version 1.1, 2010/01/26 15:22:45
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version 1.10, 2011/11/21 20:43:15
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*> \brief \b ZLALSD |
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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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*> This code makes very mild assumptions about floating point |
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*> arithmetic. It will work on machines with a guard digit in |
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*> add/subtract, or on those binary machines without guard digits |
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*> which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. |
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*> It could conceivably fail on hexadecimal or decimal machines |
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*> without guard digits, but we know of none. |
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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 at least |
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*> (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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*> \date November 2011 |
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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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* ===================================================================== |
SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, |
SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, |
$ RANK, WORK, RWORK, IWORK, INFO ) |
$ RANK, WORK, RWORK, IWORK, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.0) -- |
* -- 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 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
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COMPLEX*16 B( LDB, * ), WORK( * ) |
COMPLEX*16 B( LDB, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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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 |
|
* 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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* 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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* This code makes very mild assumptions about floating point |
|
* arithmetic. It will work on machines with a guard digit in |
|
* add/subtract, or on those binary machines without guard digits |
|
* which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. |
|
* It could conceivably fail on hexadecimal or decimal machines |
|
* without guard digits, but we know of none. |
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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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* = '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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* |
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* SMLSIZ (input) 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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* |
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* N (input) INTEGER |
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* The dimension of the bidiagonal matrix. N >= 0. |
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* |
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* NRHS (input) INTEGER |
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* The number of columns of B. NRHS must be at least 1. |
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* |
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* D (input/output) 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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* |
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* E (input/output) 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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* |
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* B (input/output) 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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* |
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* LDB (input) 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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* |
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* RCOND (input) 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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* |
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* RANK (output) 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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* |
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* WORK (workspace) COMPLEX*16 array, dimension at least |
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* (N * NRHS). |
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* |
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* RWORK (workspace) DOUBLE PRECISION array, dimension at least |
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* (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2), |
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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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* |
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* IWORK (workspace) INTEGER array, dimension at least |
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* (3*N*NLVL + 11*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 = -i, the i-th argument had an illegal value. |
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* > 0: The algorithm failed to compute an 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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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Ming Gu and Ren-Cang Li, Computer Science Division, University of |
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* California at Berkeley, USA |
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* Osni Marques, LBNL/NERSC, USA |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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END IF |
END IF |
* |
* |
* In the real version, B is passed to DLASDQ and multiplied |
* In the real version, B is passed to DLASDQ and multiplied |
* internally by Q'. Here B is complex and that product is |
* internally by Q**H. Here B is complex and that product is |
* computed below in two steps (real and imaginary parts). |
* computed below in two steps (real and imaginary parts). |
* |
* |
J = IRWB - 1 |
J = IRWB - 1 |
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* |
* |
* Since B is complex, the following call to DGEMM is performed |
* Since B is complex, the following call to DGEMM is performed |
* in two steps (real and imaginary parts). That is for V * B |
* in two steps (real and imaginary parts). That is for V * B |
* (in the real version of the code V' is stored in WORK). |
* (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, |
* CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, |
* $ WORK( NWORK ), N ) |
* $ WORK( NWORK ), N ) |
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END IF |
END IF |
* |
* |
* In the real version, B is passed to DLASDQ and multiplied |
* In the real version, B is passed to DLASDQ and multiplied |
* internally by Q'. Here B is complex and that product is |
* internally by Q**H. Here B is complex and that product is |
* computed below in two steps (real and imaginary parts). |
* computed below in two steps (real and imaginary parts). |
* |
* |
J = IRWB - 1 |
J = IRWB - 1 |