SRC/chgeqz.f(3)

NAME
SRC/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 (character job, character compq, character compz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

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, , 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 COMPQ = 'V', the unitary matrix Q1 used in the
    !>          reduction of (A,B) to generalized Hessenberg form.
    !>          On exit, if COMPQ = 'I', the unitary matrix of left Schur
    !>          vectors of (H,T), and if COMPQ = 'V', the unitary matrix of
    !>          left Schur vectors of (A,B).
    !>          Not referenced if COMPQ = '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.

Further Details:

    !>
    !>  We assume that complex ABS works as long as its value is less than
    !>  overflow.
    !>

Definition at line 281 of file chgeqz.f.

Author

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