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DGGRQF(l)			       )			     DGGRQF(l)

NAME
       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 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) 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 trapezoidal
	       matrix R; the remaining elements, with the array	 TAUA,	repre‐
	       sent the orthogonal matrix Q as a product of elementary reflec‐
	       tors (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   Further	Details).    B
	       (input/output) DOUBLE PRECISION	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	dimen‐
	       sion 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   Further   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 factoriza‐
	       tion 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 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    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INF0= -i, the i-th argument had an illegal 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.

LAPACK version 3.0		 15 June 2000			     DGGRQF(l)
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