ZHSEQR (3lapack)

compute the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors

SYNOPSIS

  SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK,
                     INFO )

      CHARACTER      COMPZ, JOB

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

      COMPLEX*16     H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE

  ZHSEQR computes the eigenvalues of a complex upper Hessenberg matrix H,
  and, optionally, the matrices T and Z from the Schur decomposition H = Z T
  Z**H, where T is an upper triangular matrix (the Schur form), and Z is the
  unitary matrix of Schur vectors.

  Optionally Z may be postmultiplied into an input unitary matrix Q, so that
  this routine can give the Schur factorization of a matrix A which has been
  reduced to the Hessenberg form H by the unitary matrix Q:  A = Q*H*Q**H =
  (QZ)*T*(QZ)**H.

ARGUMENTS

  JOB     (input) CHARACTER*1
          = 'E': compute eigenvalues only;
          = 'S': compute eigenvalues and the Schur form T.

  COMPZ   (input) CHARACTER*1
          = 'N': no Schur vectors are computed;
          = 'I': Z is initialized to the unit matrix and the matrix Z of
          Schur vectors of H is returned; = 'V': Z must contain an unitary
          matrix Q on entry, and the product Q*Z is returned.

  N       (input) INTEGER
          The order of the matrix H.  N >= 0.

  ILO     (input) INTEGER
          IHI     (input) INTEGER It is assumed that H is already upper
          triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
          normally set by a previous call to ZGEBAL, and then passed to
          CGEHRD when the matrix output by ZGEBAL is reduced to Hessenberg
          form. Otherwise ILO and IHI should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  H       (input/output) COMPLEX*16 array, dimension (LDH,N)
          On entry, the upper Hessenberg matrix H.  On exit, if JOB = 'S', H
          contains the upper triangular matrix T from the Schur decomposition
          (the Schur form). If JOB = 'E', the contents of H are unspecified
          on exit.

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

  W       (output) COMPLEX*16 array, dimension (N)
          The computed eigenvalues. If JOB = 'S', the eigenvalues are stored
          in the same order as on the diagonal of the Schur form returned in
          H, with W(i) = H(i,i).

  Z       (input/output) COMPLEX*16 array, dimension (LDZ,N)
          If COMPZ = 'N': Z is not referenced.
          If COMPZ = 'I': on entry, Z need not be set, and on exit, Z
          contains the unitary matrix Z of the Schur vectors of H.  If COMPZ
          = 'V': on entry Z must contain an N-by-N matrix Q, which is assumed
          to be equal to the unit matrix except for the submatrix
          Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z.  Normally Q is the
          unitary matrix generated by ZUNGHR after the call to ZGEHRD which
          formed the Hessenberg matrix H.

  LDZ     (input) INTEGER
          The leading dimension of the array Z.  LDZ >= max(1,N) if COMPZ =
          'I' or 'V'; LDZ >= 1 otherwise.

  WORK    (workspace) COMPLEX*16 array, dimension (N)

  LWORK   (input) INTEGER
          This argument is currently redundant.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, ZHSEQR failed to compute all the eigenvalues in
          a total of 30*(IHI-ILO+1) iterations; elements 1:ilo-1 and i+1:n of
          W contain those eigenvalues which have been successfully computed.