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

CGGRQF(l)					)					CGGRQF(l)

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

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

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

	   COMPLEX	  A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE
       CGGRQF 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 unitary matrix, Z is a P-by-P unitary 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 conjugate 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) COMPLEX 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 uni-
	       tary 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) COMPLEX array, dimension (min(M,N))
	       The scalar factors of the elementary reflectors which represent the unitary matrix
	       Q  (see Further Details).  B	  (input/output) COMPLEX 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 trian-
	       gular if P >= N); the elements below the diagonal, with the array TAUB,	represent
	       the  unitary 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) COMPLEX array, dimension (min(P,N))
	       The scalar factors of the elementary reflectors which represent the unitary matrix
	       Z  (see	Further  Details).   WORK     (workspace/output) COMPLEX 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 CUN-
	       MRQ.

	       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(m,n).

       Each H(i) has the form

	  H(i) = I - taua * v * v'

       where taua is a complex scalar, and v is a complex 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 CUNGRQ.
       To use Q to update another matrix, use LAPACK subroutine CUNMRQ.

       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 complex scalar, and v is a complex 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 CUNGQR.
       To use Z to update another matrix, use LAPACK subroutine CUNMQR.

LAPACK version 3.0			   15 June 2000 				CGGRQF(l)


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