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NAME
     sggrqf - compute a generalized RQ factorization of an M-by-N
     matrix A and a P-by-N matrix B

SYNOPSIS
     SUBROUTINE SGGRQF( M, P, N, A,  LDA,  TAUA,  B,  LDB,  TAUB,
               WORK, LWORK, INFO )

     INTEGER INFO, LDA, LDB, LWORK, M, N, P

     REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( *
               )



     #include <sunperf.h>

     void sggrqf(int m, int p, int n, float *sa, int  lda,  float
               *taua, float *sb, int ldb, float *taub, int *info)
               ;

PURPOSE
     SGGRQF computes a generalized RQ factorization of an  M-by-N
     matrix A and a P-by-N matrix B:

                 A = R*Q,        B = Z*T*Q,

     where Q is an  N-by-N  orthogonal  matrix,  Z  is  a  P-by-P
     orthogonal matrix, and R and T assume one of the forms:

     if M<=N,  R = ( 0  R12 ) M, or if M > N,  R = ( R11 ) M-N,
                    N-M  M                         ( R21 ) N
                                                      N

     where R12 or R21 is upper triangular, and

     if P>=N,  T = ( T11 ) N  , or if P < N,  T = ( T11  T12 ) P,
                   (  0  ) P-N                       P   N-P
                      N

     where T11 is upper triangular.

     In particular, if B is square and nonsingular, the GRQ  fac-
     torization  of A and B implicitly gives the RQ factorization
     of A*inv(B):

                  A*inv(B) = (R*inv(T))*Z'

     where inv(B) denotes the inverse of the  matrix  B,  and  Z'
     denotes the transpose of the matrix Z.


ARGUMENTS
     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, if  M  <=
               N,  the  upper  triangle  of the subarray A(1:M,N-
               M+1:N) contains the M-by-M upper triangular matrix
               R;  if M > N, the elements on and above the (M-N)-
               th  subdiagonal  contain  the  M-by-N  upper  tra-
               pezoidal  matrix  R;  the remaining elements, with
               the array TAUA, represent the orthogonal matrix  Q
               as a product of elementary reflectors (see Further
               Details).

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

     TAUA      (output) REAL array, dimension (min(M,N))
               The scalar factors of  the  elementary  reflectors
               which  represent  the  orthogonal  matrix  Q  (see
               Further  Details).   B        (input/output)  REAL
               array,  dimension  (LDB,N)  On  entry,  the P-by-N
               matrix B.  On exit, the elements on and above  the
               diagonal  of  the  array contain the min(P,N)-by-N
               upper trapezoidal matrix T (T is upper  triangular
               if  P >= N); the elements below the diagonal, with
               the array TAUB, represent the orthogonal matrix  Z
               as a product of elementary reflectors (see Further
               Details).  LDB      (input)  INTEGER  The  leading
               dimension of the array B. LDB >= max(1,P).

     TAUB      (output) REAL array, dimension (min(P,N))
               The scalar factors of  the  elementary  reflectors
               which  represent  the  orthogonal  matrix  Z  (see
               Further Details).  WORK    (workspace/output) REAL
               array,  dimension  (LWORK)  On  exit, if INFO = 0,
               WORK(1) returns the optimal LWORK.

     LWORK     (input) INTEGER
               The  dimension  of  the  array  WORK.   LWORK   >=
               max(1,N,M,P).   For  optimum  performance LWORK >=
               max(N,M,P)*max(NB1,NB2,NB3),  where  NB1  is   the
               optimal  blocksize  for the RQ factorization of an
               M-by-N matrix, NB2 is the  optimal  blocksize  for
               the  QR  factorization of a P-by-N matrix, and NB3
               is the optimal blocksize for a call of SORMRQ.

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  if INF0= -i, the i-th argument had an  ille-
               gal value.

FURTHER DETAILS
     The matrix Q is  represented  as  a  product  of  elementary
     reflectors

        Q = H(1) H(2) . . . H(k), where k = min(m,n).

     Each H(i) has the form

        H(i) = I - taua * v * v'

     where taua is a real scalar, and v is a real vector with
     v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on
     exit in A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
     To form Q explicitly, use LAPACK subroutine SORGRQ.
     To use Q to update another  matrix,  use  LAPACK  subroutine
     SORMRQ.

     The matrix Z is  represented  as  a  product  of  elementary
     reflectors

        Z = H(1) H(2) . . . H(k), where k = min(p,n).

     Each H(i) has the form

        H(i) = I - taub * v * v'

     where taub is a real scalar, and v is a real vector with
     v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is  stored  on  exit  in
     B(i+1:p,i), and taub in TAUB(i).
     To form Z explicitly, use LAPACK subroutine SORGQR.
     To use Z to update another  matrix,  use  LAPACK  subroutine
     SORMQR.

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