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dhseqr examples
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Man Page Details
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NAME
       DHSEQR - compute the eigenvalues of a Hessenberg matrix H  and, option-
       ally, 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.

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

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

           CHARACTER      COMPZ, JOB

           DOUBLE         PRECISION  H( LDH, * ), WI( * ), WORK( * ), WR( * ),
                          Z( LDZ, * )

PURPOSE
          DHSEQR computes the eigenvalues of a 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 DGEBAL, and then passed to SGEHRD
          when the matrix output by DGEBAL 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DORGHR after
          the call to DGEHRD 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/output) DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) 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.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, DHSEQR 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.

