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RedHat 9 (Linux i386) - man page for dggqrf (redhat section l)

DGGQRF(l)					)					DGGQRF(l)

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

SYNOPSIS
       SUBROUTINE DGGQRF( N, M, P, 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
       DGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P  matrix
       B:
		   A = Q*R,	   B = Q*T*Z,

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

       where R11 is upper triangular, and

       if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
			P-N  N				 ( T21 ) P
							    P

       where T12 or T21 is upper triangular.

       In particular, if B is square and nonsingular, the GQR factorization of A and B implicitly
       gives the QR factorization of inv(B)*A:

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

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

ARGUMENTS
       N       (input) INTEGER
	       The number of rows of the matrices A and B. N >= 0.

       M       (input) INTEGER
	       The number of columns of the matrix A.  M >= 0.

       P       (input) INTEGER
	       The number of columns of the matrix B.  P >= 0.

       A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
	       On entry, the N-by-M matrix A.  On exit, the elements on and above the diagonal of
	       the  array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper trian-
	       gular if N >= M); the elements below the diagonal, with the array TAUA,	represent
	       the  orthogonal	matrix Q as a product of min(N,M) elementary reflectors (see Fur-
	       ther Details).

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

       TAUA    (output) DOUBLE PRECISION array, dimension (min(N,M))
	       The scalar factors of the elementary reflectors	which  represent  the  orthogonal
	       matrix  Q  (see	Further Details).  B	   (input/output) DOUBLE PRECISION array,
	       dimension (LDB,P) On entry, the N-by-P matrix B.  On exit, if N <=  P,  the  upper
	       triangle  of  the  subarray  B(1:N,P-N+1:P)  contains  the N-by-N upper triangular
	       matrix T; if N > P, the elements on and above the (N-P)-th subdiagonal contain the
	       N-by-P  upper  trapezoidal  matrix T; the remaining elements, with the array TAUB,
	       represent the orthogonal matrix Z as a product of elementary reflectors (see  Fur-
	       ther Details).

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

       TAUB    (output) DOUBLE PRECISION array, dimension (min(N,P))
	       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
	       QR factorization of an N-by-M matrix, NB2 is the optimal blocksize for the RQ fac-
	       torization of an N-by-P matrix, and NB3 is the optimal blocksize  for  a  call  of
	       DORMQR.

	       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 INFO = -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(n,m).

       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(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and taua in TAUA(i).
       To form Q explicitly, use LAPACK subroutine DORGQR.
       To use Q to update another matrix, use LAPACK subroutine DORMQR.

       The matrix Z is represented as a product of elementary reflectors

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

       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(p-k+i+1:p)  =	0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in B(n-k+i,1:p-k+i-1),
       and taub in TAUB(i).
       To form Z explicitly, use LAPACK subroutine DORGRQ.
       To use Z to update another matrix, use LAPACK subroutine DORMRQ.

LAPACK version 3.0			   15 June 2000 				DGGQRF(l)


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