744 lines
25 KiB
Fortran
744 lines
25 KiB
Fortran
*> \brief <b> SGELSS solves overdetermined or underdetermined systems for GE matrices</b>
|
|
*
|
|
* =========== DOCUMENTATION ===========
|
|
*
|
|
* Online html documentation available at
|
|
* http://www.netlib.org/lapack/explore-html/
|
|
*
|
|
*> \htmlonly
|
|
*> Download SGELSS + dependencies
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgelss.f">
|
|
*> [TGZ]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgelss.f">
|
|
*> [ZIP]</a>
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgelss.f">
|
|
*> [TXT]</a>
|
|
*> \endhtmlonly
|
|
*
|
|
* Definition:
|
|
* ===========
|
|
*
|
|
* SUBROUTINE SGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
|
|
* WORK, LWORK, INFO )
|
|
*
|
|
* .. Scalar Arguments ..
|
|
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
|
|
* REAL RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
* REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
|
|
* ..
|
|
*
|
|
*
|
|
*> \par Purpose:
|
|
* =============
|
|
*>
|
|
*> \verbatim
|
|
*>
|
|
*> SGELSS computes the minimum norm solution to a real linear least
|
|
*> squares problem:
|
|
*>
|
|
*> Minimize 2-norm(| b - A*x |).
|
|
*>
|
|
*> using the singular value decomposition (SVD) of A. A is an M-by-N
|
|
*> matrix which may be rank-deficient.
|
|
*>
|
|
*> Several right hand side vectors b and solution vectors x can be
|
|
*> handled in a single call; they are stored as the columns of the
|
|
*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix
|
|
*> X.
|
|
*>
|
|
*> The effective rank of A is determined by treating as zero those
|
|
*> singular values which are less than RCOND times the largest singular
|
|
*> value.
|
|
*> \endverbatim
|
|
*
|
|
* Arguments:
|
|
* ==========
|
|
*
|
|
*> \param[in] M
|
|
*> \verbatim
|
|
*> M is INTEGER
|
|
*> The number of rows of the matrix A. M >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] N
|
|
*> \verbatim
|
|
*> N is INTEGER
|
|
*> The number of columns of the matrix A. N >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] NRHS
|
|
*> \verbatim
|
|
*> NRHS is INTEGER
|
|
*> The number of right hand sides, i.e., the number of columns
|
|
*> of the matrices B and X. NRHS >= 0.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] A
|
|
*> \verbatim
|
|
*> A is REAL array, dimension (LDA,N)
|
|
*> On entry, the M-by-N matrix A.
|
|
*> On exit, the first min(m,n) rows of A are overwritten with
|
|
*> its right singular vectors, stored rowwise.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDA
|
|
*> \verbatim
|
|
*> LDA is INTEGER
|
|
*> The leading dimension of the array A. LDA >= max(1,M).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in,out] B
|
|
*> \verbatim
|
|
*> B is REAL array, dimension (LDB,NRHS)
|
|
*> On entry, the M-by-NRHS right hand side matrix B.
|
|
*> On exit, B is overwritten by the N-by-NRHS solution
|
|
*> matrix X. If m >= n and RANK = n, the residual
|
|
*> sum-of-squares for the solution in the i-th column is given
|
|
*> by the sum of squares of elements n+1:m in that column.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LDB
|
|
*> \verbatim
|
|
*> LDB is INTEGER
|
|
*> The leading dimension of the array B. LDB >= max(1,max(M,N)).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] S
|
|
*> \verbatim
|
|
*> S is REAL array, dimension (min(M,N))
|
|
*> The singular values of A in decreasing order.
|
|
*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] RCOND
|
|
*> \verbatim
|
|
*> RCOND is REAL
|
|
*> RCOND is used to determine the effective rank of A.
|
|
*> Singular values S(i) <= RCOND*S(1) are treated as zero.
|
|
*> If RCOND < 0, machine precision is used instead.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] RANK
|
|
*> \verbatim
|
|
*> RANK is INTEGER
|
|
*> The effective rank of A, i.e., the number of singular values
|
|
*> which are greater than RCOND*S(1).
