--- rpl/lapack/lapack/zlalsd.f 2011/07/22 07:38:17 1.9 +++ rpl/lapack/lapack/zlalsd.f 2011/11/21 20:43:15 1.10 @@ -1,10 +1,197 @@ +*> \brief \b ZLALSD +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZLALSD + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, +* RANK, WORK, RWORK, IWORK, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER UPLO +* INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ +* DOUBLE PRECISION RCOND +* .. +* .. Array Arguments .. +* INTEGER IWORK( * ) +* DOUBLE PRECISION D( * ), E( * ), RWORK( * ) +* COMPLEX*16 B( LDB, * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> ZLALSD uses the singular value decomposition of A to solve the least +*> 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 +*> are N-by-NRHS. The solution X overwrites B. +*> +*> The singular values of A smaller than RCOND times the largest +*> singular value are treated as zero in solving the least squares +*> problem; in this case a minimum norm solution is returned. +*> The actual singular values are returned in D in ascending order. +*> +*> 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. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] UPLO +*> \verbatim +*> UPLO is CHARACTER*1 +*> = 'U': D and E define an upper bidiagonal matrix. +*> = 'L': D and E define a lower bidiagonal matrix. +*> \endverbatim +*> +*> \param[in] SMLSIZ +*> \verbatim +*> SMLSIZ is INTEGER +*> The maximum size of the subproblems at the bottom of the +*> computation tree. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The dimension of the bidiagonal matrix. N >= 0. +*> \endverbatim +*> +*> \param[in] NRHS +*> \verbatim +*> NRHS is INTEGER +*> The number of columns of B. NRHS must be at least 1. +*> \endverbatim +*> +*> \param[in,out] D +*> \verbatim +*> D is DOUBLE PRECISION array, dimension (N) +*> On entry D contains the main diagonal of the bidiagonal +*> matrix. On exit, if INFO = 0, D contains its singular values. +*> \endverbatim +*> +*> \param[in,out] E +*> \verbatim +*> E is DOUBLE PRECISION array, dimension (N-1) +*> Contains the super-diagonal entries of the bidiagonal matrix. +*> On exit, E has been destroyed. +*> \endverbatim +*> +*> \param[in,out] B +*> \verbatim +*> B is COMPLEX*16 array, dimension (LDB,NRHS) +*> On input, B contains the right hand sides of the least +*> squares problem. On output, B contains the solution X. +*> \endverbatim +*> +*> \param[in] LDB +*> \verbatim +*> LDB is INTEGER +*> The leading dimension of B in the calling subprogram. +*> LDB must be at least max(1,N). +*> \endverbatim +*> +*> \param[in] RCOND +*> \verbatim +*> RCOND is DOUBLE PRECISION +*> The singular values of A less than or equal to RCOND times +*> the largest singular value are treated as zero in solving +*> the least squares problem. If RCOND is negative, +*> machine precision is used instead. +*> For example, if diag(S)*X=B were the least squares problem, +*> where diag(S) is a diagonal matrix of singular values, the +*> solution would be X(i) = B(i) / S(i) if S(i) is greater than +*> RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to +*> RCOND*max(S). +*> \endverbatim +*> +*> \param[out] RANK +*> \verbatim +*> RANK is INTEGER +*> The number of singular values of A greater than RCOND times +*> the largest singular value. +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX*16 array, dimension at least +*> (N * NRHS). +*> \endverbatim +*> +*> \param[out] RWORK +*> \verbatim +*> RWORK is DOUBLE PRECISION array, dimension at least +*> (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + +*> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ), +*> where +*> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) +*> \endverbatim +*> +*> \param[out] IWORK +*> \verbatim +*> IWORK is INTEGER array, dimension at least +*> (3*N*NLVL + 11*N). +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit. +*> < 0: if INFO = -i, the i-th argument had an illegal value. +*> > 0: The algorithm failed to compute a singular value while +*> working on the submatrix lying in rows and columns +*> INFO/(N+1) through MOD(INFO,N+1). +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2011 +* +*> \ingroup complex16OTHERcomputational +* +*> \par Contributors: +* ================== +*> +*> Ming Gu and Ren-Cang Li, Computer Science Division, University of +*> California at Berkeley, USA \n +*> Osni Marques, LBNL/NERSC, USA \n +* +* ===================================================================== SUBROUTINE ZLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, $ RANK, WORK, RWORK, IWORK, INFO ) * -* -- LAPACK routine (version 3.3.1) -- +* -- LAPACK computational routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* -- April 2011 -- +* November 2011 * * .. Scalar Arguments .. CHARACTER UPLO @@ -17,101 +204,6 @@ COMPLEX*16 B( LDB, * ), WORK( * ) * .. * -* Purpose -* ======= -* -* ZLALSD uses the singular value decomposition of A to solve the least -* 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 -* are N-by-NRHS. The solution X overwrites B. -* -* The singular values of A smaller than RCOND times the largest -* singular value are treated as zero in solving the least squares -* problem; in this case a minimum norm solution is returned. -* The actual singular values are returned in D in ascending order. -* -* 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. -* -* Arguments -* ========= -* -* UPLO (input) CHARACTER*1 -* = 'U': D and E define an upper bidiagonal matrix. -* = 'L': D and E define a lower bidiagonal matrix. -* -* SMLSIZ (input) INTEGER -* The maximum size of the subproblems at the bottom of the -* computation tree. -* -* N (input) INTEGER -* The dimension of the bidiagonal matrix. N >= 0. -* -* NRHS (input) INTEGER -* The number of columns of B. NRHS must be at least 1. -* -* D (input/output) DOUBLE PRECISION array, dimension (N) -* On entry D contains the main diagonal of the bidiagonal -* matrix. On exit, if INFO = 0, D contains its singular values. -* -* E (input/output) DOUBLE PRECISION array, dimension (N-1) -* Contains the super-diagonal entries of the bidiagonal matrix. -* On exit, E has been destroyed. -* -* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) -* On input, B contains the right hand sides of the least -* squares problem. On output, B contains the solution X. -* -* LDB (input) INTEGER -* The leading dimension of B in the calling subprogram. -* LDB must be at least max(1,N). -* -* RCOND (input) DOUBLE PRECISION -* The singular values of A less than or equal to RCOND times -* the largest singular value are treated as zero in solving -* the least squares problem. If RCOND is negative, -* machine precision is used instead. -* For example, if diag(S)*X=B were the least squares problem, -* where diag(S) is a diagonal matrix of singular values, the -* solution would be X(i) = B(i) / S(i) if S(i) is greater than -* RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to -* RCOND*max(S). -* -* RANK (output) INTEGER -* The number of singular values of A greater than RCOND times -* the largest singular value. -* -* WORK (workspace) COMPLEX*16 array, dimension at least -* (N * NRHS). -* -* RWORK (workspace) DOUBLE PRECISION array, dimension at least -* (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + -* MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ), -* where -* NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) -* -* IWORK (workspace) INTEGER array, dimension at least -* (3*N*NLVL + 11*N). -* -* INFO (output) INTEGER -* = 0: successful exit. -* < 0: if INFO = -i, the i-th argument had an illegal value. -* > 0: The algorithm failed to compute a singular value while -* working on the submatrix lying in rows and columns -* INFO/(N+1) through MOD(INFO,N+1). -* -* Further Details -* =============== -* -* Based on contributions by -* Ming Gu and Ren-Cang Li, Computer Science Division, University of -* California at Berkeley, USA -* Osni Marques, LBNL/NERSC, USA -* * ===================================================================== * * .. Parameters ..