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dggrqf.f(3)				      LAPACK				      dggrqf.f(3)

NAME
       dggrqf.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dggrqf (M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO)
	   DGGRQF

Function/Subroutine Documentation
   subroutine dggrqf (integerM, integerP, integerN, double precision, dimension( lda, * )A,
       integerLDA, double precision, dimension( * )TAUA, double precision, dimension( ldb, * )B,
       integerLDB, double precision, dimension( * )TAUB, double precision, dimension( * )WORK,
       integerLWORK, integerINFO)
       DGGRQF

       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**T

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

       Parameters:
	   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 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, represent the orthogonal
		     matrix Q as a product of elementary reflectors (see Further
		     Details).

	   LDA

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

	   TAUA

		     TAUA is DOUBLE PRECISION array, dimension (min(M,N))
		     The scalar factors of the elementary reflectors which
		     represent the orthogonal matrix Q (see Further Details).

	   B

		     B is 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

		     LDB is INTEGER
		     The leading dimension of the array B. LDB >= max(1,P).

	   TAUB

		     TAUB is DOUBLE PRECISION array, dimension (min(P,N))
		     The scalar factors of the elementary reflectors which
		     represent the orthogonal matrix Z (see Further Details).

	   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. 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 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 INF0= -i, the i-th argument had an illegal value.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       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**T

	     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**T

	     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.

       Definition at line 214 of file dggrqf.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dggrqf.f(3)
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