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OpenBLAS/lapack-netlib/SRC/sgesvdx.f

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*> \brief <b> SGESVDX computes the singular value decomposition (SVD) for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGESVDX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgesvdx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgesvdx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgesvdx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGESVDX( JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU,
* $ IL, IU, NS, S, U, LDU, VT, LDVT, WORK,
* $ LWORK, IWORK, INFO )
*
*
* .. Scalar Arguments ..
* CHARACTER JOBU, JOBVT, RANGE
* INTEGER IL, INFO, IU, LDA, LDU, LDVT, LWORK, M, N, NS
* REAL VL, VU
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
* REAL A( LDA, * ), S( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGESVDX computes the singular value decomposition (SVD) of a real
*> M-by-N matrix A, optionally computing the left and/or right singular
*> vectors. The SVD is written
*>
*> A = U * SIGMA * transpose(V)
*>
*> where SIGMA is an M-by-N matrix which is zero except for its
*> min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
*> V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA
*> are the singular values of A; they are real and non-negative, and
*> are returned in descending order. The first min(m,n) columns of
*> U and V are the left and right singular vectors of A.
*>
*> SGESVDX uses an eigenvalue problem for obtaining the SVD, which
*> allows for the computation of a subset of singular values and
*> vectors. See SBDSVDX for details.
*>
*> Note that the routine returns V**T, not V.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> Specifies options for computing all or part of the matrix U:
*> = 'V': the first min(m,n) columns of U (the left singular
*> vectors) or as specified by RANGE are returned in
*> the array U;
*> = 'N': no columns of U (no left singular vectors) are
*> computed.
*> \endverbatim
*>
*> \param[in] JOBVT
*> \verbatim
*> JOBVT is CHARACTER*1
*> Specifies options for computing all or part of the matrix
*> V**T:
*> = 'V': the first min(m,n) rows of V**T (the right singular
*> vectors) or as specified by RANGE are returned in
*> the array VT;
*> = 'N': no rows of V**T (no right singular vectors) are
*> computed.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*> RANGE is CHARACTER*1
*> = 'A': all singular values will be found.
*> = 'V': all singular values in the half-open interval (VL,VU]
*> will be found.
*> = 'I': the IL-th through IU-th singular values will be found.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the input matrix A. N >= 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 contents of A are destroyed.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*> VL is REAL
*> If RANGE='V', the lower bound of the interval to
*> be searched for singular values. VU > VL.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*> VU is REAL
*> If RANGE='V', the upper bound of the interval to
*> be searched for singular values. VU > VL.
*> Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*> IL is INTEGER
*> If RANGE='I', the index of the
*> smallest singular value to be returned.
*> 1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*> IU is INTEGER
*> If RANGE='I', the index of the
*> largest singular value to be returned.
*> 1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
*> Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*> NS is INTEGER
*> The total number of singular values found,
*> 0 <= NS <= min(M,N).
*> If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is REAL array, dimension (min(M,N))
*> The singular values of A, sorted so that S(i) >= S(i+1).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is REAL array, dimension (LDU,UCOL)
*> If JOBU = 'V', U contains columns of U (the left singular
*> vectors, stored columnwise) as specified by RANGE; if
*> JOBU = 'N', U is not referenced.
*> Note: The user must ensure that UCOL >= NS; if RANGE = 'V',
*> the exact value of NS is not known in advance and an upper
*> bound must be used.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U. LDU >= 1; if
*> JOBU = 'V', LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*> VT is REAL array, dimension (LDVT,N)
*> If JOBVT = 'V', VT contains the rows of V**T (the right singular
*> vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N',
*> VT is not referenced.
*> Note: The user must ensure that LDVT >= NS; if RANGE = 'V',
*> the exact value of NS is not known in advance and an upper
*> bound must be used.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
*> The leading dimension of the array VT. LDVT >= 1; if
*> JOBVT = 'V', LDVT >= NS (see above).
*> \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 >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see
*> comments inside the code):
*> - PATH 1 (M much larger than N)
*> - PATH 1t (N much larger than M)
*> LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths.
*> 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] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (12*MIN(M,N))
*> If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0,
*> then IWORK contains the indices of the eigenvectors that failed
*> to converge in SBDSVDX/SSTEVX.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, then i eigenvectors failed to converge
*> in SBDSVDX/SSTEVX.
