[BACK]Return to zggsvd.f CVS log [TXT] [DIR] Up to [local] / rpl / lapack / lapack

Diff for /rpl/lapack/lapack/zggsvd.f between versions 1.8 and 1.9

version 1.8, 2011/07/22 07:38:14 version 1.9, 2011/11/21 20:43:10
Line 1 Line 1
   *> \brief <b> ZGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b>
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZGGSVD + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zggsvd.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zggsvd.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zggsvd.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
   *                          LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
   *                          RWORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          JOBQ, JOBU, JOBV
   *       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
   *       ..
   *       .. Array Arguments ..
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   ALPHA( * ), BETA( * ), RWORK( * )
   *       COMPLEX*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
   *      $                   U( LDU, * ), V( LDV, * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZGGSVD computes the generalized singular value decomposition (GSVD)
   *> of an M-by-N complex matrix A and P-by-N complex matrix B:
   *>
   *>       U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )
   *>
   *> where U, V and Q are unitary matrices.
   *> Let K+L = the effective numerical rank of the
   *> matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
   *> triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
   *> matrices and of the following structures, respectively:
   *>
   *> If M-K-L >= 0,
   *>
   *>                     K  L
   *>        D1 =     K ( I  0 )
   *>                 L ( 0  C )
   *>             M-K-L ( 0  0 )
   *>
   *>                   K  L
   *>        D2 =   L ( 0  S )
   *>             P-L ( 0  0 )
   *>
   *>                 N-K-L  K    L
   *>   ( 0 R ) = K (  0   R11  R12 )
   *>             L (  0    0   R22 )
   *> where
   *>
   *>   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
   *>   S = diag( BETA(K+1),  ... , BETA(K+L) ),
   *>   C**2 + S**2 = I.
   *>
   *>   R is stored in A(1:K+L,N-K-L+1:N) on exit.
   *>
   *> If M-K-L < 0,
   *>
   *>                   K M-K K+L-M
   *>        D1 =   K ( I  0    0   )
   *>             M-K ( 0  C    0   )
   *>
   *>                     K M-K K+L-M
   *>        D2 =   M-K ( 0  S    0  )
   *>             K+L-M ( 0  0    I  )
   *>               P-L ( 0  0    0  )
   *>
   *>                    N-K-L  K   M-K  K+L-M
   *>   ( 0 R ) =     K ( 0    R11  R12  R13  )
   *>               M-K ( 0     0   R22  R23  )
   *>             K+L-M ( 0     0    0   R33  )
   *>
   *> where
   *>
   *>   C = diag( ALPHA(K+1), ... , ALPHA(M) ),
   *>   S = diag( BETA(K+1),  ... , BETA(M) ),
   *>   C**2 + S**2 = I.
   *>
   *>   (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
   *>   ( 0  R22 R23 )
   *>   in B(M-K+1:L,N+M-K-L+1:N) on exit.
   *>
   *> The routine computes C, S, R, and optionally the unitary
   *> transformation matrices U, V and Q.
   *>
   *> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
   *> A and B implicitly gives the SVD of A*inv(B):
   *>                      A*inv(B) = U*(D1*inv(D2))*V**H.
   *> If ( A**H,B**H)**H has orthnormal columns, then the GSVD of A and B is also
   *> equal to the CS decomposition of A and B. Furthermore, the GSVD can
   *> be used to derive the solution of the eigenvalue problem:
   *>                      A**H*A x = lambda* B**H*B x.
   *> In some literature, the GSVD of A and B is presented in the form
   *>                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
   *> where U and V are orthogonal and X is nonsingular, and D1 and D2 are
   *> ``diagonal''.  The former GSVD form can be converted to the latter
   *> form by taking the nonsingular matrix X as
   *>
   *>                       X = Q*(  I   0    )
   *>                             (  0 inv(R) )
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBU
   *> \verbatim
   *>          JOBU is CHARACTER*1
   *>          = 'U':  Unitary matrix U is computed;
   *>          = 'N':  U is not computed.
   *> \endverbatim
   *>
   *> \param[in] JOBV
   *> \verbatim
   *>          JOBV is CHARACTER*1
   *>          = 'V':  Unitary matrix V is computed;
   *>          = 'N':  V is not computed.
   *> \endverbatim
   *>
   *> \param[in] JOBQ
   *> \verbatim
   *>          JOBQ is CHARACTER*1
   *>          = 'Q':  Unitary matrix Q is computed;
   *>          = 'N':  Q is not computed.
