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LAPACK version 3.0
zggrqf(3)

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

SYNTAX
  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( *
		     )

DESCRIPTION
  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 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, represent 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*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 factorization of a
	  P-by-N matrix, and NB3 is the optimal blocksize for a call of
	  ZUNMRQ.

	  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.