*> if INFO = N*2 + 1, an internal error occurred in
*> SBDSVDX
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup realGEsing
*
* =====================================================================
SUBROUTINE SGESVDX( JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU,
$ IL, IU, NS, S, U, LDU, VT, LDVT, WORK,
$ LWORK, IWORK, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER JOBU, JOBVT, RANGE
INTEGER IL, INFO, IU, LDA, LDU, LDVT, LWORK, M, N, NS
REAL VL, VU
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
REAL A( LDA, * ), S( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
CHARACTER JOBZ, RNGTGK
LOGICAL ALLS, INDS, LQUERY, VALS, WANTU, WANTVT
INTEGER I, ID, IE, IERR, ILQF, ILTGK, IQRF, ISCL,
$ ITAU, ITAUP, ITAUQ, ITEMP, ITGKZ, IUTGK,
$ J, MAXWRK, MINMN, MINWRK, MNTHR
REAL ABSTOL, ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
REAL DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL SBDSVDX, SGEBRD, SGELQF, SGEQRF, SLACPY,
$ SLASCL, SLASET, SORMBR, SORMLQ, SORMQR,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANGE
EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
NS = 0
INFO = 0
ABSTOL = 2*SLAMCH('S')
LQUERY = ( LWORK.EQ.-1 )
MINMN = MIN( M, N )
WANTU = LSAME( JOBU, 'V' )
WANTVT = LSAME( JOBVT, 'V' )
IF( WANTU .OR. WANTVT ) THEN
JOBZ = 'V'
ELSE
JOBZ = 'N'
END IF
ALLS = LSAME( RANGE, 'A' )
VALS = LSAME( RANGE, 'V' )
INDS = LSAME( RANGE, 'I' )
*
INFO = 0
IF( .NOT.LSAME( JOBU, 'V' ) .AND.
$ .NOT.LSAME( JOBU, 'N' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( JOBVT, 'V' ) .AND.
$ .NOT.LSAME( JOBVT, 'N' ) ) THEN
INFO = -2
ELSE IF( .NOT.( ALLS .OR. VALS .OR. INDS ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( M.GT.LDA ) THEN
INFO = -7
ELSE IF( MINMN.GT.0 ) THEN
IF( VALS ) THEN
IF( VL.LT.ZERO ) THEN
INFO = -8
ELSE IF( VU.LE.VL ) THEN
INFO = -9
END IF
ELSE IF( INDS ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, MINMN ) ) THEN
INFO = -10
ELSE IF( IU.LT.MIN( MINMN, IL ) .OR. IU.GT.MINMN ) THEN
INFO = -11
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( WANTU .AND. LDU.LT.M ) THEN
INFO = -15
ELSE IF( WANTVT ) THEN
IF( INDS ) THEN
IF( LDVT.LT.IU-IL+1 ) THEN
INFO = -17
END IF
ELSE IF( LDVT.LT.MINMN ) THEN
INFO = -17
END IF
END IF
END IF
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
IF( M.GE.N ) THEN
MNTHR = ILAENV( 6, 'SGESVD', JOBU // JOBVT, M, N, 0, 0 )
IF( M.GE.MNTHR ) THEN
*
* Path 1 (M much larger than N)
*
MAXWRK = N +
$ N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 )
MAXWRK = MAX( MAXWRK, N*(N+5) + 2*N*
$ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) )
IF (WANTU) THEN
MAXWRK = MAX(MAXWRK,N*(N*3+6)+N*
$ ILAENV( 1, 'SORMQR', ' ', N, N, -1, -1 ) )
END IF
IF (WANTVT) THEN
MAXWRK = MAX(MAXWRK,N*(N*3+6)+N*
$ ILAENV( 1, 'SORMLQ', ' ', N, N, -1, -1 ) )
END IF
MINWRK = N*(N*3+20)
ELSE
*
* Path 2 (M at least N, but not much larger)
*
MAXWRK = 4*N + ( M+N )*
$ ILAENV( 1, 'SGEBRD', ' ', M, N, -1, -1 )
IF (WANTU) THEN
MAXWRK = MAX(MAXWRK,N*(N*2+5)+N*
$ ILAENV( 1, 'SORMQR', ' ', N, N, -1, -1 ) )
END IF
IF (WANTVT) THEN
MAXWRK = MAX(MAXWRK,N*(N*2+5)+N*
$ ILAENV( 1, 'SORMLQ', ' ', N, N, -1, -1 ) )
