SGGSVP(3)      LAPACK routine of NEC Numeric Library Collection      SGGSVP(3)



NAME
       SGGSVP

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)



PURPOSE
            SGGSVP computes orthogonal matrices U, V and Q such that

                               N-K-L  K    L
             U**T*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
                            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**T*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**T,B**T)**T.

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




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

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

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

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

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

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

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

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

           TOLA      (input)
                     TOLA is REAL

           TOLB      (input)
                     TOLB is 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)
                     K is INTEGER

           L         (output)
                     L is INTEGER

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

           U         (output)
                     U is REAL array, dimension (LDU,M)
                     If JOBU = 'U', U contains the orthogonal matrix U.
                     If JOBU = 'N', U is not referenced.

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

           V         (output)
                     V is REAL array, dimension (LDV,P)
                     If JOBV = 'V', V contains the orthogonal matrix V.
                     If JOBV = 'N', V is not referenced.

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

           Q         (output)
                     Q is REAL array, dimension (LDQ,N)
                     If JOBQ = 'Q', Q contains the orthogonal matrix Q.
                     If JOBQ = 'N', Q is not referenced.

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

           IWORK     (output)
                     IWORK is INTEGER array, dimension (N)

           TAU       (output)
                     TAU is REAL array, dimension (N)

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

           INFO      (output)
                     INFO is 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.



LAPACK routine                  31 October 2017                      SGGSVP(3)