1
|
||
|
|
DO 140 JCOL = 1, NRHS
|
||
|
|
DO 130 JROW = 1, N
|
||
|
|
J = J + 1
|
||
|
|
RWORK( J ) = DIMAG( B( JROW, JCOL ) )
|
||
|
|
130 CONTINUE
|
||
|
|
140 CONTINUE
|
||
|
|
CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, RWORK( IRWVT ), N,
|
||
|
|
$ RWORK( IRWB ), N, ZERO, RWORK( IRWIB ), N )
|
||
|
|
JREAL = IRWRB - 1
|
||
|
|
JIMAG = IRWIB - 1
|
||
|
|
DO 160 JCOL = 1, NRHS
|
||
|
|
DO 150 JROW = 1, N
|
||
|
|
JREAL = JREAL + 1
|
||
|
|
JIMAG = JIMAG + 1
|
||
|
|
B( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
|
||
|
|
$ RWORK( JIMAG ) )
|
||
|
|
150 CONTINUE
|
||
|
|
160 CONTINUE
|
||
|
|
*
|
||
|
|
* Unscale.
|
||
|
|
*
|
||
|
|
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
|
||
|
|
CALL DLASRT( 'D', N, D, INFO )
|
||
|
|
CALL ZLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Book-keeping and setting up some constants.
|
||
|
|
*
|
||
|
|
NLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
|
||
|
|
*
|
||
|
|
SMLSZP = SMLSIZ + 1
|
||
|
|
*
|
||
|
|
U = 1
|
||
|
|
VT = 1 + SMLSIZ*N
|
||
|
|
DIFL = VT + SMLSZP*N
|
||
|
|
DIFR = DIFL + NLVL*N
|
||
|
|
Z = DIFR + NLVL*N*2
|
||
|
|
C = Z + NLVL*N
|
||
|
|
S = C + N
|
||
|
|
POLES = S + N
|
||
|
|
GIVNUM = POLES + 2*NLVL*N
|
||
|
|
NRWORK = GIVNUM + 2*NLVL*N
|
||
|
|
BX = 1
|
||
|
|
*
|
||
|
|
IRWRB = NRWORK
|
||
|
|
IRWIB = IRWRB + SMLSIZ*NRHS
|
||
|
|
IRWB = IRWIB + SMLSIZ*NRHS
|
||
|
|
*
|
||
|
|
SIZEI = 1 + N
|
||
|
|
K = SIZEI + N
|
||
|
|
GIVPTR = K + N
|
||
|
|
PERM = GIVPTR + N
|
||
|
|
GIVCOL = PERM + NLVL*N
|
||
|
|
IWK = GIVCOL + NLVL*N*2
|
||
|
|
*
|
||
|
|
ST = 1
|
||
|
|
SQRE = 0
|
||
|
|
ICMPQ1 = 1
|
||
|
|
ICMPQ2 = 0
|
||
|
|
NSUB = 0
|
||
|
|
*
|
||
|
|
DO 170 I = 1, N
|
||
|
|
IF( ABS( D( I ) ).LT.EPS ) THEN
|
||
|
|
D( I ) = SIGN( EPS, D( I ) )
|
||
|
|
END IF
|
||
|
|
170 CONTINUE
|
||
|
|
*
|
||
|
|
DO 240 I = 1, NM1
|
||
|
|
IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
|
||
|
|
NSUB = NSUB + 1
|
||
|
|
IWORK( NSUB ) = ST
|
||
|
|
*
|
||
|
|
* Subproblem found. First determine its size and then
|
||
|
|
* apply divide and conquer on it.
|
||
|
|
*
|
||
|
|
IF( I.LT.NM1 ) THEN
|
||
|
|
*
|
||
|
|
* A subproblem with E(I) small for I < NM1.
|
||
|
|
*
|
||
|
|
NSIZE = I - ST + 1
|
||
|
|
IWORK( SIZEI+NSUB-1 ) = NSIZE
|
||
|
|
ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
|
||
|
|
*
|
||
|
|
* A subproblem with E(NM1) not too small but I = NM1.
