This information is part of the Modelica Standard Library maintained by the Modelica Association.

Lapack documentation
    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.

    Arguments
    =========

    JOB     (input) CHARACTER*1
            = 'E': Compute eigenvalues only;
            = 'S': Compute 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 an orthogonal matrix Q1 on entry and
                   the product Q1*Q is returned.

    COMPZ   (input) 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       (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) 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;
            2-by-2 diagonal blocks (corresponding to complex conjugate
            pairs of eigenvalues) are returned in standard form, with
            H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
            If JOB = 'E', the diagonal blocks of H match those 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) 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     (input) INTEGER
            The leading dimension of the array T.  LDT >= max( 1, N ).

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

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

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

    WORK    (workspace/output) DOUBLE PRECISION 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.

    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 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.

    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.

(alphaReal, alphaImag, beta, info) = dhgeqz(A, B)

A	Type: Real[:,size(A, 1)]
B	Type: Real[size(A, 1),size(A, 1)]

alphaReal	Type: Real[size(A, 1)] Description: Real part of alpha (eigenvalue=(alphaReal+i*alphaImag)/beta)
alphaImag	Type: Real[size(A, 1)] Description: Imaginary part of alpha
beta	Type: Real[size(A, 1)] Description: Denominator of eigenvalue
info	Type: Integer