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[out] WORK
|
|
*> \verbatim
|
|
*> WORK is REAL array, dimension (MAX(1,LWORK))
|
|
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
|
*> \endverbatim
|
|
*>
|
|
*> \param[in] LWORK
|
|
*> \verbatim
|
|
*> LWORK is INTEGER
|
|
*> The dimension of the array WORK. LWORK >= 1, and also:
|
|
*> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
|
|
*> For good performance, LWORK should generally be larger.
|
|
*>
|
|
*> If LWORK = -1, then a workspace query is assumed; the routine
|
|
*> only calculates the optimal size of the WORK array, returns
|
|
*> this value as the first entry of the WORK array, and no error
|
|
*> message related to LWORK is issued by XERBLA.
|
|
*> \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 for computing the SVD failed to converge;
|
|
*> if INFO = i, i off-diagonal elements of an intermediate
|
|
*> bidiagonal form did not converge to zero.
|
|
*> \endverbatim
|
|
*
|
|
* Authors:
|
|
* ========
|
|
*
|
|
*> \author Univ. of Tennessee
|
|
*> \author Univ. of California Berkeley
|
|
*> \author Univ. of Colorado Denver
|
|
*> \author NAG Ltd.
|
|
*
|
|
*> \date November 2011
|
|
*
|
|
*> \ingroup realGEsolve
|
|
*
|
|
* =====================================================================
|
|
SUBROUTINE SGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
|
|
$ WORK, LWORK, INFO )
|
|
*
|
|
* -- LAPACK driver 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..--
|
|
* November 2011
|
|
*
|
|
* .. Scalar Arguments ..
|
|
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
|
|
REAL RCOND
|
|
* ..
|
|
* .. Array Arguments ..
|
|
REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
|
|
* ..
|
|
*
|
|
* =====================================================================
|
|
*
|
|
* .. Parameters ..
|
|
REAL ZERO, ONE
|
|
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
|
|
* ..
|
|
* .. Local Scalars ..
|
|
LOGICAL LQUERY
|
|
INTEGER BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL,
|
|
$ ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN,
|
|
$ MAXWRK, MINMN, MINWRK, MM, MNTHR
|
|
INTEGER LWORK_SGEQRF, LWORK_SORMQR, LWORK_SGEBRD,
|
|
$ LWORK_SORMBR, LWORK_SORGBR, LWORK_SORMLQ
|
|
REAL ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
|
|
* ..
|
|
* .. Local Arrays ..
|
|
REAL DUM( 1 )
|
|
* ..
|
|
* .. External Subroutines ..
|
|
EXTERNAL SBDSQR, SCOPY, SGEBRD, SGELQF, SGEMM, SGEMV,
|
|
$ SGEQRF, SLABAD, SLACPY, SLASCL, SLASET, SORGBR,
|
|
$ SORMBR, SORMLQ, SORMQR, SRSCL, XERBLA
|
|
* ..
|
|
* .. External Functions ..
|
|
INTEGER ILAENV
|
|
REAL SLAMCH, SLANGE
|
|
EXTERNAL ILAENV, SLAMCH, SLANGE
|
|
* ..
|
|
* .. Intrinsic Functions ..
|
|
INTRINSIC MAX, MIN
|
|
* ..
|
|
* .. Executable Statements ..
|
|
*
|
|
* Test the input arguments
|
|
*
|
|
INFO = 0
|
|
MINMN = MIN( M, N )
|
|
MAXMN = MAX( M, N )
|
|
LQUERY = ( LWORK.EQ.-1 )
|
|
IF( M.LT.0 ) THEN
|
|
INFO = -1
|
|
ELSE IF( N.LT.0 ) THEN
|
|
INFO = -2
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
|
INFO = -3
|
|
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
|
|
INFO = -5
|
|
ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
|
|
INFO = -7
|
|
END IF
|
|
*
|
|
* Compute workspace
|
|
* (Note: Comments in the code beginning "Workspace:" describe the
|
|
* minimal amount of workspace needed at that point in the code,
|
|
* as well as the preferred amount for good performance.
