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## chgeqz()

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

Download CHGEQZ + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  JOB

      !>          JOB is CHARACTER*1
      !>          = 'E': Compute eigenvalues only;
      !>          = 'S': Computer eigenvalues and the Schur form.
      !>

  [in]

  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.
      !>

  [in]

  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.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrices H, T, Q, and Z.  N >= 0.
      !>

  [in]

  ILO

      !>          ILO is INTEGER
      !>

  [in]

  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.
      !>

  [in,out]

  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.
      !>

  [in]

  LDH

      !>          LDH is INTEGER
      !>          The leading dimension of the array H.  LDH >= max( 1, N ).
      !>

  [in,out]

  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.
      !>

  [in]

  LDT

      !>          LDT is INTEGER
      !>          The leading dimension of the array T.  LDT >= max( 1, N ).
      !>

  [out]

  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.
      !>

  [out]

  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.
      !>

  [in,out]

  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'.
      !>

  [in]

  LDQ

      !>          LDQ is INTEGER
      !>          The leading dimension of the array Q.  LDQ >= 1.
      !>          If COMPQ='V' or 'I', then LDQ >= N.
      !>

  [in,out]

  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'.
      !>

  [in]

  LDZ

      !>          LDZ is INTEGER
      !>          The leading dimension of the array Z.  LDZ >= 1.
      !>          If COMPZ='V' or 'I', then LDZ >= N.
      !>

  [out]

  WORK

      !>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
      !>

  [in]

  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.
      !>

  [out]

  RWORK

      !>          RWORK is REAL array, dimension (N)
      !>

  [out]

  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.
      !>