*> \brief SGESVDX computes the singular value decomposition (SVD) for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGESVDX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \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 3.8.0) -- * -- 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, $ SCOPY, 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