SGGSVP (3lapack)

compute orthogonal matrices U, V and Q such that N-K-L K L U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0

SYNOPSIS

  SUBROUTINE SGGSVP( JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB,
                     K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, INFO )

      CHARACTER      JOBQ, JOBU, JOBV

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

      REAL           TOLA, TOLB

      INTEGER        IWORK( * )

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

PURPOSE

  SGGSVP computes orthogonal matrices U, V and Q such that
                L ( 0     0   A23 )
            M-K-L ( 0     0    0  )

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

                 N-K-L  K    L
   V'*B*Q =   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.  K+L = the effective numerical rank of the
  (M+P)-by-N matrix (A',B')'.  Z' denotes the transpose of Z.

  This decomposition is the preprocessing step for computing the Generalized
  Singular Value Decomposition (GSVD), see subroutine SGGSVD.

ARGUMENTS

  JOBU    (input) CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed.

  JOBV    (input) CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed.

  JOBQ    (input) CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = '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.

  A       (input/output) REAL array, dimension (LDA,N)
          On entry, the M-by-N matrix A.  On exit, A contains the triangular
          (or trapezoidal) matrix described in the Purpose section.

  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, B contains the triangular
          matrix described in the Purpose section.

  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 thresholds to determine
          the effective numerical rank of matrix B and a subblock of A.
          Generally, they are set to TOLA = MAX(M,N)*norm(A)*MACHEPS, TOLB =
          MAX(P,N)*norm(B)*MACHEPS.  The size of TOLA and TOLB may affect the
          size of backward errors of the decomposition.

  K       (output) INTEGER
          L       (output) INTEGER On exit, K and L specify the dimension of
          the subblocks described in Purpose.  K + L = effective numerical
          rank of (A',B')'.

  U       (output) REAL array, dimension (LDU,M)
          If JOBU = 'U', U contains the orthogonal matrix 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       (output) REAL array, dimension (LDV,M)
          If JOBV = 'V', V contains the orthogonal matrix 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       (output) REAL array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the orthogonal matrix 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.

  IWORK   (workspace) INTEGER array, dimension (N)

  TAU     (workspace) REAL array, dimension (N)

  WORK    (workspace) REAL array, dimension (max(3*N,M,P))

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

FURTHER DETAILS

  The subroutine uses LAPACK subroutine SGEQPF for the QR factorization with
  column pivoting to detect the effective numerical rank of the a matrix. It
  may be replaced by a better rank determination strategy.