CXML

STGSJA (3lapack)


SYNOPSIS

  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 )

      CHARACTER      JOBQ, JOBU, JOBV

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

      REAL           TOLA, TOLB

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

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'*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,

                      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.

ARGUMENTS

  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 STGSJA.  See Further
          details.

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

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

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

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

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

  WORK    (workspace) REAL array, dimension (2*N)

  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.

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

          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'*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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