Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3 - 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 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.

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 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       (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 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) 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 3.3.1)             April 2011                            CHGEQZ(3lapack)