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

NAME

       LAPACK-3 - computes the eigenvalues of a real matrix pair (H,T),

SYNOPSIS

       SUBROUTINE DHGEQZ( JOB,  COMPQ,  COMPZ, N, ILO, IHI, H, LDH, T, LDT, ALPHAR, ALPHAI, BETA,
                          Q, LDQ, Z, LDZ, WORK, LWORK, INFO )

           CHARACTER      COMPQ, COMPZ, JOB

           INTEGER        IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N

           DOUBLE         PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), H( LDH, * ), Q(  LDQ,  *
                          ), T( LDT, * ), WORK( * ), Z( LDZ, * )

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.
                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.

 LAPACK routine (version 3.3.1)             April 2011                            DHGEQZ(3lapack)