END IF
MINWRK = MAX(N*(N*2+19),4*N+M)
END IF
ELSE
MNTHR = ILAENV( 6, 'SGESVD', JOBU // JOBVT, M, N, 0, 0 )
IF( N.GE.MNTHR ) THEN
*
* Path 1t (N much larger than M)
*
MAXWRK = M +
$ M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 )
MAXWRK = MAX( MAXWRK, M*(M+5) + 2*M*
$ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) )
IF (WANTU) THEN
MAXWRK = MAX(MAXWRK,M*(M*3+6)+M*
$ ILAENV( 1, 'SORMQR', ' ', M, M, -1, -1 ) )
END IF
IF (WANTVT) THEN
MAXWRK = MAX(MAXWRK,M*(M*3+6)+M*
$ ILAENV( 1, 'SORMLQ', ' ', M, M, -1, -1 ) )
END IF
MINWRK = M*(M*3+20)
ELSE
*
* Path 2t (N at least M, but not much larger)
*
MAXWRK = 4*M + ( M+N )*
$ ILAENV( 1, 'SGEBRD', ' ', M, N, -1, -1 )
IF (WANTU) THEN
MAXWRK = MAX(MAXWRK,M*(M*2+5)+M*
$ ILAENV( 1, 'SORMQR', ' ', M, M, -1, -1 ) )
END IF
IF (WANTVT) THEN
MAXWRK = MAX(MAXWRK,M*(M*2+5)+M*
$ ILAENV( 1, 'SORMLQ', ' ', M, M, -1, -1 ) )
END IF
MINWRK = MAX(M*(M*2+19),4*M+N)
END IF
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
WORK( 1 ) = REAL( MAXWRK )
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGESVDX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
* Set singular values indices accord to RANGE.
*
IF( ALLS ) THEN
RNGTGK = 'I'
ILTGK = 1
IUTGK = MIN( M, N )
ELSE IF( INDS ) THEN
RNGTGK = 'I'
ILTGK = IL
IUTGK = IU
ELSE
RNGTGK = 'V'
ILTGK = 0
IUTGK = 0
END IF
*
* Get machine constants
*
EPS = SLAMCH( 'P' )
SMLNUM = SQRT( SLAMCH( 'S' ) ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = SLANGE( 'M', M, N, A, LDA, DUM )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ISCL = 1
CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
ELSE IF( ANRM.GT.BIGNUM ) THEN
ISCL = 1
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
END IF
*
IF( M.GE.N ) THEN
*
* A has at least as many rows as columns. If A has sufficiently
* more rows than columns, first reduce A using the QR
* decomposition.
*
IF( M.GE.MNTHR ) THEN
*
* Path 1 (M much larger than N):
* A = Q * R = Q * ( QB * B * PB**T )
* = Q * ( QB * ( UB * S * VB**T ) * PB**T )
* U = Q * QB * UB; V**T = VB**T * PB**T
*
* Compute A=Q*R
* (Workspace: need 2*N, prefer N+N*NB)
*
ITAU = 1
ITEMP = ITAU + N
CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
*
* Copy R into WORK and bidiagonalize it:
* (Workspace: need N*N+5*N, prefer N*N+4*N+2*N*NB)
*
IQRF = ITEMP
ID = IQRF + N*N
IE = ID + N
ITAUQ = IE + N
ITAUP = ITAUQ + N
ITEMP = ITAUP + N
CALL SLACPY( 'U', N, N, A, LDA, WORK( IQRF ), N )
CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IQRF+1 ), N )
CALL SGEBRD( N, N, WORK( IQRF ), N, WORK( ID ), WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
*
* Solve eigenvalue problem TGK*Z=Z*S.
* (Workspace: need 14*N + 2*N*(N+1))
*
ITGKZ = ITEMP
ITEMP = ITGKZ + N*(N*2+1)
CALL SBDSVDX( 'U', JOBZ, RNGTGK, N, WORK( ID ), WORK( IE ),
$ VL, VU, ILTGK, IUTGK, NS, S, WORK( ITGKZ ),
$ N*2, WORK( ITEMP ), IWORK, INFO)
*
* If needed, compute left singular vectors.
*
IF( WANTU ) THEN
J = ITGKZ
DO I = 1, NS
CALL SCOPY( N, WORK( J ), 1, U( 1,I ), 1 )
J = J + N*2
END DO
CALL SLASET( 'A', M-N, NS, ZERO, ZERO, U( N+1,1 ), LDU )
*
* Call SORMBR to compute QB*UB.