   *> \endverbatim
   *>
   *> \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 matrices A and B.  N >= 0.
   *> \endverbatim
   *>
   *> \param[in] P
   *> \verbatim
   *>          P is INTEGER
   *>          The number of rows of the matrix B.  P >= 0.
   *> \endverbatim
   *>
   *> \param[out] K
   *> \verbatim
   *>          K is INTEGER
   *> \endverbatim
   *>
   *> \param[out] L
   *> \verbatim
   *>          L is INTEGER
   *>
   *>          On exit, K and L specify the dimension of the subblocks
   *>          described in Purpose.
   *>          K + L = effective numerical rank of (A**H,B**H)**H.
   *> \endverbatim
   *>
   *> \param[in,out] A
   *> \verbatim
   *>          A is COMPLEX*16 array, dimension (LDA,N)
   *>          On entry, the M-by-N matrix A.
   *>          On exit, A contains the triangular matrix R, or part of R.
   *>          See Purpose for details.
   *> \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 COMPLEX*16 array, dimension (LDB,N)
   *>          On entry, the P-by-N matrix B.
   *>          On exit, B contains part of the triangular matrix R if
   *>          M-K-L < 0.  See Purpose for details.
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,P).
   *> \endverbatim
   *>
   *> \param[out] ALPHA
   *> \verbatim
   *>          ALPHA is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] BETA
   *> \verbatim
   *>          BETA is DOUBLE PRECISION array, dimension (N)
   *>
   *>          On exit, ALPHA and BETA contain the generalized singular
   *>          value pairs of A and B;
   *>            ALPHA(1:K) = 1,
   *>            BETA(1:K)  = 0,
   *>          and if M-K-L >= 0,
   *>            ALPHA(K+1:K+L) = C,
   *>            BETA(K+1:K+L)  = S,
   *>          or if M-K-L < 0,
   *>            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
   *>            BETA(K+1:M) =S, BETA(M+1:K+L) =1
   *>          and
   *>            ALPHA(K+L+1:N) = 0
   *>            BETA(K+L+1:N)  = 0
   *> \endverbatim
   *>
   *> \param[out] U
   *> \verbatim
   *>          U is COMPLEX*16 array, dimension (LDU,M)
   *>          If JOBU = 'U', U contains the M-by-M unitary matrix U.
   *>          If JOBU = 'N', U is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDU
   *> \verbatim
   *>          LDU is INTEGER
   *>          The leading dimension of the array U. LDU >= max(1,M) if
   *>          JOBU = 'U'; LDU >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[out] V
   *> \verbatim
   *>          V is COMPLEX*16 array, dimension (LDV,P)
   *>          If JOBV = 'V', V contains the P-by-P unitary matrix V.
   *>          If JOBV = 'N', V is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDV
   *> \verbatim
   *>          LDV is INTEGER
   *>          The leading dimension of the array V. LDV >= max(1,P) if
   *>          JOBV = 'V'; LDV >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[out] Q
   *> \verbatim
   *>          Q is COMPLEX*16 array, dimension (LDQ,N)
   *>          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
   *>          If JOBQ = 'N', Q is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDQ
   *> \verbatim
   *>          LDQ is INTEGER
   *>          The leading dimension of the array Q. LDQ >= max(1,N) if
   *>          JOBQ = 'Q'; LDQ >= 1 otherwise.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension (max(3*N,M,P)+N)
   *> \endverbatim
   *>
   *> \param[out] RWORK
   *> \verbatim
   *>          RWORK is DOUBLE PRECISION array, dimension (2*N)
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (N)
   *>          On exit, IWORK stores the sorting information. More
   *>          precisely, the following loop will sort ALPHA
   *>             for I = K+1, min(M,K+L)
   *>                 swap ALPHA(I) and ALPHA(IWORK(I))
   *>             endfor
   *>          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
   *> \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 = 1, the Jacobi-type procedure failed to
   *>                converge.  For further details, see subroutine ZTGSJA.