|
|
* NB refers to the optimal block size for the immediately
|
|
* following subroutine, as returned by ILAENV.)
|
|
*
|
|
IF( INFO.EQ.0 ) THEN
|
|
MINWRK = 1
|
|
MAXWRK = 1
|
|
IF( MINMN.GT.0 ) THEN
|
|
MM = M
|
|
MNTHR = ILAENV( 6, 'SGELSS', ' ', M, N, NRHS, -1 )
|
|
IF( M.GE.N .AND. M.GE.MNTHR ) THEN
|
|
*
|
|
* Path 1a - overdetermined, with many more rows than
|
|
* columns
|
|
*
|
|
* Compute space needed for SGEQRF
|
|
CALL SGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, INFO )
|
|
LWORK_SGEQRF=DUM(1)
|
|
* Compute space needed for SORMQR
|
|
CALL SORMQR( 'L', 'T', M, NRHS, N, A, LDA, DUM(1), B,
|
|
$ LDB, DUM(1), -1, INFO )
|
|
LWORK_SORMQR=DUM(1)
|
|
MM = N
|
|
MAXWRK = MAX( MAXWRK, N + LWORK_SGEQRF )
|
|
MAXWRK = MAX( MAXWRK, N + LWORK_SORMQR )
|
|
END IF
|
|
IF( M.GE.N ) THEN
|
|
*
|
|
* Path 1 - overdetermined or exactly determined
|
|
*
|
|
* Compute workspace needed for SBDSQR
|
|
*
|
|
BDSPAC = MAX( 1, 5*N )
|
|
* Compute space needed for SGEBRD
|
|
CALL SGEBRD( MM, N, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, INFO )
|
|
LWORK_SGEBRD=DUM(1)
|
|
* Compute space needed for SORMBR
|
|
CALL SORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, DUM(1),
|
|
$ B, LDB, DUM(1), -1, INFO )
|
|
LWORK_SORMBR=DUM(1)
|
|
* Compute space needed for SORGBR
|
|
CALL SORGBR( 'P', N, N, N, A, LDA, DUM(1),
|
|
$ DUM(1), -1, INFO )
|
|
LWORK_SORGBR=DUM(1)
|
|
* Compute total workspace needed
|
|
MAXWRK = MAX( MAXWRK, 3*N + LWORK_SGEBRD )
|
|
MAXWRK = MAX( MAXWRK, 3*N + LWORK_SORMBR )
|
|
MAXWRK = MAX( MAXWRK, 3*N + LWORK_SORGBR )
|
|
MAXWRK = MAX( MAXWRK, BDSPAC )
|
|
MAXWRK = MAX( MAXWRK, N*NRHS )
|
|
MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC )
|
|
MAXWRK = MAX( MINWRK, MAXWRK )
|
|
END IF
|
|
IF( N.GT.M ) THEN
|
|
*
|
|
* Compute workspace needed for SBDSQR
|
|
*
|
|
BDSPAC = MAX( 1, 5*M )
|
|
MINWRK = MAX( 3*M+NRHS, 3*M+N, BDSPAC )
|
|
IF( N.GE.MNTHR ) THEN
|
|
*
|
|
* Path 2a - underdetermined, with many more columns
|
|
* than rows
|
|
*
|
|
* Compute space needed for SGEBRD
|
|
CALL SGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, INFO )
|
|
LWORK_SGEBRD=DUM(1)
|
|
* Compute space needed for SORMBR
|
|
CALL SORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
|
|
$ DUM(1), B, LDB, DUM(1), -1, INFO )
|
|
LWORK_SORMBR=DUM(1)
|
|
* Compute space needed for SORGBR
|
|
CALL SORGBR( 'P', M, M, M, A, LDA, DUM(1),
|
|
$ DUM(1), -1, INFO )
|
|
LWORK_SORGBR=DUM(1)
|
|
* Compute space needed for SORMLQ
|
|
CALL SORMLQ( 'L', 'T', N, NRHS, M, A, LDA, DUM(1),
