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

SGGRQF(l)					)					SGGRQF(l)

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( * )

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 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) REAL array, dimension (LDA,N)
	       On entry, the M-by-N matrix A.  On exit, if M <= N, the upper triangle of the sub-
	       array  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) 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 Fur-
	       ther 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 fac-
	       torization of a P-by-N matrix, and NB3 is the optimal  blocksize  for  a  call  of
	       SORMRQ.

	       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 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.

LAPACK version 3.0			   15 June 2000 				SGGRQF(l)


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