*> \brief \b SORCSD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SORCSD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * RECURSIVE SUBROUTINE SORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, * SIGNS, M, P, Q, X11, LDX11, X12, * LDX12, X21, LDX21, X22, LDX22, THETA, * U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, * LDV2T, WORK, LWORK, IWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS * INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12, * $ LDX21, LDX22, LWORK, M, P, Q * .. * .. Array Arguments .. * INTEGER IWORK( * ) * REAL THETA( * ) * REAL U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), * $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ), * $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22, * $ * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SORCSD computes the CS decomposition of an M-by-M partitioned *> orthogonal matrix X: *> *> [ I 0 0 | 0 0 0 ] *> [ 0 C 0 | 0 -S 0 ] *> [ X11 | X12 ] [ U1 | ] [ 0 0 0 | 0 0 -I ] [ V1 | ]**T *> X = [-----------] = [---------] [---------------------] [---------] . *> [ X21 | X22 ] [ | U2 ] [ 0 0 0 | I 0 0 ] [ | V2 ] *> [ 0 S 0 | 0 C 0 ] *> [ 0 0 I | 0 0 0 ] *> *> X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P, *> (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are *> R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in *> which R = MIN(P,M-P,Q,M-Q). *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBU1 *> \verbatim *> JOBU1 is CHARACTER *> = 'Y': U1 is computed; *> otherwise: U1 is not computed. *> \endverbatim *> *> \param[in] JOBU2 *> \verbatim *> JOBU2 is CHARACTER *> = 'Y': U2 is computed; *> otherwise: U2 is not computed. *> \endverbatim *> *> \param[in] JOBV1T *> \verbatim *> JOBV1T is CHARACTER *> = 'Y': V1T is computed; *> otherwise: V1T is not computed. *> \endverbatim *> *> \param[in] JOBV2T *> \verbatim *> JOBV2T is CHARACTER *> = 'Y': V2T is computed; *> otherwise: V2T is not computed. *> \endverbatim *> *> \param[in] TRANS *> \verbatim *> TRANS is CHARACTER *> = 'T': X, U1, U2, V1T, and V2T are stored in row-major *> order; *> otherwise: X, U1, U2, V1T, and V2T are stored in column- *> major order. *> \endverbatim *> *> \param[in] SIGNS *> \verbatim *> SIGNS is CHARACTER *> = 'O': The lower-left block is made nonpositive (the *> "other" convention); *> otherwise: The upper-right block is made nonpositive (the *> "default" convention). *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows and columns in X. *> \endverbatim *> *> \param[in] P *> \verbatim *> P is INTEGER *> The number of rows in X11 and X12. 0 <= P <= M. *> \endverbatim *> *> \param[in] Q *> \verbatim *> Q is INTEGER *> The number of columns in X11 and X21. 