   *> \endverbatim
   *
   *> \par Internal Parameters:
   *  =========================
   *>
   *> \verbatim
   *>  TOLA    DOUBLE PRECISION
   *>  TOLB    DOUBLE PRECISION
   *>          TOLA and TOLB are the thresholds to determine the effective
   *>          rank of (A**H,B**H)**H. Generally, they are set to
   *>                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,
   *>                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
   *>          The size of TOLA and TOLB may affect the size of backward
   *>          errors of the decomposition.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16OTHERsing
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ming Gu and Huan Ren, Computer Science Division, University of
   *>     California at Berkeley, USA
   *>
   *  =====================================================================
       SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,        SUBROUTINE ZGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B,
      $                   LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,       $                   LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK,
      $                   RWORK, IWORK, INFO )       $                   RWORK, IWORK, INFO )
 *  *
 *  -- LAPACK driver routine (version 3.3.1) --  *  -- LAPACK driver routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          JOBQ, JOBU, JOBV        CHARACTER          JOBQ, JOBU, JOBV
Line 18 Line 351
      $                   U( LDU, * ), V( LDV, * ), WORK( * )       $                   U( LDU, * ), V( LDV, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZGGSVD computes the generalized singular value decomposition (GSVD)  
 *  of an M-by-N complex matrix A and P-by-N complex matrix B:  
 *  
 *        U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )  
 *  
 *  where U, V and Q are unitary matrices.  
 *  Let K+L = the effective numerical rank of the  
 *  matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper  
 *  triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"  
 *  matrices and of the following structures, respectively:  
 *  
 *  If M-K-L >= 0,  
 *  
 *                      K  L  
 *         D1 =     K ( I  0 )  
 *                  L ( 0  C )  
 *              M-K-L ( 0  0 )  
 *  
 *                    K  L  
 *         D2 =   L ( 0  S )  
 *              P-L ( 0  0 )  
 *  
 *                  N-K-L  K    L  
 *    ( 0 R ) = K (  0   R11  R12 )  
 *              L (  0    0   R22 )  
 *  where  
 *  
 *    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),  
 *    S = diag( BETA(K+1),  ... , BETA(K+L) ),  
 *    C**2 + S**2 = I.  
 *  
 *    R is stored in A(1:K+L,N-K-L+1:N) on exit.  
 *  
 *  If M-K-L < 0,  
 *  
 *                    K M-K K+L-M  
 *         D1 =   K ( I  0    0   )  
 *              M-K ( 0  C    0   )  
 *  
 *                      K M-K K+L-M  
 *         D2 =   M-K ( 0  S    0  )  
 *              K+L-M ( 0  0    I  )  
 *                P-L ( 0  0    0  )  
 *  
 *                     N-K-L  K   M-K  K+L-M  
 *    ( 0 R ) =     K ( 0    R11  R12  R13  )  
 *                M-K ( 0     0   R22  R23  )  
 *              K+L-M ( 0     0    0   R33  )  
 *  
 *  where  
 *  
 *    C = diag( ALPHA(K+1), ... , ALPHA(M) ),  
 *    S = diag( BETA(K+1),  ... , BETA(M) ),  
 *    C**2 + S**2 = I.  
 *  
 *    (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored  
 *    ( 0  R22 R23 )  
 *    in B(M-K+1:L,N+M-K-L+1:N) on exit.  
 *  
 *  The routine computes C, S, R, and optionally the unitary  
 *  transformation matrices U, V and Q.  
 *  
 *  In particular, if B is an N-by-N nonsingular matrix, then the GSVD of  
 *  A and B implicitly gives the SVD of A*inv(B):  
 *                       A*inv(B) = U*(D1*inv(D2))*V**H.  
 *  If ( A**H,B**H)**H has orthnormal columns, then the GSVD of A and B is also  
 *  equal to the CS decomposition of A and B. Furthermore, the GSVD can  
 *  be used to derive the solution of the eigenvalue problem:  
 *                       A**H*A x = lambda* B**H*B x.  
 *  In some literature, the GSVD of A and B is presented in the form  
 *                   U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )  
 *  where U and V are orthogonal and X is nonsingular, and D1 and D2 are  
 *  ``diagonal''.  The former GSVD form can be converted to the latter  
 *  form by taking the nonsingular matrix X as  
 *  
 *                        X = Q*(  I   0    )  
 *                              (  0 inv(R) )  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBU    (input) CHARACTER*1  
 *          = 'U':  Unitary matrix U is computed;  
 *          = 'N':  U is not computed.  
 *  
 *  JOBV    (input) CHARACTER*1  
 *          = 'V':  Unitary matrix V is computed;  
 *          = 'N':  V is not computed.  
 *  
 *  JOBQ    (input) CHARACTER*1  
 *          = 'Q':  Unitary matrix Q is computed;  
 *          = 'N':  Q is not computed.  
 *  
 *  M       (input) INTEGER  
 *          The number of rows of the matrix A.  M >= 0.  
 *  
 *  N       (input) INTEGER  
 *          The number of columns of the matrices A and B.  N >= 0.  
