CXML
        	    
LAPACK version 3.0
cggqrf(3)

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

SYNTAX
  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( *
		     )

DESCRIPTION
  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 subdiagonal 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 (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 factorization 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.