* (Workspace in WORK( ITEMP ): need N, prefer N*NB)
*
CALL SORMBR( 'Q', 'L', 'N', N, NS, N, WORK( IQRF ), N,
$ WORK( ITAUQ ), U, LDU, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
*
* Call SORMQR to compute Q*(QB*UB).
* (Workspace in WORK( ITEMP ): need N, prefer N*NB)
*
CALL SORMQR( 'L', 'N', M, NS, N, A, LDA,
$ WORK( ITAU ), U, LDU, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
END IF
*
* If needed, compute right singular vectors.
*
IF( WANTVT) THEN
J = ITGKZ + N
DO I = 1, NS
CALL SCOPY( N, WORK( J ), 1, VT( I,1 ), LDVT )
J = J + N*2
END DO
*
* Call SORMBR to compute VB**T * PB**T
* (Workspace in WORK( ITEMP ): need N, prefer N*NB)
*
CALL SORMBR( 'P', 'R', 'T', NS, N, N, WORK( IQRF ), N,
$ WORK( ITAUP ), VT, LDVT, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
END IF
ELSE
*
* Path 2 (M at least N, but not much larger)
* Reduce A to bidiagonal form without QR decomposition
* A = QB * B * PB**T = QB * ( UB * S * VB**T ) * PB**T
* U = QB * UB; V**T = VB**T * PB**T
*
* Bidiagonalize A
* (Workspace: need 4*N+M, prefer 4*N+(M+N)*NB)
*
ID = 1
IE = ID + N
ITAUQ = IE + N
ITAUP = ITAUQ + N
ITEMP = ITAUP + N
CALL SGEBRD( M, N, A, LDA, WORK( ID ), WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
*
* Solve eigenvalue problem TGK*Z=Z*S.
* (Workspace: need 14*N + 2*N*(N+1))
*
ITGKZ = ITEMP
ITEMP = ITGKZ + N*(N*2+1)
CALL SBDSVDX( 'U', JOBZ, RNGTGK, N, WORK( ID ), WORK( IE ),
$ VL, VU, ILTGK, IUTGK, NS, S, WORK( ITGKZ ),
$ N*2, WORK( ITEMP ), IWORK, INFO)
*
* If needed, compute left singular vectors.
*
IF( WANTU ) THEN
J = ITGKZ
DO I = 1, NS
CALL SCOPY( N, WORK( J ), 1, U( 1,I ), 1 )
J = J + N*2
END DO
CALL SLASET( 'A', M-N, NS, ZERO, ZERO, U( N+1,1 ), LDU )
*
* Call SORMBR to compute QB*UB.
* (Workspace in WORK( ITEMP ): need N, prefer N*NB)
*
CALL SORMBR( 'Q', 'L', 'N', M, NS, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( ITEMP ),
$ LWORK-ITEMP+1, IERR )
END IF
*
* If needed, compute right singular vectors.
*
IF( WANTVT) THEN
J = ITGKZ + N
DO I = 1, NS
CALL SCOPY( N, WORK( J ), 1, VT( I,1 ), LDVT )
J = J + N*2
END DO
*
* Call SORMBR to compute VB**T * PB**T
* (Workspace in WORK( ITEMP ): need N, prefer N*NB)
*
CALL SORMBR( 'P', 'R', 'T', NS, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( ITEMP ),
$ LWORK-ITEMP+1, IERR )
END IF
END IF
ELSE
*
* A has more columns than rows. If A has sufficiently more
* columns than rows, first reduce A using the LQ decomposition.
*
IF( N.GE.MNTHR ) THEN
*
* Path 1t (N much larger than M):
* A = L * Q = ( QB * B * PB**T ) * Q
* = ( QB * ( UB * S * VB**T ) * PB**T ) * Q
* U = QB * UB ; V**T = VB**T * PB**T * Q
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M+M*NB)
*
ITAU = 1
ITEMP = ITAU + M
CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
* Copy L into WORK and bidiagonalize it:
* (Workspace in WORK( ITEMP ): need M*M+5*N, prefer M*M+4*M+2*M*NB)
*
ILQF = ITEMP
ID = ILQF + M*M
IE = ID + M
ITAUQ = IE + M
ITAUP = ITAUQ + M
ITEMP = ITAUP + M
CALL SLACPY( 'L', M, M, A, LDA, WORK( ILQF ), M )
CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( ILQF+M ), M )
CALL SGEBRD( M, M, WORK( ILQF ), M, WORK( ID ), WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
*
* Solve eigenvalue problem TGK*Z=Z*S.
