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CentOS 7.0 - man page for zhgeqz (centos section 3)

zhgeqz.f(3)				      LAPACK				      zhgeqz.f(3)

NAME
       zhgeqz.f -

SYNOPSIS
   Functions/Subroutines
       subroutine zhgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHA, BETA, Q, LDQ, Z,
	   LDZ, WORK, LWORK, RWORK, INFO)
	   ZHGEQZ

Function/Subroutine Documentation
   subroutine zhgeqz (characterJOB, characterCOMPQ, characterCOMPZ, integerN, integerILO,
       integerIHI, complex*16, dimension( ldh, * )H, integerLDH, complex*16, dimension( ldt, *
       )T, integerLDT, complex*16, dimension( * )ALPHA, complex*16, dimension( * )BETA,
       complex*16, dimension( ldq, * )Q, integerLDQ, complex*16, dimension( ldz, * )Z,
       integerLDZ, complex*16, dimension( * )WORK, integerLWORK, double precision, dimension( *
       )RWORK, integerINFO)
       ZHGEQZ

       Purpose:

	    ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
	    where H is an upper Hessenberg matrix and T is upper triangular,
	    using the single-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 ZGGHRD.

	    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
	    factorization 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 ZGGHRD 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
	    eigenvalue 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.

       Parameters:
	   JOB

		     JOB is CHARACTER*1
		     = 'E': Compute eigenvalues only;
		     = 'S': Computer eigenvalues and the Schur form.

	   COMPQ

		     COMPQ is 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

		     COMPZ is 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

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

	   ILO

		     ILO is INTEGER

	   IHI

		     IHI is 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

		     H is COMPLEX*16 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

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

	   T

		     T is COMPLEX*16 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

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

	   ALPHA

		     ALPHA is COMPLEX*16 array, dimension (N)
		     The complex scalars alpha that define the eigenvalues of
		     GNEP.  ALPHA(i) = S(i,i) in the generalized Schur
		     factorization.

	   BETA

		     BETA is COMPLEX*16 array, dimension (N)
		     The real non-negative scalars beta that define the
		     eigenvalues of GNEP.  BETA(i) = P(i,i) in the generalized
		     Schur factorization.

		     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

		     Q is COMPLEX*16 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

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

	   Z

		     Z is COMPLEX*16 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

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

	   WORK

		     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
		     On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

	   LWORK

		     LWORK is 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

		     RWORK is DOUBLE PRECISION array, dimension (N)

	   INFO

		     INFO is 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.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   April 2012

       Further Details:

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

       Definition at line 283 of file zhgeqz.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      zhgeqz.f(3)


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