|
|
$ B, LDB, DUM(1), -1, INFO )
|
|
LWORK_SORMLQ=DUM(1)
|
|
* Compute total workspace needed
|
|
MAXWRK = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1,
|
|
$ -1 )
|
|
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_SGEBRD )
|
|
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_SORMBR )
|
|
MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_SORGBR )
|
|
MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC )
|
|
IF( NRHS.GT.1 ) THEN
|
|
MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
|
|
ELSE
|
|
MAXWRK = MAX( MAXWRK, M*M + 2*M )
|
|
END IF
|
|
MAXWRK = MAX( MAXWRK, M + LWORK_SORMLQ )
|
|
ELSE
|
|
*
|
|
* Path 2 - underdetermined
|
|
*
|
|
* Compute space needed for SGEBRD
|
|
CALL SGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
|
|
$ DUM(1), DUM(1), -1, INFO )
|
|
LWORK_SGEBRD=DUM(1)
|
|
* Compute space needed for SORMBR
|
|
CALL SORMBR( 'Q', 'L', 'T', M, NRHS, M, A, LDA,
|
|
$ DUM(1), B, LDB, DUM(1), -1, INFO )
|
|
LWORK_SORMBR=DUM(1)
|
|
* Compute space needed for SORGBR
|
|
CALL SORGBR( 'P', M, N, M, A, LDA, DUM(1),
|
|
$ DUM(1), -1, INFO )
|
|
LWORK_SORGBR=DUM(1)
|
|
MAXWRK = 3*M + LWORK_SGEBRD
|
|
MAXWRK = MAX( MAXWRK, 3*M + LWORK_SORMBR )
|
|
MAXWRK = MAX( MAXWRK, 3*M + LWORK_SORGBR )
|
|
MAXWRK = MAX( MAXWRK, BDSPAC )
|
|
MAXWRK = MAX( MAXWRK, N*NRHS )
|
|
END IF
|
|
END IF
|
|
MAXWRK = MAX( MINWRK, MAXWRK )
|
|
END IF
|
|
WORK( 1 ) = MAXWRK
|
|
*
|
|
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY )
|
|
$ INFO = -12
|
|
END IF
|
|
*
|
|
IF( INFO.NE.0 ) THEN
|
|
CALL XERBLA( 'SGELSS', -INFO )
|
|
RETURN
|
|
ELSE IF( LQUERY ) THEN
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Quick return if possible
|
|
*
|
|
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
|
|
RANK = 0
|
|
RETURN
|
|
END IF
|
|
*
|
|
* Get machine parameters
|
|
*
|
|
EPS = SLAMCH( 'P' )
|
|
SFMIN = SLAMCH( 'S' )
|
|
SMLNUM = SFMIN / EPS
|
|
BIGNUM = ONE / SMLNUM
|
|
CALL SLABAD( SMLNUM, BIGNUM )
|
|
*
|
|
* Scale A if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
ANRM = SLANGE( 'M', M, N, A, LDA, WORK )
|
|
IASCL = 0
|
|
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
|
|
*
|
|
* Scale matrix norm up to SMLNUM
|
|
*
|
|
CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
|
|
IASCL = 1
|
|
ELSE IF( ANRM.GT.BIGNUM ) THEN
|
|
*
|
|
* Scale matrix norm down to BIGNUM
|
|
*
|
|
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
|
|
IASCL = 2
|
|
ELSE IF( ANRM.EQ.ZERO ) THEN
|
|
*
|
|
* Matrix all zero. Return zero solution.
