cggqrf man page on Scientific

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CGGQRF(1)		 LAPACK routine (version 3.2)		     CGGQRF(1)

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

SYNOPSIS
       SUBROUTINE CGGQRF( N, M, P, 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
       CGGQRF 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 unitary matrix, Z is a P-by-P unitary 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
       conjugate transpose of 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) COMPLEX 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 triangular if  N  >=  M);  the
	       elements below the diagonal, with the array TAUA, represent the
	       unitary matrix Q as a product of min(N,M) elementary reflectors
	       (see Further Details).

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

       TAUA    (output) COMPLEX array, dimension (min(N,M))
	       The scalar factors of the elementary reflectors which represent
	       the   unitary   matrix	Q   (see    Further    Details).     B
	       (input/output)  COMPLEX	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 sub‐
	       diagonal	 contain  the  N-by-P  upper trapezoidal matrix T; the
	       remaining elements, 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,N).

       TAUB    (output) COMPLEX array, dimension (min(N,P))
	       The scalar factors of the elementary reflectors which represent
	       the   unitary   matrix	Z   (see   Further   Details).	  WORK
	       (workspace/output) COMPLEX array, dimension  (MAX(1,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 factoriza‐
	       tion  of an N-by-P matrix, and NB3 is the optimal blocksize for
	       a call of CUNMQR.  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 complex scalar, and v is a complex 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 CUNGQR.
       To use Q to update another matrix, use LAPACK subroutine	 CUNMQR.   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  complex scalar, and v is a complex 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 CUNGRQ.
       To use Z to update another matrix, use LAPACK subroutine CUNMRQ.

 LAPACK routine (version 3.2)	 November 2008			     CGGQRF(1)
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