IRIX 6.2 Development Libraries/SGI IRIX 6.2 Development Libraries.iso/dist/complib.idb/usr/share/catman/p_man/cat3/complib/dtgsja.z/dtgsja

Text File | 1996-03-14 | 11KB | 331 lines

DDDDTTTTGGGGSSSSJJJJAAAA((((3333FFFF))))                                                          DDDDTTTTGGGGSSSSJJJJAAAA((((3333FFFF))))



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     DTGSJA - compute the generalized singular value decomposition (GSVD) of
     two real upper triangular (or trapezoidal) matrices A and B

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     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 )

         CHARACTER      JOBQ, JOBU, JOBV

         INTEGER        INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, NCYCLE, P

         DOUBLE         PRECISION TOLA, TOLB

         DOUBLE         PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA(
                        * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

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

     where U, V and Q are orthogonal matrices, Z' denotes the transpose of Z,
     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,



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                         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

               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.






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

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

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

     K       (input) INTEGER
             L       (input) 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       (input/output) 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     (input) INTEGER
             The leading dimension of the array A. LDA >= max(1,M).

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





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     TOLA    (input) DOUBLE PRECISION
             TOLB    (input) 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) DOUBLE PRECISION array, dimension (N)
             BETA    (output) 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) 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     (input) INTEGER
             The leading dimension of the array U. LDU >= max(1,M) if JOBU =
             'U'; LDU >= 1 otherwise.

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

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

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






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DDDDTTTTGGGGSSSSJJJJAAAA((((3333FFFF))))                                                          DDDDTTTTGGGGSSSSJJJJAAAA((((3333FFFF))))



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

     INFO    (output) 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.

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     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 ===============

             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'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,

             where U1, V1 and Q1 are orthogonal matrix, and Z' 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.



























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