0 <= Q <= M. *> \endverbatim *> *> \param[in,out] X11 *> \verbatim *> X11 is REAL array, dimension (LDX11,Q) *> On entry, part of the orthogonal matrix whose CSD is desired. *> \endverbatim *> *> \param[in] LDX11 *> \verbatim *> LDX11 is INTEGER *> The leading dimension of X11. LDX11 >= MAX(1,P). *> \endverbatim *> *> \param[in,out] X12 *> \verbatim *> X12 is REAL array, dimension (LDX12,M-Q) *> On entry, part of the orthogonal matrix whose CSD is desired. *> \endverbatim *> *> \param[in] LDX12 *> \verbatim *> LDX12 is INTEGER *> The leading dimension of X12. LDX12 >= MAX(1,P). *> \endverbatim *> *> \param[in,out] X21 *> \verbatim *> X21 is REAL array, dimension (LDX21,Q) *> On entry, part of the orthogonal matrix whose CSD is desired. *> \endverbatim *> *> \param[in] LDX21 *> \verbatim *> LDX21 is INTEGER *> The leading dimension of X11. LDX21 >= MAX(1,M-P). *> \endverbatim *> *> \param[in,out] X22 *> \verbatim *> X22 is REAL array, dimension (LDX22,M-Q) *> On entry, part of the orthogonal matrix whose CSD is desired. *> \endverbatim *> *> \param[in] LDX22 *> \verbatim *> LDX22 is INTEGER *> The leading dimension of X11. LDX22 >= MAX(1,M-P). *> \endverbatim *> *> \param[out] THETA *> \verbatim *> THETA is REAL array, dimension (R), in which R = *> MIN(P,M-P,Q,M-Q). *> C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and *> S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ). *> \endverbatim *> *> \param[out] U1 *> \verbatim *> U1 is REAL array, dimension (LDU1,P) *> If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1. *> \endverbatim *> *> \param[in] LDU1 *> \verbatim *> LDU1 is INTEGER *> The leading dimension of U1. If JOBU1 = 'Y', LDU1 >= *> MAX(1,P). *> \endverbatim *> *> \param[out] U2 *> \verbatim *> U2 is REAL array, dimension (LDU2,M-P) *> If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal *> matrix U2. *> \endverbatim *> *> \param[in] LDU2 *> \verbatim *> LDU2 is INTEGER *> The leading dimension of U2. If JOBU2 = 'Y', LDU2 >= *> MAX(1,M-P). *> \endverbatim *> *> \param[out] V1T *> \verbatim *> V1T is REAL array, dimension (LDV1T,Q) *> If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal *> matrix V1**T. *> \endverbatim *> *> \param[in] LDV1T *> \verbatim *> LDV1T is INTEGER *> The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >= *> MAX(1,Q). *> \endverbatim *> *> \param[out] V2T *> \verbatim *> V2T is REAL array, dimension (LDV2T,M-Q) *> If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal *> matrix V2**T. *> \endverbatim *> *> \param[in] LDV2T *> \verbatim *> LDV2T is INTEGER *> The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >= *> MAX(1,M-Q). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> If INFO > 0 on exit, WORK(2:R) contains the values PHI(1), *> ..., PHI(R-1) that, together with THETA(1), ..., THETA(R), *> define the matrix in intermediate bidiagonal-block form *> remaining after nonconvergence. INFO specifies the number *> of nonzero PHI's. