zhgeqz.f(3)

NAME
zhgeqz.f -

SYNOPSIS

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

Function/Subroutine Documentation

  subroutine ZHGEQZ (CHARACTERJOB, CHARACTERCOMPQ, CHARACTERCOMPZ, INTEGERN, INTEGERILO, INTEGERIHI, COMPLEX*16, dimension( ldh, * )H, INTEGERLDH, COMPLEX*16, dimension( ldt, * )T, INTEGERLDT, COMPLEX*16, dimension( * )ALPHA, COMPLEX*16, dimension( * )BETA, COMPLEX*16, dimension( ldq, * )Q, INTEGERLDQ, COMPLEX*16, dimension( ldz, * )Z, INTEGERLDZ, COMPLEX*16, dimension( * )WORK, INTEGERLWORK, DOUBLE PRECISION, dimension( * )RWORK, INTEGERINFO)

ZHGEQZ

Purpose:

     ZHGEQZ 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 ZGGHRD.

     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 ZGGHRD 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*16 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*16 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*16 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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 zhgeqz.f'.

Author

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