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NAME

       stgsja.f -

SYNOPSIS

   Functions/Subroutines
       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

Function/Subroutine Documentation

   subroutine stgsja (characterJOBU, characterJOBV, characterJOBQ, integerM, integerP, integerN,
       integerK, integerL, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B,
       integerLDB, realTOLA, realTOLB, real, dimension( * )ALPHA, real, dimension( * )BETA, real,
       dimension( ldu, * )U, integerLDU, real, dimension( ldv, * )V, integerLDV, real, dimension(
       ldq, * )Q, integerLDQ, real, dimension( * )WORK, integerNCYCLE, integerINFO)
       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.

       Date:
           November 2011

       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.

       Definition at line 377 of file stgsja.f.

Author

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