IRIX Base Documentation 2002 November/SGI IRIX Base Documentation 2002 November.iso/usr/share/catman/p_man/cat3/SCSL/dggsvd.z/dggsvd

Text File | 2002-10-03 | 10.3 KB | 331 lines

DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))                                                          DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))



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     DGGSVD - compute the generalized singular value decomposition (GSVD) of
     an M-by-N real matrix A and P-by-N real matrix B

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     SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB,
                        ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO
                        )

         CHARACTER      JOBQ, JOBU, JOBV

         INTEGER        INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P

         INTEGER        IWORK( * )

         DOUBLE         PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA(
                        * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

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     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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     DGGSVD computes the generalized singular value decomposition (GSVD) of an
     M-by-N real matrix A and P-by-N real matrix B:
         U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )

     where U, V and Q are orthogonal matrices, and Z' is the transpose of Z.
     Let K+L = the effective numerical rank of the matrix (A',B')', 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 )



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DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))                                                          DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))



                     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 orthogonal
     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'.
     If ( A',B')' has orthonormal 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'*A x = lambda* B'*B x.
     In some literature, the GSVD of A and B is presented in the form
                      U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
     where U and V are orthogonal and X is nonsingular, D1 and D2 are
     ``diagonal''.  The former GSVD form can be converted to the latter form



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DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))                                                          DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))



     by taking the nonsingular matrix X as

                          X = Q*( I   0    )
                                ( 0 inv(R) ).


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     JOBU    (input) CHARACTER*1
             = 'U':  Orthogonal matrix U is computed;
             = 'N':  U is not computed.

     JOBV    (input) CHARACTER*1
             = 'V':  Orthogonal matrix V is computed;
             = 'N':  V is not computed.

     JOBQ    (input) CHARACTER*1
             = 'Q':  Orthogonal 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 the Purpose section.  K + L =
             effective numerical rank of (A',B')'.

     A       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDB,N)
             On entry, the P-by-N matrix B.  On exit, B contains the
             triangular matrix R if M-K-L < 0.  See Purpose for details.

     LDB     (input) INTEGER
             The leading dimension of the array B. LDA >= 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,



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DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))                                                          DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))



             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) DOUBLE PRECISION array, dimension (LDU,M)
             If JOBU = 'U', U contains the M-by-M orthogonal 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) DOUBLE PRECISION array, dimension (LDV,P)
             If JOBV = 'V', V contains the P-by-P orthogonal 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) DOUBLE PRECISION array, dimension (LDQ,N)
             If JOBQ = 'Q', Q contains the N-by-N orthogonal 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) DOUBLE PRECISION array,
             dimension (max(3*N,M,P)+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 DTGSJA.

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     TOLA    DOUBLE PRECISION
             TOLB    DOUBLE PRECISION TOLA and TOLB are the thresholds to
             determine the effective rank of (A',B')'. 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.




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DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))                                                          DDDDGGGGGGGGSSSSVVVVDDDD((((3333SSSS))))



             Further Details ===============

             2-96 Based on modifications by Ming Gu and Huan Ren, Computer
             Science Division, University of California at Berkeley, USA

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     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.














































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