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NAME

       chgeqz.f -

SYNOPSIS

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

Function/Subroutine Documentation

   subroutine chgeqz (characterJOB, characterCOMPQ, characterCOMPZ, integerN, integerILO,
       integerIHI, complex, dimension( ldh, * )H, integerLDH, complex, dimension( ldt, * )T,
       integerLDT, complex, dimension( * )ALPHA, complex, dimension( * )BETA, complex, dimension(
       ldq, * )Q, integerLDQ, complex, dimension( ldz, * )Z, integerLDZ, complex, dimension( *
       )WORK, integerLWORK, real, dimension( * )RWORK, integerINFO)
       CHGEQZ

       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 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 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
            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 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
            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 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 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 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 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 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 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 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 REAL 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 chgeqz.f.

Author

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