|
|
*
|
|
CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
|
|
CALL SLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
|
|
RANK = 0
|
|
GO TO 70
|
|
END IF
|
|
*
|
|
* Scale B if max element outside range [SMLNUM,BIGNUM]
|
|
*
|
|
BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK )
|
|
IBSCL = 0
|
|
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
|
|
*
|
|
* Scale matrix norm up to SMLNUM
|
|
*
|
|
CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
|
|
IBSCL = 1
|
|
ELSE IF( BNRM.GT.BIGNUM ) THEN
|
|
*
|
|
* Scale matrix norm down to BIGNUM
|
|
*
|
|
CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
|
|
IBSCL = 2
|
|
END IF
|
|
*
|
|
* Overdetermined case
|
|
*
|
|
IF( M.GE.N ) THEN
|
|
*
|
|
* Path 1 - overdetermined or exactly determined
|
|
*
|
|
MM = M
|
|
IF( M.GE.MNTHR ) THEN
|
|
*
|
|
* Path 1a - overdetermined, with many more rows than columns
|
|
*
|
|
MM = N
|
|
ITAU = 1
|
|
IWORK = ITAU + N
|
|
*
|
|
* Compute A=Q*R
|
|
* (Workspace: need 2*N, prefer N+N*NB)
|
|
*
|
|
CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, INFO )
|
|
*
|
|
* Multiply B by transpose(Q)
|
|
* (Workspace: need N+NRHS, prefer N+NRHS*NB)
|
|
*
|
|
CALL SORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
|
|
$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
|
|
*
|
|
* Zero out below R
|
|
*
|
|
IF( N.GT.1 )
|
|
$ CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
|
|
END IF
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
IWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in A
|
|
* (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
|
|
*
|
|
CALL SGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors of R
|
|
* (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
|
|
*
|
|
CALL SORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
|
|
$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
|
|
*
|
|
* Generate right bidiagonalizing vectors of R in A
|
|
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
|
|
*
|
|
CALL SORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
|
|
IWORK = IE + N
|
|
*
|
|
* Perform bidiagonal QR iteration
|
|
* multiply B by transpose of left singular vectors
|
|
* compute right singular vectors in A
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL SBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
|
|
$ 1, B, LDB, WORK( IWORK ), INFO )
|
|
IF( INFO.NE.0 )
|
|
$ GO TO 70
|
|
*
|
|
* Multiply B by reciprocals of singular values
|
|
*
|
|
THR = MAX( RCOND*S( 1 ), SFMIN )
|
|
IF( RCOND.LT.ZERO )
|
|
$ THR = MAX( EPS*S( 1 ), SFMIN )
|
|
RANK = 0
|
|
DO 10 I = 1, N
|
|
IF( S( I ).GT.THR ) THEN
|
|
CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB )
|
|
RANK = RANK + 1
|
|
ELSE
|
|
CALL SLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
|
|
END IF
|
|
10 CONTINUE
|
|
*
|
|
* Multiply B by right singular vectors
|
|
* (Workspace: need N, prefer N*NRHS)
|
|
*
|
|
IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
|
|
CALL SGEMM( 'T', 'N', N, NRHS, N, ONE, A, LDA, B, LDB, ZERO,
|
|
$ WORK, LDB )
|
|
CALL SLACPY( 'G', N, NRHS, WORK, LDB, B, LDB )
|
|
ELSE IF( NRHS.GT.1 ) THEN
|
|
CHUNK = LWORK / N
|
|
DO 20 I = 1, NRHS, CHUNK
|
|
BL = MIN( NRHS-I+1, CHUNK )
|
|
