ZGGSVD(3)      LAPACK routine of NEC Numeric Library Collection      ZGGSVD(3)



NAME
       ZGGSVD

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)



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

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

           JOBQ      (input)
                     JOBQ is CHARACTER*1
                     = 'Q':  Unitary matrix Q is computed;
                     = 'N':  Q is not computed.

           M         (input)
                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N         (input)
                     N is INTEGER
                     The number of columns of the matrices A and B.  N >= 0.

           P         (input)
                     P is INTEGER
                     The number of rows of the matrix B.  P >= 0.

           K         (output)
                     K is INTEGER

           L         (output)
                     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.

           A         (input/output)
                     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.

           LDA       (input)
                     LDA is INTEGER
                     The leading dimension of the array A. LDA >= max(1,M).

           B         (input/output)
                     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.

           LDB       (input)
                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,P).

           ALPHA     (output)
                     ALPHA is DOUBLE PRECISION array, dimension (N)

           BETA      (output)
                     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

           U         (output)
                     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.

           LDU       (input)
                     LDU is INTEGER
                     The leading dimension of the array U. LDU >= max(1,M) if
                     JOBU = 'U'; LDU >= 1 otherwise.

           V         (output)
                     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.

           LDV       (input)
                     LDV is INTEGER
                     The leading dimension of the array V. LDV >= max(1,P) if
                     JOBV = 'V'; LDV >= 1 otherwise.

           Q         (output)
                     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.

           LDQ       (input)
                     LDQ is INTEGER
                     The leading dimension of the array Q. LDQ >= max(1,N) if
                     JOBQ = 'Q'; LDQ >= 1 otherwise.

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (max(3*N,M,P)+N)

           RWORK     (output)
                     RWORK is DOUBLE PRECISION array, dimension (2*N)

           IWORK     (output)
                     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).

           INFO      (output)
                     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.



       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.



LAPACK routine                  31 October 2017                      ZGGSVD(3)