cggrqf man page on YellowDog

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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  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,	repre‐
	       sent the unitary 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 diago‐
	       nal 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 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 factoriza‐
	       tion of a P-by-N matrix, and NB3 is the optimal blocksize for a
	       call of CUNMRQ.

	       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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