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



NAME
       ZTGSJA

SYNOPSIS
       SUBROUTINE ZTGSJA (JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB,
           TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE,
           INFO)



PURPOSE
            ZTGSJA computes the generalized singular value decomposition (GSVD)
            of two complex upper triangular (or trapezoidal) matrices A and B.

            On entry, it is assumed that matrices A and B have the following
            forms, which may be obtained by the preprocessing subroutine ZGGSVP
            from a general M-by-N matrix A and P-by-N matrix B:

                         N-K-L  K    L
               A =    K ( 0    A12  A13 ) if M-K-L >= 0;
                      L ( 0     0   A23 )
                  M-K-L ( 0     0    0  )

                       N-K-L  K    L
               A =  K ( 0    A12  A13 ) if M-K-L < 0;
                  M-K ( 0     0   A23 )

                       N-K-L  K    L
               B =  L ( 0     0   B13 )
                  P-L ( 0     0    0  )

            where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
            upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
            otherwise A23 is (M-K)-by-L upper trapezoidal.

            On exit,

                   U**H *A*Q = D1*( 0 R ),    V**H *B*Q = D2*( 0 R ),

            where U, V and Q are unitary matrices.
            R is a nonsingular upper triangular matrix, and D1
            and D2 are ``diagonal'' matrices, which are of the following
            structures:

            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 ) K
                        L (  0    0   R22 ) L

            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.

            R = ( 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 computation of the unitary transformation matrices U, V or Q
            is optional.  These matrices may either be formed explicitly, or they
            may be postmultiplied into input matrices U1, V1, or Q1.




ARGUMENTS
           JOBU      (input)
                     JOBU is CHARACTER*1
                     = 'U':  U must contain a unitary matrix U1 on entry, and
                             the product U1*U is returned;
                     = 'I':  U is initialized to the unit matrix, and the
                             unitary matrix U is returned;
                     = 'N':  U is not computed.

           JOBV      (input)
                     JOBV is CHARACTER*1
                     = 'V':  V must contain a unitary matrix V1 on entry, and
                             the product V1*V is returned;
                     = 'I':  V is initialized to the unit matrix, and the
                             unitary matrix V is returned;
                     = 'N':  V is not computed.

           JOBQ      (input)
                     JOBQ is CHARACTER*1
                     = 'Q':  Q must contain a unitary matrix Q1 on entry, and
                             the product Q1*Q is returned;
                     = 'I':  Q is initialized to the unit matrix, and the
                             unitary matrix Q is returned;
                     = 'N':  Q is not computed.

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

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

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

           K         (input)
                     K is INTEGER

           L         (input)
                     L is INTEGER

                     K and L specify the subblocks in the input matrices A and B:
                     A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
                     of A and B, whose GSVD is going to be computed by ZTGSJA.
                     See Further Details.

           A         (input/output)
                     A is COMPLEX*16 array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, A(N-K+1:N,1:MIN(K+L,M) ) 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, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
                     a part of R.  See Purpose for details.

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

           TOLA      (input)
                     TOLA is DOUBLE PRECISION

           TOLB      (input)
                     TOLB is DOUBLE PRECISION

                     TOLA and TOLB are the convergence criteria for the Jacobi-
                     Kogbetliantz iteration procedure. Generally, they are the
                     same as used in the preprocessing step, say
                         TOLA = MAX(M,N)*norm(A)*MAZHEPS,
                         TOLB = MAX(P,N)*norm(B)*MAZHEPS.

           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) = diag(C),
                       BETA(K+1:K+L)  = diag(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.
                     Furthermore, if K+L < N,
                       ALPHA(K+L+1:N) = 0 and
                       BETA(K+L+1:N)  = 0.

           U         (input/output)
                     U is COMPLEX*16 array, dimension (LDU,M)
                     On entry, if JOBU = 'U', U must contain a matrix U1 (usually
                     the unitary matrix returned by ZGGSVP).
                     On exit,
                     if JOBU = 'I', U contains the unitary matrix U;
                     if JOBU = 'U', U contains the product U1*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         (input/output)
                     V is COMPLEX*16 array, dimension (LDV,P)
                     On entry, if JOBV = 'V', V must contain a matrix V1 (usually
                     the unitary matrix returned by ZGGSVP).
                     On exit,
                     if JOBV = 'I', V contains the unitary matrix V;
                     if JOBV = 'V', V contains the product V1*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         (input/output)
                     Q is COMPLEX*16 array, dimension (LDQ,N)
                     On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
                     the unitary matrix returned by ZGGSVP).
                     On exit,
                     if JOBQ = 'I', Q contains the unitary matrix Q;
                     if JOBQ = 'Q', Q contains the product Q1*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 (2*N)

           NCYCLE    (output)
                     NCYCLE is INTEGER
                     The number of cycles required for convergence.

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     = 1:  the procedure does not converge after MAXIT cycles.



       Internal Parameters:


             MAXIT   INTEGER
                     MAXIT specifies the total loops that the iterative procedure
                     may take. If after MAXIT cycles, the routine fails to
                     converge, we return INFO = 1.






FURTHER DETAILS
             ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
             min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
             matrix B13 to the form:

                      U1**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,

             where U1, V1 and Q1 are unitary matrix.
             C1 and S1 are diagonal matrices satisfying

                           C1**2 + S1**2 = I,

             and R1 is an L-by-L nonsingular upper triangular matrix.



LAPACK routine                  31 October 2017                      ZTGSJA(3)