CALL SGEMM( 'T', 'N', N, BL, N, ONE, A, LDA, B( 1, I ),
|
|
$ LDB, ZERO, WORK, N )
|
|
CALL SLACPY( 'G', N, BL, WORK, N, B( 1, I ), LDB )
|
|
20 CONTINUE
|
|
ELSE
|
|
CALL SGEMV( 'T', N, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
|
|
CALL SCOPY( N, WORK, 1, B, 1 )
|
|
END IF
|
|
*
|
|
ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
|
|
$ MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN
|
|
*
|
|
* Path 2a - underdetermined, with many more columns than rows
|
|
* and sufficient workspace for an efficient algorithm
|
|
*
|
|
LDWORK = M
|
|
IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
|
|
$ M*LDA+M+M*NRHS ) )LDWORK = LDA
|
|
ITAU = 1
|
|
IWORK = M + 1
|
|
*
|
|
* Compute A=L*Q
|
|
* (Workspace: need 2*M, prefer M+M*NB)
|
|
*
|
|
CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, INFO )
|
|
IL = IWORK
|
|
*
|
|
* Copy L to WORK(IL), zeroing out above it
|
|
*
|
|
CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
|
|
CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
|
|
$ LDWORK )
|
|
IE = IL + LDWORK*M
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IL)
|
|
* (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
|
|
*
|
|
CALL SGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( IWORK ),
|
|
$ LWORK-IWORK+1, INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors of L
|
|
* (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
|
|
*
|
|
CALL SORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
|
|
$ WORK( ITAUQ ), B, LDB, WORK( IWORK ),
|
|
$ LWORK-IWORK+1, INFO )
|
|
*
|
|
* Generate right bidiagonalizing vectors of R in WORK(IL)
|
|
* (Workspace: need M*M+5*M-1, prefer M*M+4*M+(M-1)*NB)
|
|
*
|
|
CALL SORGBR( 'P', M, M, M, WORK( IL ), LDWORK, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration,
|
|
* computing right singular vectors of L in WORK(IL) and
|
|
* multiplying B by transpose of left singular vectors
|
|
* (Workspace: need M*M+M+BDSPAC)
|
|
*
|
|
CALL SBDSQR( 'U', M, M, 0, NRHS, S, WORK( IE ), WORK( IL ),
|
|
$ LDWORK, A, LDA, B, LDB, WORK( IWORK ), INFO )
|
|
IF( INFO.NE.0 )
|
|
$ GO TO 70
|
|
*
|
|
* Multiply B by reciprocals of singular values
|
|
*
|
|
THR = MAX( RCOND*S( 1 ), SFMIN )
|
|
IF( RCOND.LT.ZERO )
|
|
$ THR = MAX( EPS*S( 1 ), SFMIN )
|
|
RANK = 0
|
|
DO 30 I = 1, M
|
|
IF( S( I ).GT.THR ) THEN
|
|
CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB )
|
|
RANK = RANK + 1
|
|
ELSE
|
|
CALL SLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
|
|
END IF
|
|
30 CONTINUE
|
|
IWORK = IE
|
|
*
|
|
* Multiply B by right singular vectors of L in WORK(IL)
|
|
* (Workspace: need M*M+2*M, prefer M*M+M+M*NRHS)
|
|
*
|
|
IF( LWORK.GE.LDB*NRHS+IWORK-1 .AND. NRHS.GT.1 ) THEN
|
|
CALL SGEMM( 'T', 'N', M, NRHS, M, ONE, WORK( IL ), LDWORK,
|
|
$ B, LDB, ZERO, WORK( IWORK ), LDB )
|
|
CALL SLACPY( 'G', M, NRHS, WORK( IWORK ), LDB, B, LDB )
|
|
ELSE IF( NRHS.GT.1 ) THEN
|
|
CHUNK = ( LWORK-IWORK+1 ) / M
|
|
DO 40 I = 1, NRHS, CHUNK
|
|
BL = MIN( NRHS-I+1, CHUNK )
|
|
CALL SGEMM( 'T', 'N', M, BL, M, ONE, WORK( IL ), LDWORK,
|
|
$ B( 1, I ), LDB, ZERO, WORK( IWORK ), M )
|
|
CALL SLACPY( 'G', M, BL, WORK( IWORK ), M, B( 1, I ),
|
|
$ LDB )
|
|
40 CONTINUE
|
|
ELSE
|
|
CALL SGEMV( 'T', M, M, ONE, WORK( IL ), LDWORK, B( 1, 1 ),