 *  
 *  P       (input) INTEGER  
 *          The number of rows of the matrix B.  P >= 0.  
 *  
 *  K       (output) INTEGER  
 *  L       (output) INTEGER  
 *          On exit, K and L specify the dimension of the subblocks  
 *          described in Purpose.  
 *          K + L = effective numerical rank of (A**H,B**H)**H.  
 *  
 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)  
 *          On entry, the M-by-N matrix A.  
 *          On exit, A contains the triangular matrix R, or part of R.  
 *          See Purpose for details.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1,M).  
 *  
 *  B       (input/output) COMPLEX*16 array, dimension (LDB,N)  
 *          On entry, the P-by-N matrix B.  
 *          On exit, B contains part of the triangular matrix R if  
 *          M-K-L < 0.  See Purpose for details.  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B. LDB >= max(1,P).  
 *  
 *  ALPHA   (output) DOUBLE PRECISION array, dimension (N)  
 *  BETA    (output) DOUBLE PRECISION array, dimension (N)  
 *          On exit, ALPHA and BETA contain the generalized singular  
 *          value pairs of A and B;  
 *            ALPHA(1:K) = 1,  
 *            BETA(1:K)  = 0,  
 *          and if M-K-L >= 0,  
 *            ALPHA(K+1:K+L) = C,  
 *            BETA(K+1:K+L)  = S,  
 *          or if M-K-L < 0,  
 *            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0  
 *            BETA(K+1:M) = S, BETA(M+1:K+L) = 1  
 *          and  
 *            ALPHA(K+L+1:N) = 0  
 *            BETA(K+L+1:N)  = 0  
 *  
 *  U       (output) COMPLEX*16 array, dimension (LDU,M)  
 *          If JOBU = 'U', U contains the M-by-M unitary matrix U.  
 *          If JOBU = 'N', U is not referenced.  
 *  
 *  LDU     (input) INTEGER  
 *          The leading dimension of the array U. LDU >= max(1,M) if  
 *          JOBU = 'U'; LDU >= 1 otherwise.  
 *  
 *  V       (output) COMPLEX*16 array, dimension (LDV,P)  
 *          If JOBV = 'V', V contains the P-by-P unitary matrix V.  
 *          If JOBV = 'N', V is not referenced.  
 *  
 *  LDV     (input) INTEGER  
 *          The leading dimension of the array V. LDV >= max(1,P) if  
 *          JOBV = 'V'; LDV >= 1 otherwise.  
 *  
 *  Q       (output) COMPLEX*16 array, dimension (LDQ,N)  
 *          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.  
 *          If JOBQ = 'N', Q is not referenced.  
 *  
 *  LDQ     (input) INTEGER  
 *          The leading dimension of the array Q. LDQ >= max(1,N) if  
 *          JOBQ = 'Q'; LDQ >= 1 otherwise.  
 *  
 *  WORK    (workspace) COMPLEX*16 array, dimension (max(3*N,M,P)+N)  
 *  
 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)  
 *  
 *  IWORK   (workspace/output) INTEGER array, dimension (N)  
 *          On exit, IWORK stores the sorting information. More  
 *          precisely, the following loop will sort ALPHA  
 *             for I = K+1, min(M,K+L)  
 *                 swap ALPHA(I) and ALPHA(IWORK(I))  
 *             endfor  
 *          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).  
 *  
 *  INFO    (output) INTEGER  
 *          = 0:  successful exit.  
 *          < 0:  if INFO = -i, the i-th argument had an illegal value.  
 *          > 0:  if INFO = 1, the Jacobi-type procedure failed to  
 *                converge.  For further details, see subroutine ZTGSJA.  
 *  
 *  Internal Parameters  
 *  ===================  
 *  
 *  TOLA    DOUBLE PRECISION  
 *  TOLB    DOUBLE PRECISION  
 *          TOLA and TOLB are the thresholds to determine the effective  
 *          rank of (A**H,B**H)**H. Generally, they are set to  
 *                   TOLA = MAX(M,N)*norm(A)*MAZHEPS,  
 *                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.  
 *          The size of TOLA and TOLB may affect the size of backward  
 *          errors of the decomposition.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  2-96 Based on modifications by  
 *     Ming Gu and Huan Ren, Computer Science Division, University of  
 *     California at Berkeley, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Local Scalars ..  *     .. Local Scalars ..
Removed from v.1.8  
changed lines
  Added in v.1.9

CVSweb interface <joel.bertrand@systella.fr>