NAME

       sggsvp3.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sggsvp3 (JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU,
           V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
           SGGSVP3

Function/Subroutine Documentation

   subroutine sggsvp3 (character JOBU, character JOBV, character JOBQ, integer M, integer P,
       integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer
       LDB, real TOLA, real TOLB, integer K, integer L, real, dimension( ldu, * ) U, integer LDU,
       real, dimension( ldv, * ) V, integer LDV, real, dimension( ldq, * ) Q, integer LDQ,
       integer, dimension( * ) IWORK, real, dimension( * ) TAU, real, dimension( * ) WORK,
       integer LWORK, integer INFO)
       SGGSVP3

       Purpose:

            SGGSVP3 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
            SGGSVD3.

       Parameters:
           JOBU

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

           JOBV

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

           JOBQ

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

           A

                     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

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

           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 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

                     K is INTEGER

           L

                     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

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

           IWORK

                     IWORK is INTEGER array, dimension (N)

           TAU

                     TAU is REAL array, dimension (N)

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is issued by XERBLA.

           INFO

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           August 2015

       Further Details:

             The subroutine uses LAPACK subroutine SGEQP3 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.

             SGGSVP3 replaces the deprecated subroutine SGGSVP.

Author

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