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dhgeqz.f(3)				      LAPACK				      dhgeqz.f(3)

NAME
       dhgeqz.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dhgeqz (JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA,
	   Q, LDQ, Z, LDZ, WORK, LWORK, INFO)
	   DHGEQZ

Function/Subroutine Documentation
   subroutine dhgeqz (characterJOB, characterCOMPQ, characterCOMPZ, integerN, integerILO,
       integerIHI, double precision, dimension( ldh, * )H, integerLDH, double precision,
       dimension( ldt, * )T, integerLDT, double precision, dimension( * )ALPHAR, double
       precision, dimension( * )ALPHAI, double precision, dimension( * )BETA, double precision,
       dimension( ldq, * )Q, integerLDQ, double precision, dimension( ldz, * )Z, integerLDZ,
       double precision, dimension( * )WORK, integerLWORK, integerINFO)
       DHGEQZ

       Purpose:

	    DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
	    where H is an upper Hessenberg matrix and T is upper triangular,
	    using the double-shift QZ method.
	    Matrix pairs of this type are produced by the reduction to
	    generalized upper Hessenberg form of a real matrix pair (A,B):

	       A = Q1*H*Z1**T,	B = Q1*T*Z1**T,

	    as computed by DGGHRD.

	    If JOB='S', then the Hessenberg-triangular pair (H,T) is
	    also reduced to generalized Schur form,

	       H = Q*S*Z**T,  T = Q*P*Z**T,

	    where Q and Z are orthogonal matrices, P is an upper triangular
	    matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
	    diagonal blocks.

	    The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
	    (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
	    eigenvalues.

	    Additionally, the 2-by-2 upper triangular diagonal blocks of P
	    corresponding to 2-by-2 blocks of S are reduced to positive diagonal
	    form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
	    P(j,j) > 0, and P(j+1,j+1) > 0.

	    Optionally, the orthogonal matrix Q from the generalized Schur
	    factorization may be postmultiplied into an input matrix Q1, and the
	    orthogonal matrix Z may be postmultiplied into an input matrix Z1.
	    If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
	    the matrix pair (A,B) to generalized upper Hessenberg form, then the
	    output matrices Q1*Q and Z1*Z are the orthogonal factors from the
	    generalized Schur factorization of (A,B):

	       A = (Q1*Q)*S*(Z1*Z)**T,	B = (Q1*Q)*P*(Z1*Z)**T.

	    To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
	    of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
	    complex and beta real.
	    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.
	    Real eigenvalues 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': Compute 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 an orthogonal 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': Z is initialized to the unit matrix and the matrix Z
			    of right Schur vectors of (H,T) is returned;
		     = 'V': Z must contain an orthogonal 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 DOUBLE PRECISION array, dimension (LDH, N)
		     On entry, the N-by-N upper Hessenberg matrix H.
		     On exit, if JOB = 'S', H contains the upper quasi-triangular
		     matrix S from the generalized Schur factorization.
		     If JOB = 'E', the diagonal blocks of H match those 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 DOUBLE PRECISION 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;
		     2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
		     are reduced to positive diagonal form, i.e., if H(j+1,j) is
		     non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
		     T(j+1,j+1) > 0.
		     If JOB = 'E', the diagonal blocks of T match those of P, but
		     the rest of T is unspecified.

	   LDT

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

	   ALPHAR

		     ALPHAR is DOUBLE PRECISION array, dimension (N)
		     The real parts of each scalar alpha defining an eigenvalue
		     of GNEP.

	   ALPHAI

		     ALPHAI is DOUBLE PRECISION array, dimension (N)
		     The imaginary parts of each scalar alpha defining an
		     eigenvalue of GNEP.
		     If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
		     positive, then the j-th and (j+1)-st eigenvalues are a
		     complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).

	   BETA

		     BETA is DOUBLE PRECISION array, dimension (N)
		     The scalars beta that define the eigenvalues of GNEP.
		     Together, the quantities alpha = (ALPHAR(j),ALPHAI(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 DOUBLE PRECISION array, dimension (LDQ, N)
		     On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
		     the reduction of (A,B) to generalized Hessenberg form.
		     On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
		     vectors of (H,T), and if COMPZ = 'V', the orthogonal 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 DOUBLE PRECISION array, dimension (LDZ, N)
		     On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
		     the reduction of (A,B) to generalized Hessenberg form.
		     On exit, if COMPZ = 'I', the orthogonal matrix of
		     right Schur vectors of (H,T), and if COMPZ = 'V', the
		     orthogonal 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 DOUBLE PRECISION 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.

	   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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(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:

	     Iteration counters:

	     JITER  -- counts iterations.
	     IITER  -- counts iterations run since ILAST was last
		       changed.  This is therefore reset only when a 1-by-1 or
		       2-by-2 block deflates off the bottom.

       Definition at line 303 of file dhgeqz.f.

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

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