Provided by: liblapack-doc_3.12.0-3build1.1_all bug

NAME

       tgsja - tgsja: generalized SVD of trapezoidal matrices, step in ggsvd3

SYNOPSIS

   Functions
       subroutine ctgsja (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)
           CTGSJA
       subroutine dtgsja (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)
           DTGSJA
       subroutine stgsja (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)
           STGSJA
       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)
           ZTGSJA

Detailed Description

Function Documentation

   subroutine ctgsja (character jobu, character jobv, character jobq, integer m, integer p,
       integer n, integer k, integer l, complex, dimension( lda, * ) a, integer lda, complex,
       dimension( ldb, * ) b, integer ldb, real tola, real tolb, real, dimension( * ) alpha,
       real, dimension( * ) beta, complex, dimension( ldu, * ) u, integer ldu, complex,
       dimension( ldv, * ) v, integer ldv, complex, dimension( ldq, * ) q, integer ldq, complex,
       dimension( * ) work, integer ncycle, integer info)
       CTGSJA

       Purpose:

            CTGSJA 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 CGGSVP
            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.

       Parameters
           JOBU

                     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

                     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

                     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

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

           P

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

           N

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

           K

                     K is INTEGER

           L

                     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 CTGSJA.
                     See Further Details.

           A

                     A is COMPLEX 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

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

           B

                     B is COMPLEX 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

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

           TOLA

                     TOLA is REAL

           TOLB

                     TOLB is REAL

                     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)*MACHEPS,
                         TOLB = MAX(P,N)*norm(B)*MACHEPS.

           ALPHA

                     ALPHA is REAL array, dimension (N)

           BETA

                     BETA is REAL 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
                       BETA(K+L+1:N)  = 0.

           U

                     U is COMPLEX array, dimension (LDU,M)
                     On entry, if JOBU = 'U', U must contain a matrix U1 (usually
                     the unitary matrix returned by CGGSVP).
                     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

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

           V

                     V is COMPLEX array, dimension (LDV,P)
                     On entry, if JOBV = 'V', V must contain a matrix V1 (usually
                     the unitary matrix returned by CGGSVP).
                     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

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

           Q

                     Q is COMPLEX array, dimension (LDQ,N)
                     On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
                     the unitary matrix returned by CGGSVP).
                     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

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

           WORK

                     WORK is COMPLEX array, dimension (2*N)

           NCYCLE

                     NCYCLE is INTEGER
                     The number of cycles required for convergence.

           INFO

                     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.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             CTGSJA 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.

   subroutine dtgsja (character jobu, character jobv, character jobq, integer m, integer p,
       integer n, integer k, integer l, double precision, dimension( lda, * ) a, integer lda,
       double precision, dimension( ldb, * ) b, integer ldb, double precision tola, double
       precision tolb, double precision, dimension( * ) alpha, double precision, dimension( * )
       beta, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension(
       ldv, * ) v, integer ldv, double precision, dimension( ldq, * ) q, integer ldq, double
       precision, dimension( * ) work, integer ncycle, integer info)
       DTGSJA

       Purpose:

            DTGSJA computes the generalized singular value decomposition (GSVD)
            of two real 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 DGGSVP
            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**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),

            where U, V and Q are orthogonal 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 orthogonal 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.

       Parameters
           JOBU

                     JOBU is CHARACTER*1
                     = 'U':  U must contain an orthogonal matrix U1 on entry, and
                             the product U1*U is returned;
                     = 'I':  U is initialized to the unit matrix, and the
                             orthogonal matrix U is returned;
                     = 'N':  U is not computed.

           JOBV

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

           JOBQ

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

           M

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

           P

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

           N

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

           K

                     K is INTEGER

           L

                     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 DTGSJA.
                     See Further Details.

           A

                     A is DOUBLE PRECISION 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

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

           B

                     B is DOUBLE PRECISION 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

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

           TOLA

                     TOLA is DOUBLE PRECISION

           TOLB

                     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

                     ALPHA is DOUBLE PRECISION array, dimension (N)

           BETA

                     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

                     U is DOUBLE PRECISION array, dimension (LDU,M)
                     On entry, if JOBU = 'U', U must contain a matrix U1 (usually
                     the orthogonal matrix returned by DGGSVP).
                     On exit,
                     if JOBU = 'I', U contains the orthogonal matrix U;
                     if JOBU = 'U', U contains the product U1*U.
                     If JOBU = 'N', U is not referenced.

           LDU

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

           V

                     V is DOUBLE PRECISION array, dimension (LDV,P)
                     On entry, if JOBV = 'V', V must contain a matrix V1 (usually
                     the orthogonal matrix returned by DGGSVP).
                     On exit,
                     if JOBV = 'I', V contains the orthogonal matrix V;
                     if JOBV = 'V', V contains the product V1*V.
                     If JOBV = 'N', V is not referenced.