|
||
|
|
*
|
||
|
|
NSIZE = N - ST + 1
|
||
|
|
IWORK( SIZEI+NSUB-1 ) = NSIZE
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* A subproblem with E(NM1) small. This implies an
|
||
|
|
* 1-by-1 subproblem at D(N), which is not solved
|
||
|
|
* explicitly.
|
||
|
|
*
|
||
|
|
NSIZE = I - ST + 1
|
||
|
|
IWORK( SIZEI+NSUB-1 ) = NSIZE
|
||
|
|
NSUB = NSUB + 1
|
||
|
|
IWORK( NSUB ) = N
|
||
|
|
IWORK( SIZEI+NSUB-1 ) = 1
|
||
|
|
CALL ZCOPY( NRHS, B( N, 1 ), LDB, WORK( BX+NM1 ), N )
|
||
|
|
END IF
|
||
|
|
ST1 = ST - 1
|
||
|
|
IF( NSIZE.EQ.1 ) THEN
|
||
|
|
*
|
||
|
|
* This is a 1-by-1 subproblem and is not solved
|
||
|
|
* explicitly.
|
||
|
|
*
|
||
|
|
CALL ZCOPY( NRHS, B( ST, 1 ), LDB, WORK( BX+ST1 ), N )
|
||
|
|
ELSE IF( NSIZE.LE.SMLSIZ ) THEN
|
||
|
|
*
|
||
|
|
* This is a small subproblem and is solved by DLASDQ.
|
||
|
|
*
|
||
|
|
CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE,
|
||
|
|
$ RWORK( VT+ST1 ), N )
|
||
|
|
CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE,
|
||
|
|
$ RWORK( U+ST1 ), N )
|
||
|
|
CALL DLASDQ( 'U', 0, NSIZE, NSIZE, NSIZE, 0, D( ST ),
|
||
|
|
$ E( ST ), RWORK( VT+ST1 ), N, RWORK( U+ST1 ),
|
||
|
|
$ N, RWORK( NRWORK ), 1, RWORK( NRWORK ),
|
||
|
|
$ INFO )
|
||
|
|
IF( INFO.NE.0 ) THEN
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* In the real version, B is passed to DLASDQ and multiplied
|
||
|
|
* internally by Q**H. Here B is complex and that product is
|
||
|
|
* computed below in two steps (real and imaginary parts).
|
||
|
|
*
|
||
|
|
J = IRWB - 1
|
||
|
|
DO 190 JCOL = 1, NRHS
|
||
|
|
DO 180 JROW = ST, ST + NSIZE - 1
|
||
|
|
J = J + 1
|
||
|
|
RWORK( J ) = DBLE( B( JROW, JCOL ) )
|
||
|
|
180 CONTINUE
|
||
|
|
190 CONTINUE
|
||
|
|
CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
|
||
|
|
$ RWORK( U+ST1 ), N, RWORK( IRWB ), NSIZE,
|
||
|
|
$ ZERO, RWORK( IRWRB ), NSIZE )
|
||
|
|
J = IRWB - 1
|
||
|
|
DO 210 JCOL = 1, NRHS
|
||
|
|
DO 200 JROW = ST, ST + NSIZE - 1
|
||
|
|
J = J + 1
|
||
|
|
RWORK( J ) = DIMAG( B( JROW, JCOL ) )
|
||
|
|
200 CONTINUE
|
||
|
|
210 CONTINUE
|
||
|
|
CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
|
||
|
|
$ RWORK( U+ST1 ), N, RWORK( IRWB ), NSIZE,
|
||
|
|
$ ZERO, RWORK( IRWIB ), NSIZE )
|
||
|
|
JREAL = IRWRB - 1
|
||
|
|
JIMAG = IRWIB - 1
|
||
|
|
DO 230 JCOL = 1, NRHS
|
||
|
|
DO 220 JROW = ST, ST + NSIZE - 1
|
||
|
|
JREAL = JREAL + 1
|
||
|
|
JIMAG = JIMAG + 1
|
||
|
|
B( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
|
||
|
|
$ RWORK( JIMAG ) )
|
||
|
|
220 CONTINUE
|
||
|
|
230 CONTINUE
|
||
|
|
*
|
||
|
|
CALL ZLACPY( 'A', NSIZE, NRHS, B( ST, 1 ), LDB,
|
||
|
|
$ WORK( BX+ST1 ), N )
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* A large problem. Solve it using divide and conquer.