|
|
$ 1, ZERO, WORK( IWORK ), 1 )
|
|
CALL SCOPY( M, WORK( IWORK ), 1, B( 1, 1 ), 1 )
|
|
END IF
|
|
*
|
|
* Zero out below first M rows of B
|
|
*
|
|
CALL SLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
|
|
IWORK = ITAU + M
|
|
*
|
|
* Multiply transpose(Q) by B
|
|
* (Workspace: need M+NRHS, prefer M+NRHS*NB)
|
|
*
|
|
CALL SORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
|
|
$ LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
|
|
*
|
|
ELSE
|
|
*
|
|
* Path 2 - remaining underdetermined cases
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
IWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize A
|
|
* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
|
|
*
|
|
CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
|
|
$ INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors
|
|
* (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
|
|
*
|
|
CALL SORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
|
|
$ B, LDB, WORK( IWORK ), LWORK-IWORK+1, INFO )
|
|
*
|
|
* Generate right bidiagonalizing vectors in A
|
|
* (Workspace: need 4*M, prefer 3*M+M*NB)
|
|
*
|
|
CALL SORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
|
|
$ WORK( IWORK ), LWORK-IWORK+1, INFO )
|
|
IWORK = IE + M
|
|
*
|
|
* Perform bidiagonal QR iteration,
|
|
* computing right singular vectors of A in A and
|
|
* multiplying B by transpose of left singular vectors
|
|
* (Workspace: need BDSPAC)
|
|
*
|
|
CALL SBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
|
|
$ 1, B, LDB, WORK( IWORK ), INFO )
|
|
IF( INFO.NE.0 )
|
|
$ GO TO 70
|
|
*
|
|
* Multiply B by reciprocals of singular values
|
|
*
|
|
THR = MAX( RCOND*S( 1 ), SFMIN )
|
|
IF( RCOND.LT.ZERO )
|
|
$ THR = MAX( EPS*S( 1 ), SFMIN )
|
|
RANK = 0
|
|
DO 50 I = 1, M
|
|
IF( S( I ).GT.THR ) THEN
|
|
CALL SRSCL( NRHS, S( I ), B( I, 1 ), LDB )
|
|
RANK = RANK + 1
|
|
ELSE
|
|
CALL SLASET( 'F', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
|
|
END IF
|
|
50 CONTINUE
|
|
*
|
|
* Multiply B by right singular vectors of A
|
|
* (Workspace: need N, prefer N*NRHS)
|
|
*
|
|
IF( LWORK.GE.LDB*NRHS .AND. NRHS.GT.1 ) THEN
|
|
CALL SGEMM( 'T', 'N', N, NRHS, M, ONE, A, LDA, B, LDB, ZERO,
|
|
$ WORK, LDB )
|
|
CALL SLACPY( 'F', N, NRHS, WORK, LDB, B, LDB )
|
|
ELSE IF( NRHS.GT.1 ) THEN
|
|
CHUNK = LWORK / N
|
|
DO 60 I = 1, NRHS, CHUNK
|
|
BL = MIN( NRHS-I+1, CHUNK )
|
|
CALL SGEMM( 'T', 'N', N, BL, M, ONE, A, LDA, B( 1, I ),
|
|
$ LDB, ZERO, WORK, N )
|
|
CALL SLACPY( 'F', N, BL, WORK, N, B( 1, I ), LDB )
|
|
60 CONTINUE
|
|
ELSE
|
|
CALL SGEMV( 'T', M, N, ONE, A, LDA, B, 1, ZERO, WORK, 1 )
|
|
CALL SCOPY( N, WORK, 1, B, 1 )
|
|
END IF
|
|
END IF
|
|
*
|
|
* Undo scaling
|
|
*
|
|
IF( IASCL.EQ.1 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
|
|
CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ INFO )
|
|
ELSE IF( IASCL.EQ.2 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
|
|
CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ INFO )
|
|
END IF
|
|
IF( IBSCL.EQ.1 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
|
|
ELSE IF( IBSCL.EQ.2 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
|
|
END IF
|
|
*
|
|
70 CONTINUE
|
|
WORK( 1 ) = MAXWRK
|
|
RETURN
|
|
*
|
|
* End of SGELSS
|
|
*
|
|
END
|