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

ZGGRQF(l)					)					ZGGRQF(l)

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

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

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

	   COMPLEX*16	  A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( * )

PURPOSE
       ZGGRQF 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*16 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*16 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*16 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 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*16 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*16	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 ZUN-
	       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 ZUNGRQ.
       To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.

       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 ZUNGQR.
       To use Z to update another matrix, use LAPACK subroutine ZUNMQR.

LAPACK version 3.0			   15 June 2000 				ZGGRQF(l)


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