|
||
|
|
*
|
||
|
|
CALL DLASDA( ICMPQ1, SMLSIZ, NSIZE, SQRE, D( ST ),
|
||
|
|
$ E( ST ), RWORK( U+ST1 ), N, RWORK( VT+ST1 ),
|
||
|
|
$ IWORK( K+ST1 ), RWORK( DIFL+ST1 ),
|
||
|
|
$ RWORK( DIFR+ST1 ), RWORK( Z+ST1 ),
|
||
|
|
$ RWORK( POLES+ST1 ), IWORK( GIVPTR+ST1 ),
|
||
|
|
$ IWORK( GIVCOL+ST1 ), N, IWORK( PERM+ST1 ),
|
||
|
|
$ RWORK( GIVNUM+ST1 ), RWORK( C+ST1 ),
|
||
|
|
$ RWORK( S+ST1 ), RWORK( NRWORK ),
|
||
|
|
$ IWORK( IWK ), INFO )
|
||
|
|
IF( INFO.NE.0 ) THEN
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
BXST = BX + ST1
|
||
|
|
CALL ZLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, B( ST, 1 ),
|
||
|
|
$ LDB, WORK( BXST ), N, RWORK( U+ST1 ), N,
|
||
|
|
$ RWORK( VT+ST1 ), IWORK( K+ST1 ),
|
||
|
|
$ RWORK( DIFL+ST1 ), RWORK( DIFR+ST1 ),
|
||
|
|
$ RWORK( Z+ST1 ), RWORK( POLES+ST1 ),
|
||
|
|
$ IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
|
||
|
|
$ IWORK( PERM+ST1 ), RWORK( GIVNUM+ST1 ),
|
||
|
|
$ RWORK( C+ST1 ), RWORK( S+ST1 ),
|
||
|
|
$ RWORK( NRWORK ), IWORK( IWK ), INFO )
|
||
|
|
IF( INFO.NE.0 ) THEN
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
ST = I + 1
|
||
|
|
END IF
|
||
|
|
240 CONTINUE
|
||
|
|
*
|
||
|
|
* Apply the singular values and treat the tiny ones as zero.
|
||
|
|
*
|
||
|
|
TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
|
||
|
|
*
|
||
|
|
DO 250 I = 1, N
|
||
|
|
*
|
||
|
|
* Some of the elements in D can be negative because 1-by-1
|
||
|
|
* subproblems were not solved explicitly.
|
||
|
|
*
|
||
|
|
IF( ABS( D( I ) ).LE.TOL ) THEN
|
||
|
|
CALL ZLASET( 'A', 1, NRHS, CZERO, CZERO, WORK( BX+I-1 ), N )
|
||
|
|
ELSE
|
||
|
|
RANK = RANK + 1
|
||
|
|
CALL ZLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS,
|
||
|
|
$ WORK( BX+I-1 ), N, INFO )
|
||
|
|
END IF
|
||
|
|
D( I ) = ABS( D( I ) )
|
||
|
|
250 CONTINUE
|
||
|
|
*
|
||
|
|
* Now apply back the right singular vectors.
|
||
|
|
*
|
||
|
|
ICMPQ2 = 1
|
||
|
|
DO 320 I = 1, NSUB
|
||
|
|
ST = IWORK( I )
|
||
|
|
ST1 = ST - 1
|
||
|
|
NSIZE = IWORK( SIZEI+I-1 )
|
||
|
|
BXST = BX + ST1
|
||
|
|
IF( NSIZE.EQ.1 ) THEN
|
||
|
|
CALL ZCOPY( NRHS, WORK( BXST ), N, B( ST, 1 ), LDB )
|
||
|
|
ELSE IF( NSIZE.LE.SMLSIZ ) THEN
|
||
|
|
*
|
||
|
|
* Since B and BX are complex, the following call to DGEMM
|
||
|
|
* is performed in two steps (real and imaginary parts).
