numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
| .. | |||
| lapack/SRC/zgesvdx.f | 29013B | -rw-r--r-- |
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*> \brief <b> ZGESVDX 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 ZGESVDX + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgesvdx.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgesvdx.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgesvdx.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGESVDX( JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU, * $ IL, IU, NS, S, U, LDU, VT, LDVT, WORK, * $ LWORK, RWORK, IWORK, INFO ) * * * .. Scalar Arguments .. * CHARACTER JOBU, JOBVT, RANGE * INTEGER IL, INFO, IU, LDA, LDU, LDVT, LWORK, M, N, NS * DOUBLE PRECISION VL, VU * .. * .. Array Arguments .. * INTEGER IWORK( * ) * DOUBLE PRECISION S( * ), RWORK( * ) * COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), * $ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGESVDX computes the singular value decomposition (SVD) of a complex *> 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 unitary matrix, and *> V is an N-by-N unitary 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. *> *> ZGESVDX uses an eigenvalue problem for obtaining the SVD, which *> allows for the computation of a subset of singular values and *> vectors. See DBDSVDX 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 COMPLEX*16 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 DOUBLE PRECISION *> 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 DOUBLE PRECISION *> 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 DOUBLE PRECISION 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) *> LRWORK >= MIN(M,N)*(MIN(M,N)*2+15*MIN(M,N)). *> \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 DBDSVDX/DSTEVX. *> \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 DBDSVDX/DSTEVX. *> if INFO = N*2 + 1, an internal error occurred in *> DBDSVDX *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup gesvdx * * ===================================================================== SUBROUTINE ZGESVDX( JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU, $ IL, IU, NS, S, U, LDU, VT, LDVT, WORK, $ LWORK, RWORK, IWORK, INFO ) * * -- LAPACK driver routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER JOBU, JOBVT, RANGE INTEGER IL, INFO, IU, LDA, LDU, LDVT, LWORK, M, N, NS DOUBLE PRECISION VL, VU * .. * .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION S( * ), RWORK( * ) COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), $ WORK( * ) * .. * * ===================================================================== * * .. Parameters .. COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), $ CONE = ( 1.0D0, 0.0D0 ) ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. 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, ITEMPR, ITGKZ, $ IUTGK, J, K, MAXWRK, MINMN, MINWRK, MNTHR DOUBLE PRECISION ABSTOL, ANRM, BIGNUM, EPS, SMLNUM * .. * .. Local Arrays .. DOUBLE PRECISION DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL ZGEBRD, ZGELQF, ZGEQRF, ZLASCL, ZLASET, $ ZLACPY, $ ZUNMLQ, ZUNMBR, ZUNMQR, DBDSVDX, DLASCL, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT * .. * .. Executable Statements .. * * Test the input arguments. * NS = 0 INFO = 0 ABSTOL = 2*DLAMCH('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, 'ZGESVD', JOBU // JOBVT, M, N, 0, $ 0 ) IF( M.GE.MNTHR ) THEN * * Path 1 (M much larger than N) * MINWRK = N*(N+5) MAXWRK = N + N*ILAENV(1,'ZGEQRF',' ',M,N,-1,-1) MAXWRK = MAX(MAXWRK, $ N*N+2*N+2*N*ILAENV(1,'ZGEBRD',' ',N,N,-1, $ -1)) IF (WANTU .OR. WANTVT) THEN MAXWRK = MAX(MAXWRK, $ N*N+2*N+N*ILAENV(1,'ZUNMQR','LN',N,N,N, $ -1)) END IF ELSE * * Path 2 (M at least N, but not much larger) * MINWRK = 3*N + M MAXWRK = 2*N + (M+N)*ILAENV(1,'ZGEBRD',' ',M,N,-1, $ -1) IF (WANTU .OR. WANTVT) THEN MAXWRK = MAX(MAXWRK, $ 2*N+N*ILAENV(1,'ZUNMQR','LN',N,N,N,-1)) END IF END IF ELSE MNTHR = ILAENV( 6, 'ZGESVD', JOBU // JOBVT, M, N, 0, $ 0 ) IF( N.GE.MNTHR ) THEN * * Path 1t (N much larger than M) * MINWRK = M*(M+5) MAXWRK = M + M*ILAENV(1,'ZGELQF',' ',M,N,-1,-1) MAXWRK = MAX(MAXWRK, $ M*M+2*M+2*M*ILAENV(1,'ZGEBRD',' ',M,M,-1, $ -1)) IF (WANTU .OR. WANTVT) THEN MAXWRK = MAX(MAXWRK, $ M*M+2*M+M*ILAENV(1,'ZUNMQR','LN',M,M,M, $ -1)) END IF ELSE * * Path 2t (N greater than M, but not much larger) * * MINWRK = 3*M + N MAXWRK = 2*M + (M+N)*ILAENV(1,'ZGEBRD',' ',M,N,-1, $ -1) IF (WANTU .OR. WANTVT) THEN MAXWRK = MAX(MAXWRK, $ 2*M+M*ILAENV(1,'ZUNMQR','LN',M,M,M,-1)) END IF END IF END IF END IF MAXWRK = MAX( MAXWRK, MINWRK ) WORK( 1 ) = DCMPLX( DBLE( MAXWRK ), ZERO ) * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -19 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGESVDX', -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='A'. * 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 = DLAMCH( 'P' ) SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS BIGNUM = ONE / SMLNUM * * Scale A if max element outside range [SMLNUM,BIGNUM] * ANRM = ZLANGE( 'M', M, N, A, LDA, DUM ) ISCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ISCL = 1 CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) ELSE IF( ANRM.GT.BIGNUM ) THEN ISCL = 1 CALL ZLASCL( '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 ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( ITEMP ), $ LWORK-ITEMP+1, INFO ) * * Copy R into WORK and bidiagonalize it: * (Workspace: need N*N+3*N, prefer N*N+N+2*N*NB) * IQRF = ITEMP ITAUQ = ITEMP + N*N ITAUP = ITAUQ + N ITEMP = ITAUP + N ID = 1 IE = ID + N ITGKZ = IE + N CALL ZLACPY( 'U', N, N, A, LDA, WORK( IQRF ), N ) CALL ZLASET( 'L', N-1, N-1, CZERO, CZERO, $ WORK( IQRF+1 ), N ) CALL ZGEBRD( N, N, WORK( IQRF ), N, RWORK( ID ), $ RWORK( IE ), WORK( ITAUQ ), WORK( ITAUP ), $ WORK( ITEMP ), LWORK-ITEMP+1, INFO ) ITEMPR = ITGKZ + N*(N*2+1) * * Solve eigenvalue problem TGK*Z=Z*S. * (Workspace: need 2*N*N+14*N) * CALL DBDSVDX( 'U', JOBZ, RNGTGK, N, RWORK( ID ), $ RWORK( IE ), VL, VU, ILTGK, IUTGK, NS, S, $ RWORK( ITGKZ ), N*2, RWORK( ITEMPR ), $ IWORK, INFO) * * If needed, compute left singular vectors. * IF( WANTU ) THEN K = ITGKZ DO I = 1, NS DO J = 1, N U( J, I ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + N END DO CALL ZLASET( 'A', M-N, NS, CZERO, CZERO, U( N+1,1 ), $ LDU) * * Call ZUNMBR to compute QB*UB. * (Workspace in WORK( ITEMP ): need N, prefer N*NB) * CALL ZUNMBR( 'Q', 'L', 'N', N, NS, N, WORK( IQRF ), N, $ WORK( ITAUQ ), U, LDU, WORK( ITEMP ), $ LWORK-ITEMP+1, INFO ) * * Call ZUNMQR to compute Q*(QB*UB). * (Workspace in WORK( ITEMP ): need N, prefer N*NB) * CALL ZUNMQR( '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 K = ITGKZ + N DO I = 1, NS DO J = 1, N VT( I, J ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + N END DO * * Call ZUNMBR to compute VB**T * PB**T * (Workspace in WORK( ITEMP ): need N, prefer N*NB) * CALL ZUNMBR( 'P', 'R', 'C', 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 2*N+M, prefer 2*N+(M+N)*NB) * ITAUQ = 1 ITAUP = ITAUQ + N ITEMP = ITAUP + N ID = 1 IE = ID + N ITGKZ = IE + N CALL ZGEBRD( M, N, A, LDA, RWORK( ID ), RWORK( IE ), $ WORK( ITAUQ ), WORK( ITAUP ), WORK( ITEMP ), $ LWORK-ITEMP+1, INFO ) ITEMPR = ITGKZ + N*(N*2+1) * * Solve eigenvalue problem TGK*Z=Z*S. * (Workspace: need 2*N*N+14*N) * CALL DBDSVDX( 'U', JOBZ, RNGTGK, N, RWORK( ID ), $ RWORK( IE ), VL, VU, ILTGK, IUTGK, NS, S, $ RWORK( ITGKZ ), N*2, RWORK( ITEMPR ), $ IWORK, INFO) * * If needed, compute left singular vectors. * IF( WANTU ) THEN K = ITGKZ DO I = 1, NS DO J = 1, N U( J, I ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + N END DO CALL ZLASET( 'A', M-N, NS, CZERO, CZERO, U( N+1,1 ), $ LDU) * * Call ZUNMBR to compute QB*UB. * (Workspace in WORK( ITEMP ): need N, prefer N*NB) * CALL ZUNMBR( '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 K = ITGKZ + N DO I = 1, NS DO J = 1, N VT( I, J ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + N END DO * * Call ZUNMBR to compute VB**T * PB**T * (Workspace in WORK( ITEMP ): need N, prefer N*NB) * CALL ZUNMBR( 'P', 'R', 'C', 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 ZGELQF( 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+3*M, prefer M*M+M+2*M*NB) * ILQF = ITEMP ITAUQ = ILQF + M*M ITAUP = ITAUQ + M ITEMP = ITAUP + M ID = 1 IE = ID + M ITGKZ = IE + M CALL ZLACPY( 'L', M, M, A, LDA, WORK( ILQF ), M ) CALL ZLASET( 'U', M-1, M-1, CZERO, CZERO, $ WORK( ILQF+M ), M ) CALL ZGEBRD( M, M, WORK( ILQF ), M, RWORK( ID ), $ RWORK( IE ), WORK( ITAUQ ), WORK( ITAUP ), $ WORK( ITEMP ), LWORK-ITEMP+1, INFO ) ITEMPR = ITGKZ + M*(M*2+1) * * Solve eigenvalue problem TGK*Z=Z*S. * (Workspace: need 2*M*M+14*M) * CALL DBDSVDX( 'U', JOBZ, RNGTGK, M, RWORK( ID ), $ RWORK( IE ), VL, VU, ILTGK, IUTGK, NS, S, $ RWORK( ITGKZ ), M*2, RWORK( ITEMPR ), $ IWORK, INFO) * * If needed, compute left singular vectors. * IF( WANTU ) THEN K = ITGKZ DO I = 1, NS DO J = 1, M U( J, I ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + M END DO * * Call ZUNMBR to compute QB*UB. * (Workspace in WORK( ITEMP ): need M, prefer M*NB) * CALL ZUNMBR( '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 K = ITGKZ + M DO I = 1, NS DO J = 1, M VT( I, J ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + M END DO CALL ZLASET( 'A', NS, N-M, CZERO, CZERO, $ VT( 1,M+1 ), LDVT ) * * Call ZUNMBR to compute (VB**T)*(PB**T) * (Workspace in WORK( ITEMP ): need M, prefer M*NB) * CALL ZUNMBR( 'P', 'R', 'C', NS, M, M, WORK( ILQF ), M, $ WORK( ITAUP ), VT, LDVT, WORK( ITEMP ), $ LWORK-ITEMP+1, INFO ) * * Call ZUNMLQ to compute ((VB**T)*(PB**T))*Q. * (Workspace in WORK( ITEMP ): need M, prefer M*NB) * CALL ZUNMLQ( '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 2*M+N, prefer 2*M+(M+N)*NB) * ITAUQ = 1 ITAUP = ITAUQ + M ITEMP = ITAUP + M ID = 1 IE = ID + M ITGKZ = IE + M CALL ZGEBRD( M, N, A, LDA, RWORK( ID ), RWORK( IE ), $ WORK( ITAUQ ), WORK( ITAUP ), WORK( ITEMP ), $ LWORK-ITEMP+1, INFO ) ITEMPR = ITGKZ + M*(M*2+1) * * Solve eigenvalue problem TGK*Z=Z*S. * (Workspace: need 2*M*M+14*M) * CALL DBDSVDX( 'L', JOBZ, RNGTGK, M, RWORK( ID ), $ RWORK( IE ), VL, VU, ILTGK, IUTGK, NS, S, $ RWORK( ITGKZ ), M*2, RWORK( ITEMPR ), $ IWORK, INFO) * * If needed, compute left singular vectors. * IF( WANTU ) THEN K = ITGKZ DO I = 1, NS DO J = 1, M U( J, I ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + M END DO * * Call ZUNMBR to compute QB*UB. * (Workspace in WORK( ITEMP ): need M, prefer M*NB) * CALL ZUNMBR( '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 K = ITGKZ + M DO I = 1, NS DO J = 1, M VT( I, J ) = DCMPLX( RWORK( K ), ZERO ) K = K + 1 END DO K = K + M END DO CALL ZLASET( 'A', NS, N-M, CZERO, CZERO, $ VT( 1,M+1 ), LDVT ) * * Call ZUNMBR to compute VB**T * PB**T * (Workspace in WORK( ITEMP ): need M, prefer M*NB) * CALL ZUNMBR( 'P', 'R', 'C', 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 DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, $ S, MINMN, INFO ) IF( ANRM.LT.SMLNUM ) $ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, $ S, MINMN, INFO ) END IF * * Return optimal workspace in WORK(1) * WORK( 1 ) = DCMPLX( DBLE( MAXWRK ), ZERO ) * RETURN * * End of ZGESVDX * END