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CHGEQZ(1)		 LAPACK routine (version 3.2)		     CHGEQZ(1)

NAME
       CHGEQZ - computes the eigenvalues of a complex matrix pair (H,T),

SYNOPSIS
       SUBROUTINE CHGEQZ( JOB,	COMPQ,	COMPZ,	N,  ILO,  IHI, H, LDH, T, LDT,
			  ALPHA, BETA, Q, LDQ, Z,  LDZ,	 WORK,	LWORK,	RWORK,
			  INFO )

	   CHARACTER	  COMPQ, COMPZ, JOB

	   INTEGER	  IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N

	   REAL		  RWORK( * )

	   COMPLEX	  ALPHA(  * ), BETA( * ), H( LDH, * ), Q( LDQ, * ), T(
			  LDT, * ), WORK( * ), Z( LDZ, * )

PURPOSE
       CHGEQZ computes the eigenvalues of a complex matrix pair (H,T), where H
       is an upper Hessenberg matrix and T is upper triangular, using the sin‐
       gle-shift QZ method.
       Matrix pairs of this type are produced by the reduction to  generalized
       upper Hessenberg form of a complex matrix pair (A,B):
	  A = Q1*H*Z1**H,  B = Q1*T*Z1**H,
       as computed by CGGHRD.
       If JOB='S', then the Hessenberg-triangular pair (H,T) is
       also reduced to generalized Schur form,
	  H = Q*S*Z**H,	 T = Q*P*Z**H,
       where Q and Z are unitary matrices and S and P are upper triangular.
       Optionally,  the unitary matrix Q from the generalized Schur factoriza‐
       tion may be postmultiplied into an input matrix	Q1,  and  the  unitary
       matrix  Z  may be postmultiplied into an input matrix Z1.  If Q1 and Z1
       are the unitary matrices from CGGHRD that reduced the matrix pair (A,B)
       to  generalized Hessenberg form, then the output matrices Q1*Q and Z1*Z
       are the unitary factors from the	 generalized  Schur  factorization  of
       (A,B):
	  A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.
       To avoid overflow, eigenvalues of the matrix pair (H,T)
       (equivalently,  of  (A,B))  are	computed  as  a pair of complex values
       (alpha,beta).  If beta is nonzero, lambda = alpha / beta is  an	eigen‐
       value of the generalized nonsymmetric eigenvalue problem (GNEP)
	  A*x = lambda*B*x
       and  if	alpha  is  nonzero,  mu = beta / alpha is an eigenvalue of the
       alternate form of the GNEP
	  mu*A*y = B*y.
       The values of alpha and beta  for  the  i-th  eigenvalue	 can  be  read
       directly	 from  the  generalized	 Schur	form:	alpha = S(i,i), beta =
       P(i,i).
       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 eigenvalues only;
	       = 'S': Computer eigenvalues and the Schur form.

       COMPQ   (input) CHARACTER*1
	       = 'N': Left Schur vectors (Q) are not computed;
	       = 'I': Q is initialized to the unit matrix and the matrix Q  of
	       left  Schur vectors of (H,T) is returned; = 'V': Q must contain
	       a unitary matrix Q1 on entry and the product Q1*Q is returned.

       COMPZ   (input) CHARACTER*1
	       = 'N': Right Schur vectors (Z) are not computed;
	       = 'I': Q is initialized to the unit matrix and the matrix Z  of
	       right Schur vectors of (H,T) is returned; = 'V': Z must contain
	       a unitary matrix Z1 on entry and the product Z1*Z is returned.

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

       ILO     (input) INTEGER
	       IHI     (input) INTEGER ILO and IHI mark the rows  and  columns
	       of  H  which  are  in Hessenberg form.  It is assumed that A is
	       already upper  triangular  in  rows  and	 columns  1:ILO-1  and
	       IHI+1:N.	  If  N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and
	       IHI=0.

       H       (input/output) COMPLEX array, dimension (LDH, N)
	       On entry, the N-by-N upper Hessenberg matrix H.	 On  exit,  if
	       JOB  =  'S',  H contains the upper triangular matrix S from the
	       generalized Schur factorization.	 If JOB = 'E', the diagonal of
	       H matches that of S, but the rest of H is unspecified.

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

       T       (input/output) COMPLEX array, dimension (LDT, N)
	       On  entry,  the	N-by-N upper triangular matrix T.  On exit, if
	       JOB = 'S', T contains the upper triangular matrix  P  from  the
	       generalized Schur factorization.	 If JOB = 'E', the diagonal of
	       T matches that of P, but the rest of T is unspecified.

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

       ALPHA   (output) COMPLEX array, dimension (N)
	       The complex scalars alpha that define the eigenvalues of	 GNEP.
	       ALPHA(i) = S(i,i) in the generalized Schur factorization.

       BETA    (output) COMPLEX array, dimension (N)
	       The  real non-negative scalars beta that define the eigenvalues
	       of GNEP.	 BETA(i) = P(i,i) in the generalized Schur  factoriza‐
	       tion.   Together,  the  quantities  alpha = ALPHA(j) and beta =
	       BETA(j) represent the j-th eigenvalue of the matrix pair (A,B),
	       in  one	of  the	 forms lambda = alpha/beta or mu = beta/alpha.
	       Since either lambda or mu may overflow,	they  should  not,  in
	       general, be computed.

       Q       (input/output) COMPLEX array, dimension (LDQ, N)
	       On  entry,  if  COMPZ  = 'V', the unitary matrix Q1 used in the
	       reduction of (A,B) to generalized Hessenberg form.  On exit, if
	       COMPZ = 'I', the unitary matrix of left Schur vectors of (H,T),
	       and if COMPZ = 'V', the unitary matrix of left Schur vectors of
	       (A,B).  Not referenced if COMPZ = 'N'.

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

       Z       (input/output) COMPLEX array, dimension (LDZ, N)
	       On entry, if COMPZ = 'V', the unitary matrix  Z1	 used  in  the
	       reduction of (A,B) to generalized Hessenberg form.  On exit, if
	       COMPZ = 'I', the unitary	 matrix	 of  right  Schur  vectors  of
	       (H,T),  and  if	COMPZ = 'V', the unitary matrix of right Schur
	       vectors of (A,B).  Not referenced if COMPZ = 'N'.

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

       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).  If LWORK
	       = -1, then a workspace query is assumed; the routine only  cal‐
	       culates	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) REAL 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.  (H,T) 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.	 (H,T)
	       is not  in  Schur  form,	 but  ALPHA(i)	and  BETA(i),  i=INFO-
	       N+1,...,N should be correct.

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

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