* (Workspace: need 2*M*M+14*M)
*
ITGKZ = ITEMP
ITEMP = ITGKZ + M*(M*2+1)
CALL SBDSVDX( 'U', JOBZ, RNGTGK, M, WORK( ID ), WORK( IE ),
$ VL, VU, ILTGK, IUTGK, NS, S, WORK( ITGKZ ),
$ M*2, WORK( ITEMP ), IWORK, INFO)
*
* If needed, compute left singular vectors.
*
IF( WANTU ) THEN
J = ITGKZ
DO I = 1, NS
CALL SCOPY( M, WORK( J ), 1, U( 1,I ), 1 )
J = J + M*2
END DO
*
* Call SORMBR to compute QB*UB.
* (Workspace in WORK( ITEMP ): need M, prefer M*NB)
*
CALL SORMBR( 'Q', 'L', 'N', M, NS, M, WORK( ILQF ), M,
$ WORK( ITAUQ ), U, LDU, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
END IF
*
* If needed, compute right singular vectors.
*
IF( WANTVT) THEN
J = ITGKZ + M
DO I = 1, NS
CALL SCOPY( M, WORK( J ), 1, VT( I,1 ), LDVT )
J = J + M*2
END DO
CALL SLASET( 'A', NS, N-M, ZERO, ZERO, VT( 1,M+1 ), LDVT)
*
* Call SORMBR to compute (VB**T)*(PB**T)
* (Workspace in WORK( ITEMP ): need M, prefer M*NB)
*
CALL SORMBR( 'P', 'R', 'T', NS, M, M, WORK( ILQF ), M,
$ WORK( ITAUP ), VT, LDVT, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
*
* Call SORMLQ to compute ((VB**T)*(PB**T))*Q.
* (Workspace in WORK( ITEMP ): need M, prefer M*NB)
*
CALL SORMLQ( 'R', 'N', NS, N, M, A, LDA,
$ WORK( ITAU ), VT, LDVT, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
END IF
ELSE
*
* Path 2t (N greater than M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
* A = QB * B * PB**T = QB * ( UB * S * VB**T ) * PB**T
* U = QB * UB; V**T = VB**T * PB**T
*
* Bidiagonalize A
* (Workspace: need 4*M+N, prefer 4*M+(M+N)*NB)
*
ID = 1
IE = ID + M
ITAUQ = IE + M
ITAUP = ITAUQ + M
ITEMP = ITAUP + M
CALL SGEBRD( M, N, A, LDA, WORK( ID ), WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
*
* Solve eigenvalue problem TGK*Z=Z*S.
* (Workspace: need 2*M*M+14*M)
*
ITGKZ = ITEMP
ITEMP = ITGKZ + M*(M*2+1)
CALL SBDSVDX( 'L', JOBZ, RNGTGK, M, WORK( ID ), WORK( IE ),
$ VL, VU, ILTGK, IUTGK, NS, S, WORK( ITGKZ ),
$ M*2, WORK( ITEMP ), IWORK, INFO)
*
* If needed, compute left singular vectors.
*
IF( WANTU ) THEN
J = ITGKZ
DO I = 1, NS
CALL SCOPY( M, WORK( J ), 1, U( 1,I ), 1 )
J = J + M*2
END DO
*
* Call SORMBR to compute QB*UB.
* (Workspace in WORK( ITEMP ): need M, prefer M*NB)
*
CALL SORMBR( 'Q', 'L', 'N', M, NS, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
END IF
*
* If needed, compute right singular vectors.
*
IF( WANTVT) THEN
J = ITGKZ + M
DO I = 1, NS
CALL SCOPY( M, WORK( J ), 1, VT( I,1 ), LDVT )
J = J + M*2
END DO
CALL SLASET( 'A', NS, N-M, ZERO, ZERO, VT( 1,M+1 ), LDVT)
*
* Call SORMBR to compute VB**T * PB**T
* (Workspace in WORK( ITEMP ): need M, prefer M*NB)
*
CALL SORMBR( 'P', 'R', 'T', NS, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( ITEMP ),
$ LWORK-ITEMP+1, INFO )
END IF
END IF
END IF
*
* Undo scaling if necessary
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM )
$ CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1,
$ S, MINMN, INFO )
IF( ANRM.LT.SMLNUM )
$ CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1,
$ S, MINMN, INFO )
END IF
*
* Return optimal workspace in WORK(1)
*
WORK( 1 ) = REAL( MAXWRK )
*
RETURN
*
* End of SGESVDX
*
END
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