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


SYNOPSIS

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

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

           DOUBLE         PRECISION  A(  LDA,  *  ), B( LDB, * ),
                          TAUA( * ), TAUB( * ), WORK( * )


PURPOSE

       DGGRQF 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
       factorization  of  A and B implicitly gives the RQ factor­
       ization 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.

               The  number  of columns of the matrices A and B. N
               >= 0.

       A       (input/output) DOUBLE PRECISION 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 trape­
               zoidal 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) DOUBLE PRECISION array, dimension
               (min(M,N))
               The  scalar  factors  of the elementary reflectors
               which represent the orthogonal matrix Q (see  Fur­
               ther Details).  B       (input/output) DOUBLE PRE­
               CISION 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 triangu­
               lar 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) DOUBLE PRECISION array, dimension
               (min(P,N))
               The scalar factors of  the  elementary  reflectors
               which  represent the orthogonal matrix Z (see Fur­
               ther Details).  WORK    (workspace/output)  DOUBLE
               PRECISION  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 DORMRQ.

               If LWORK = -1, then a workspace query is  assumed;

               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    (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 DORGRQ.
       To  use  Q to update another matrix, use LAPACK subroutine
       DORMRQ.

       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 DORGQR.
       To use Z to update another matrix, use  LAPACK  subroutine
       DORMQR.


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