|
||
|
|
*
|
||
|
|
* CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
|
||
|
|
* $ RWORK( VT+ST1 ), N, RWORK( BXST ), N, ZERO,
|
||
|
|
* $ B( ST, 1 ), LDB )
|
||
|
|
*
|
||
|
|
J = BXST - N - 1
|
||
|
|
JREAL = IRWB - 1
|
||
|
|
DO 270 JCOL = 1, NRHS
|
||
|
|
J = J + N
|
||
|
|
DO 260 JROW = 1, NSIZE
|
||
|
|
JREAL = JREAL + 1
|
||
|
|
RWORK( JREAL ) = DBLE( WORK( J+JROW ) )
|
||
|
|
260 CONTINUE
|
||
|
|
270 CONTINUE
|
||
|
|
CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
|
||
|
|
$ RWORK( VT+ST1 ), N, RWORK( IRWB ), NSIZE, ZERO,
|
||
|
|
$ RWORK( IRWRB ), NSIZE )
|
||
|
|
J = BXST - N - 1
|
||
|
|
JIMAG = IRWB - 1
|
||
|
|
DO 290 JCOL = 1, NRHS
|
||
|
|
J = J + N
|
||
|
|
DO 280 JROW = 1, NSIZE
|
||
|
|
JIMAG = JIMAG + 1
|
||
|
|
RWORK( JIMAG ) = DIMAG( WORK( J+JROW ) )
|
||
|
|
280 CONTINUE
|
||
|
|
290 CONTINUE
|
||
|
|
CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
|
||
|
|
$ RWORK( VT+ST1 ), N, RWORK( IRWB ), NSIZE, ZERO,
|
||
|
|
$ RWORK( IRWIB ), NSIZE )
|
||
|
|
JREAL = IRWRB - 1
|
||
|
|
JIMAG = IRWIB - 1
|
||
|
|
DO 310 JCOL = 1, NRHS
|
||
|
|
DO 300 JROW = ST, ST + NSIZE - 1
|
||
|
|
JREAL = JREAL + 1
|
||
|
|
JIMAG = JIMAG + 1
|
||
|
|
B( JROW, JCOL ) = DCMPLX( RWORK( JREAL ),
|
||
|
|
$ RWORK( JIMAG ) )
|
||
|
|
300 CONTINUE
|
||
|
|
310 CONTINUE
|
||
|
|
ELSE
|
||
|
|
CALL ZLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, WORK( BXST ), N,
|
||
|
|
$ B( ST, 1 ), LDB, RWORK( U+ST1 ), N,
|
||
|
|
$ RWORK( VT+ST1 ), IWORK( K+ST1 ),
|
||
|
|
$ RWORK( DIFL+ST1 ), RWORK( DIFR+ST1 ),
|
||
|
|
$ RWORK( Z+ST1 ), RWORK( POLES+ST1 ),
|
||
|
|
$ IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
|
||
|
|
$ IWORK( PERM+ST1 ), RWORK( GIVNUM+ST1 ),
|
||
|
|
$ RWORK( C+ST1 ), RWORK( S+ST1 ),
|
||
|
|
$ RWORK( NRWORK ), IWORK( IWK ), INFO )
|
||
|
|
IF( INFO.NE.0 ) THEN
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
320 CONTINUE
|
||
|
|
*
|
||
|
|
* Unscale and sort the singular values.
|
||
|
|
*
|
||
|
|
CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
|
||
|
|
CALL DLASRT( 'D', N, D, INFO )
|
||
|
|
CALL ZLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of ZLALSD
|
||
|
|
*
|
||
|
|
END
|