CXML
        	    
LAPACK version 3.0
zhgeqz(3)

PURPOSE
  ZHGEQZ - implement a single-shift version of the QZ method for finding the
  generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation	 det( A -
  w(i) B ) = 0	If JOB='S', then the pair (A,B) is simultaneously reduced to
  Schur form (i.e., A and B are both upper triangular) by applying one
  unitary tranformation (usually called Q) on the left and another (usually
  called Z) on the right

SYNTAX
  SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA,
		     BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO )

      CHARACTER	     COMPQ, COMPZ, JOB

      INTEGER	     IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N

      DOUBLE	     PRECISION RWORK( * )

      COMPLEX*16     A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ,
		     * ), WORK( * ), Z( LDZ, * )

DESCRIPTION
  ZHGEQZ implements a single-shift version of the QZ method for finding the
  generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation det( A - w(i)
  B ) = 0 If JOB='S', then the pair (A,B) is simultaneously reduced to Schur
  form (i.e., A and B are both upper triangular) by applying one unitary
  tranformation (usually called Q) on the left and another (usually called Z)
  on the right. The diagonal elements of A are then ALPHA(1),...,ALPHA(N),
  and of B are BETA(1),...,BETA(N).

  If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary
  transformations used to reduce (A,B) are accumulated into the arrays Q and
  Z s.t.:

       Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
       Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*

  Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
       Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
       pp. 241--256.

ARGUMENTS
  JOB	  (input) CHARACTER*1
	  = 'E': compute only ALPHA and BETA.  A and B will not necessarily
	  be put into generalized Schur form.  = 'S': put A and B into
	  generalized Schur form, as well as computing ALPHA and BETA.

  COMPQ	  (input) CHARACTER*1
	  = 'N': do not modify Q.
	  = 'V': multiply the array Q on the right by the conjugate transpose
	  of the unitary tranformation that is applied to the left side of A
	  and B to reduce them to Schur form.  = 'I': like COMPQ='V', except
	  that Q will be initialized to the identity first.

  COMPZ	  (input) CHARACTER*1
	  = 'N': do not modify Z.
	  = 'V': multiply the array Z on the right by the unitary
	  tranformation that is applied to the right side of A and B to
	  reduce them to Schur form.  = 'I': like COMPZ='V', except that Z
	  will be initialized to the identity first.

  N	  (input) INTEGER
	  The order of the matrices A, B, Q, and Z.  N >= 0.

  ILO	  (input) INTEGER
	  IHI	  (input) INTEGER It is assumed that A is already upper
	  triangular in rows and columns 1:ILO-1 and IHI+1:N.  1 <= ILO <=
	  IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  A	  (input/output) COMPLEX*16 array, dimension (LDA, N)
	  On entry, the N-by-N upper Hessenberg matrix A.  Elements below the
	  subdiagonal must be zero.  If JOB='S', then on exit A and B will
	  have been simultaneously reduced to upper triangular form.  If
	  JOB='E', then on exit A will have been destroyed.

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

  B	  (input/output) COMPLEX*16 array, dimension (LDB, N)
	  On entry, the N-by-N upper triangular matrix B.  Elements below the
	  diagonal must be zero.  If JOB='S', then on exit A and B will have
	  been simultaneously reduced to upper triangular form.	 If JOB='E',
	  then on exit B will have been destroyed.

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

  ALPHA	  (output) COMPLEX*16 array, dimension (N)
	  The diagonal elements of A when the pair (A,B) has been reduced to
	  Schur form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
	  eigenvalues.

  BETA	  (output) COMPLEX*16 array, dimension (N)
	  The diagonal elements of B when the pair (A,B) has been reduced to
	  Schur form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
	  eigenvalues.	A and B are normalized so that BETA(1),...,BETA(N)
	  are non-negative real numbers.

  Q	  (input/output) COMPLEX*16 array, dimension (LDQ, N)
	  If COMPQ='N', then Q will not be referenced.	If COMPQ='V' or 'I',
	  then the conjugate transpose of the unitary transformations which
	  are applied to A and B on the left will be applied to the array Q
	  on the right.

  LDQ	  (input) INTEGER
	  The leading dimension of the array Q.	 LDQ >= 1.  If COMPQ='V' or
	  'I', then LDQ >= N.

  Z	  (input/output) COMPLEX*16 array, dimension (LDZ, N)
	  If COMPZ='N', then Z will not be referenced.	If COMPZ='V' or 'I',
	  then the unitary transformations which are applied to A and B on
	  the right will be applied to the array Z on the right.

  LDZ	  (input) INTEGER
	  The leading dimension of the array Z.	 LDZ >= 1.  If COMPZ='V' or
	  'I', then LDZ >= N.

  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).

	  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.

  RWORK	  (workspace) DOUBLE PRECISION array, dimension (N)

  INFO	  (output) INTEGER
	  = 0: successful exit
	  < 0: if INFO = -i, the i-th argument had an illegal value
	  = 1,...,N: the QZ iteration did not converge.	 (A,B) is not in
	  Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be
	  correct.  = N+1,...,2*N: the shift calculation failed.  (A,B) is
	  not in Schur form, but ALPHA(i) and BETA(i), i=INFO-N+1,...,N
	  should be correct.  > 2*N:	 various "impossible" errors.

FURTHER DETAILS
  We assume that complex ABS works as long as its value is less than
  overflow.