           LDV

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

           Q

                     Q is DOUBLE PRECISION array, dimension (LDQ,N)
                     On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
                     the orthogonal matrix returned by DGGSVP).
                     On exit,
                     if JOBQ = 'I', Q contains the orthogonal matrix Q;
                     if JOBQ = 'Q', Q contains the product Q1*Q.
                     If JOBQ = 'N', Q is not referenced.

           LDQ

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

           WORK

                     WORK is DOUBLE PRECISION array, dimension (2*N)

           NCYCLE

                     NCYCLE is INTEGER
                     The number of cycles required for convergence.

           INFO

                     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..fi

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             DTGSJA 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**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

             where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
             of Z.  C1 and S1 are diagonal matrices satisfying

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

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

   subroutine stgsja (character jobu, character jobv, character jobq, integer m, integer p,
       integer n, integer k, integer l, real, dimension( lda, * ) a, integer lda, real,
       dimension( ldb, * ) b, integer ldb, real tola, real tolb, real, dimension( * ) alpha,
       real, dimension( * ) beta, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv,
       * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) work,
       integer ncycle, integer info)
       STGSJA

       Purpose:

            STGSJA computes the generalized singular value decomposition (GSVD)
            of two real 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 SGGSVP
            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**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),

            where U, V and Q are orthogonal 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 orthogonal 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.

       Parameters
           JOBU

                     JOBU is CHARACTER*1
                     = 'U':  U must contain an orthogonal matrix U1 on entry, and
                             the product U1*U is returned;
                     = 'I':  U is initialized to the unit matrix, and the
                             orthogonal matrix U is returned;
                     = 'N':  U is not computed.

           JOBV

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

           JOBQ

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

           M

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

           P

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

           N

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

           K

                     K is INTEGER

           L

                     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 STGSJA.
                     See Further Details.

           A

                     A is REAL 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

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

           B

                     B is REAL 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

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

           TOLA

                     TOLA is REAL

           TOLB

                     TOLB is REAL

                     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)*MACHEPS,
                         TOLB = max(P,N)*norm(B)*MACHEPS.

           ALPHA

                     ALPHA is REAL array, dimension (N)

           BETA

                     BETA is REAL 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

                     U is REAL array, dimension (LDU,M)
                     On entry, if JOBU = 'U', U must contain a matrix U1 (usually
                     the orthogonal matrix returned by SGGSVP).
                     On exit,
                     if JOBU = 'I', U contains the orthogonal matrix U;
                     if JOBU = 'U', U contains the product U1*U.
                     If JOBU = 'N', U is not referenced.

           LDU

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

           V

                     V is REAL array, dimension (LDV,P)
                     On entry, if JOBV = 'V', V must contain a matrix V1 (usually
                     the orthogonal matrix returned by SGGSVP).
                     On exit,
                     if JOBV = 'I', V contains the orthogonal matrix V;
                     if JOBV = 'V', V contains the product V1*V.
                     If JOBV = 'N', V is not referenced.

           LDV

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

           Q

                     Q is REAL array, dimension (LDQ,N)
                     On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
                     the orthogonal matrix returned by SGGSVP).
                     On exit,
                     if JOBQ = 'I', Q contains the orthogonal matrix Q;
                     if JOBQ = 'Q', Q contains the product Q1*Q.
                     If JOBQ = 'N', Q is not referenced.

           LDQ

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

           WORK

                     WORK is REAL array, dimension (2*N)

           NCYCLE

                     NCYCLE is INTEGER
                     The number of cycles required for convergence.

           INFO

                     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..fi

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:

             STGSJA 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**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

             where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose
             of Z.  C1 and S1 are diagonal matrices satisfying

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

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

   subroutine ztgsja (character jobu, character jobv, character jobq, integer m, integer p,
       integer n, integer k, integer l, complex*16, dimension( lda, * ) a, integer lda,
       complex*16, dimension( ldb, * ) b, integer ldb, double precision tola, double precision
       tolb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta,
       complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer
       ldv, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) work,
       integer ncycle, integer info)
       ZTGSJA

       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.

       Parameters
           JOBU

                     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

                     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

                     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

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

           P

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

           N

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

           K

                     K is INTEGER

           L

                     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

                     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

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

           B

                     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

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

           TOLA

                     TOLA is DOUBLE PRECISION

           TOLB

                     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

                     ALPHA is DOUBLE PRECISION array, dimension (N)

           BETA

                     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

                     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

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

           V

                     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

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

           Q

                     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

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

           WORK

                     WORK is COMPLEX*16 array, dimension (2*N)

           NCYCLE

                     NCYCLE is INTEGER
                     The number of cycles required for convergence.

           INFO

                     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.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       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.

Author

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