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> *> 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 (M-MIN(P, M-P, Q, M-Q)) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, the i-th argument had an illegal value. *> > 0: SBBCSD did not converge. See the description of WORK *> above for details. *> \endverbatim * *> \par References: * ================ *> *> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer. *> Algorithms, 50(1):33-65, 2009. * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realOTHERcomputational * * ===================================================================== RECURSIVE SUBROUTINE SORCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, $ SIGNS, M, P, Q, X11, LDX11, X12, $ LDX12, X21, LDX21, X22, LDX22, THETA, $ U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, $ LDV2T, WORK, LWORK, IWORK, INFO ) * * -- LAPACK computational 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 JOBU1, JOBU2, JOBV1T, JOBV2T, SIGNS, TRANS INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LDX11, LDX12, $ LDX21, LDX22, LWORK, M, P, Q * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL THETA( * ) REAL U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), $ V2T( LDV2T, * ), WORK( * ), X11( LDX11, * ), $ X12( LDX12, * ), X21( LDX21, * ), X22( LDX22, $ * ) * .. * * =================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER ( ONE = 1.0E+0, $ ZERO = 0.0E+0 ) * .. * .. Local Arrays .. REAL DUMMY(1) * .. * .. Local Scalars .. CHARACTER TRANST, SIGNST INTEGER CHILDINFO, I, IB11D, IB11E, IB12D, IB12E, $ IB21D, IB21E, IB22D, IB22E, IBBCSD, IORBDB, $ IORGLQ, IORGQR, IPHI, ITAUP1, ITAUP2, ITAUQ1, $ ITAUQ2, J, LBBCSDWORK, LBBCSDWORKMIN, $ LBBCSDWORKOPT, LORBDBWORK, LORBDBWORKMIN, $ LORBDBWORKOPT, LORGLQWORK, LORGLQWORKMIN, $ LORGLQWORKOPT, LORGQRWORK, LORGQRWORKMIN, $ LORGQRWORKOPT, LWORKMIN, LWORKOPT LOGICAL COLMAJOR, DEFAULTSIGNS, LQUERY, WANTU1, WANTU2, $ WANTV1T, WANTV2T * .. * .. External Subroutines .. EXTERNAL SBBCSD, SLACPY, SLAPMR, SLAPMT, $ SORBDB, SORGLQ, SORGQR, XERBLA * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. Intrinsic Functions INTRINSIC INT, MAX, MIN * .. * .. Executable Statements .. * * Test input arguments * INFO = 0 WANTU1 = LSAME( JOBU1, 'Y' ) WANTU2 = LSAME( JOBU2, 'Y' ) WANTV1T = LSAME( JOBV1T, 'Y' ) WANTV2T = LSAME( JOBV2T, 'Y' ) COLMAJOR = .NOT. LSAME( TRANS, 'T' ) DEFAULTSIGNS = .NOT. LSAME( SIGNS, 'O' ) LQUERY = LWORK .EQ. -1 IF( M .LT. 0 ) THEN INFO = -7 ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN INFO = -8 ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN INFO = -9 ELSE IF ( COLMAJOR .AND. LDX11 .LT. MAX( 1, P ) ) THEN INFO = -11 ELSE IF (.NOT. COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN INFO = -11 ELSE IF (COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN INFO = -13 ELSE IF (.NOT. COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN INFO = -13 ELSE IF (COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN INFO = -15 ELSE IF (.NOT. COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN INFO = -15 ELSE IF (COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN INFO = -17 ELSE IF (.NOT. COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN INFO = -17 ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN INFO = -20 ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN INFO = -22 ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN INFO = -24 ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN INFO = -26 END IF * * Work with transpose if convenient * IF( INFO .EQ. 0 .AND. MIN( P, M-P ) .LT. MIN( Q, M-Q ) ) THEN IF( COLMAJOR ) THEN TRANST = 'T' ELSE TRANST = 'N' END IF IF( DEFAULTSIGNS ) THEN SIGNST = 'O' ELSE SIGNST = 'D' END IF CALL SORCSD( JOBV1T, JOBV2T, JOBU1, JOBU2, TRANST, SIGNST, M, $ Q, P, X11, LDX11, X21, LDX21, X12, LDX12, X22, $ LDX22, THETA, V1T, LDV1T, V2T, LDV2T, U1, LDU1, $ U2, LDU2, WORK, LWORK, IWORK, INFO ) RETURN END IF * * Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if * convenient * IF( INFO .EQ. 0 .AND. M-Q .LT. Q ) THEN IF( DEFAULTSIGNS ) THEN SIGNST = 'O' ELSE SIGNST = 'D' END IF CALL SORCSD( JOBU2, JOBU1, JOBV2T, JOBV1T, TRANS, SIGNST, M, $ M-P, M-Q, X22, LDX22, X21, LDX21, X12, LDX12, X11, $ LDX11, THETA, U2, LDU2, U1, LDU1, V2T, LDV2T, V1T, $ LDV1T, WORK, LWORK, IWORK, INFO ) RETURN END IF * * Compute workspace * IF( INFO .EQ. 0 ) THEN * IPHI = 2 ITAUP1 = IPHI + MAX( 1, Q - 1 ) ITAUP2 = ITAUP1 + MAX( 1, P ) ITAUQ1 = ITAUP2 + MAX( 1, M - P ) ITAUQ2 = ITAUQ1 + MAX( 1, Q ) IORGQR = ITAUQ2 + MAX( 1, M - Q ) CALL SORGQR( M-Q, M-Q, M-Q, DUMMY, MAX(1,M-Q), DUMMY, WORK, -1, $ CHILDINFO ) LORGQRWORKOPT = INT( WORK(1) ) LORGQRWORKMIN = MAX( 1, M - Q ) IORGLQ = ITAUQ2 + MAX( 1, M - Q ) CALL SORGLQ( M-Q, M-Q, M-Q, DUMMY, MAX(1,M-Q), DUMMY, WORK, -1, $ CHILDINFO ) LORGLQWORKOPT = INT( WORK(1) ) LORGLQWORKMIN = MAX( 1, M - Q ) IORBDB = ITAUQ2 + MAX( 1, M - Q ) CALL SORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, $ X21, LDX21, X22, LDX22, DUMMY, DUMMY, DUMMY, DUMMY, DUMMY, $ DUMMY,WORK,-1,CHILDINFO ) LORBDBWORKOPT = INT( WORK(1) ) LORBDBWORKMIN = LORBDBWORKOPT IB11D = ITAUQ2 + MAX( 1, M - Q ) IB11E = IB11D + MAX( 1, Q ) IB12D = IB11E + MAX( 1, Q - 1 ) IB12E = IB12D + MAX( 1, Q ) IB21D = IB12E + MAX( 1, Q - 1 ) IB21E = IB21D + MAX( 1, Q ) IB22D = IB21E + MAX( 1, Q - 1 ) IB22E = IB22D + MAX( 1, Q ) IBBCSD = IB22E + MAX( 1, Q - 1 ) CALL SBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, $ DUMMY, DUMMY, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, $ LDV2T, DUMMY, DUMMY, DUMMY, DUMMY, DUMMY, DUMMY, $ DUMMY, DUMMY, WORK, -1, CHILDINFO ) LBBCSDWORKOPT = INT( WORK(1) ) LBBCSDWORKMIN = LBBCSDWORKOPT LWORKOPT = MAX( IORGQR + LORGQRWORKOPT, IORGLQ + LORGLQWORKOPT, $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKOPT ) - 1 LWORKMIN = MAX( IORGQR + LORGQRWORKMIN, IORGLQ + LORGLQWORKMIN, $ IORBDB + LORBDBWORKOPT, IBBCSD + LBBCSDWORKMIN ) - 1 WORK(1) = MAX(LWORKOPT,LWORKMIN) * IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN INFO = -22 ELSE LORGQRWORK = LWORK - IORGQR + 1 LORGLQWORK = LWORK - IORGLQ + 1 LORBDBWORK = LWORK - IORBDB + 1 LBBCSDWORK = LWORK - IBBCSD + 1 END IF END IF * * Abort if any illegal arguments * IF( INFO .NE. 0 ) THEN CALL XERBLA( 'SORCSD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Transform to bidiagonal block form * CALL SORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, $ LDX21, X22, LDX22, THETA, WORK(IPHI), WORK(ITAUP1), $ WORK(ITAUP2), WORK(ITAUQ1), WORK(ITAUQ2), $ WORK(IORBDB), LORBDBWORK, CHILDINFO ) * * Accumulate Householder reflectors * IF( COLMAJOR ) THEN IF( WANTU1 .AND. P .GT. 0 ) THEN CALL SLACPY( 'L', P, Q, X11, LDX11, U1, LDU1 ) CALL SORGQR( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGQR), $ LORGQRWORK, INFO) END IF IF( WANTU2 .AND. M-P .GT. 0 ) THEN CALL SLACPY( 'L', M-P, Q, X21, LDX21, U2, LDU2 ) CALL SORGQR( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2), $ WORK(IORGQR), LORGQRWORK, INFO ) END IF IF( WANTV1T .AND. Q .GT. 0 ) THEN CALL SLACPY( 'U', Q-1, Q-1, X11(1,2), LDX11, V1T(2,2), $ LDV1T ) V1T(1, 1) = ONE DO J = 2, Q V1T(1,J) = ZERO V1T(J,1) = ZERO END DO CALL SORGLQ( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1), $ WORK(IORGLQ), LORGLQWORK, INFO ) END IF IF( WANTV2T .AND. M-Q .GT. 0 ) THEN CALL SLACPY( 'U', P, M-Q, X12, LDX12, V2T, LDV2T ) CALL SLACPY( 'U', M-P-Q, M-P-Q, X22(Q+1,P+1), LDX22, $ V2T(P+1,P+1), LDV2T ) CALL SORGLQ( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2), $ WORK(IORGLQ), LORGLQWORK, INFO ) END IF ELSE IF( WANTU1 .AND. P .GT. 0 ) THEN CALL SLACPY( 'U', Q, P, X11, LDX11, U1, LDU1 ) CALL SORGLQ( P, P, Q, U1, LDU1, WORK(ITAUP1), WORK(IORGLQ), $ LORGLQWORK, INFO) END IF IF( WANTU2 .AND. M-P .GT. 0 ) THEN CALL SLACPY( 'U', Q, M-P, X21, LDX21, U2, LDU2 ) CALL SORGLQ( M-P, M-P, Q, U2, LDU2, WORK(ITAUP2), $ WORK(IORGLQ), LORGLQWORK, INFO ) END IF IF( WANTV1T .AND. Q .GT. 0 ) THEN CALL SLACPY( 'L', Q-1, Q-1, X11(2,1), LDX11, V1T(2,2), $ LDV1T ) V1T(1, 1) = ONE DO J = 2, Q V1T(1,J) = ZERO V1T(J,1) = ZERO END DO CALL SORGQR( Q-1, Q-1, Q-1, V1T(2,2), LDV1T, WORK(ITAUQ1), $ WORK(IORGQR), LORGQRWORK, INFO ) END IF IF( WANTV2T .AND. M-Q .GT. 0 ) THEN CALL SLACPY( 'L', M-Q, P, X12, LDX12, V2T, LDV2T ) CALL SLACPY( 'L', M-P-Q, M-P-Q, X22(P+1,Q+1), LDX22, $ V2T(P+1,P+1), LDV2T ) CALL SORGQR( M-Q, M-Q, M-Q, V2T, LDV2T, WORK(ITAUQ2), $ WORK(IORGQR), LORGQRWORK, INFO ) END IF END IF * * Compute the CSD of the matrix in bidiagonal-block form * CALL SBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, $ WORK(IPHI), U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, $ LDV2T, WORK(IB11D), WORK(IB11E), WORK(IB12D), $ WORK(IB12E), WORK(IB21D), WORK(IB21E), WORK(IB22D), $ WORK(IB22E), WORK(IBBCSD), LBBCSDWORK, INFO ) * * Permute rows and columns to place identity submatrices in top- * left corner of (1,1)-block and/or bottom-right corner of (1,2)- * block and/or bottom-right corner of (2,1)-block and/or top-left * corner of (2,2)-block * IF( Q .GT. 0 .AND. WANTU2 ) THEN DO I = 1, Q IWORK(I) = M - P - Q + I END DO DO I = Q + 1, M - P IWORK(I) = I - Q END DO IF( COLMAJOR ) THEN CALL SLAPMT( .FALSE., M-P, M-P, U2, LDU2, IWORK ) ELSE CALL SLAPMR( .FALSE., M-P, M-P, U2, LDU2, IWORK ) END IF END IF IF( M .GT. 0 .AND. WANTV2T ) THEN DO I = 1, P IWORK(I) = M - P - Q + I END DO DO I = P + 1, M - Q IWORK(I) = I - P END DO IF( .NOT. COLMAJOR ) THEN CALL SLAPMT( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK ) ELSE CALL SLAPMR( .FALSE., M-Q, M-Q, V2T, LDV2T, IWORK ) END IF END IF * RETURN * * End SORCSD * END