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NAME
     sggsvp - 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( * )



     #include <sunperf.h>

     void sggsvp(char jobu, char jobv, char jobq, int m,  int  p,
               int  n,  float  *sa,  int lda, float *sb, int ldb,
               float tola, float tolb, int *k, int *l, float *su,
               int  ldu,  float  *v,  int ldv, float *q, int ldq,
               float *tau, int *info) ;

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  non-
     singular 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  con-
               tains   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  con-
               tains  the triangular matrix described in the Pur-
               pose 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  thres-
               holds 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 Pur-
               pose.   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 ille-
               gal value.

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

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