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zggsvd (3)
  • >> zggsvd (3) ( Solaris man: Библиотечные вызовы )
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    NAME
         zggsvd - compute the generalized singular  value  decomposi-
         tion (GSVD) of an M-by-N complex matrix A and P-by-N complex
         matrix B
    
    SYNOPSIS
         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 )
    
         CHARACTER JOBQ, JOBU, JOBV
    
         INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P
    
         INTEGER IWORK( * )
    
         DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
    
         COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU,  *
                   ), V( LDV, * ), WORK( * )
    
    
    
         #include <sunperf.h>
    
         void zggsvd(char jobu, char jobv, char jobq, int m,  int  n,
                   int p, int *k, int *l, doublecomplex *za, int lda,
                   doublecomplex *zb, int ldb, double *dalpha, double
                   *dbeta,  doublecomplex *zu, int ldu, doublecomplex
                   *v,  int  ldv,  doublecomplex  *q,  int  ldq,  int
                   *info);
    
    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'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )
    
         where U, V and Q are unitary matrices, and Z' means the con-
         jugate  transpose  of  Z.  Let K+L = the effective numerical
         rank of the matrix (A',B')', then R is a (K+L)-by-(K+L) non-
         singular  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'.
         If ( A',B')' has orthnormal columns, then the GSVD of A  and
         B is also equal to the CS decomposition of A and B. Further-
         more, 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,  and  D1
         and  D2  are ``diagonal''.  The former GSVD form can be con-
         verted 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 Pur-
                   pose.   K  +  L  =  effective  numerical  rank  of
                   (A',B')'.
    
         A         (input/output) COMPLEX*16 array, dimension (LDA,N)
                   On entry, the M-by-N matrix A.  On  exit,  A  con-
                   tains  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  con-
                   tains  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 general-
                   ized 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) INTEGER array, dimension (N)
         INFO      (output)INTEGER
                   = 0:  successful exit.
                   < 0:  if INFO = -i, the i-th argument had an ille-
                   gal value.
                   > 0:  if  INFO  =  1,  the  Jacobi-type  procedure
                   failed to converge.  For further details, see sub-
                   routine ZTGSJA.
    
    PARAMETERS
         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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