SHSEQR (3lapack)

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

SYNOPSIS

  SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK,
                     LWORK, INFO )

      CHARACTER      COMPZ, JOB

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

      REAL           H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, * )

PURPOSE

  SHSEQR computes the eigenvalues of a real upper Hessenberg matrix H and,
  optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T,
  where T is an upper quasi-triangular matrix (the Schur form), and Z is the
  orthogonal matrix of Schur vectors.

  Optionally Z may be postmultiplied into an input orthogonal 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 orthogonal matrix Q:  A =
  Q*H*Q**T = (QZ)*T*(QZ)**T.

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
          orthogonal 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 SGEBAL, and then passed to
          SGEHRD when the matrix output by SGEBAL 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) REAL array, dimension (LDH,N)
          On entry, the upper Hessenberg matrix H.  On exit, if JOB = 'S', H
          contains the upper quasi-triangular matrix T from the Schur
          decomposition (the Schur form); 2-by-2 diagonal blocks
          (corresponding to complex conjugate pairs of eigenvalues) are
          returned in standard form, with H(i,i) = H(i+1,i+1) and
          H(i+1,i)*H(i,i+1) < 0. 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).

  WR      (output) REAL array, dimension (N)
          WI      (output) REAL array, dimension (N) The real and imaginary
          parts, respectively, of the computed eigenvalues. If two
          eigenvalues are computed as a complex conjugate pair, they are
          stored in consecutive elements of WR and WI, say the i-th and
          (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the
          eigenvalues are stored in the same order as on the diagonal of the
          Schur form returned in H, with WR(i) = H(i,i) and, if
          H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) =
          sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

  Z       (input/output) REAL 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 orthogonal 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
          orthogonal matrix generated by SORGHR after the call to SGEHRD
          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) REAL 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, SHSEQR failed to compute all of the eigenvalues
          in a total of 30*(IHI-ILO+1) iterations; elements 1:ilo-1 and i+1:n
          of WR and WI contain those eigenvalues which have been successfully
          computed.