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shseqr


 NAME
      SHSEQR - 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 factoriza-
      tion 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 pre-
              vious 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 <= max(1,IHI); IHI <= N.

      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 com-
              plex 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 consecu-
              tive 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.  Nor-
              mally 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) itera-
              tions; elements 1:ilo-1 and i+1:n of WR and WI con-
              tain those eigenvalues which have been successfully
              computed.