dhseqr(3F)

NAME
     DHSEQR - 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 DHSEQR( 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

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

PURPOSE
     DHSEQR 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 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) DOUBLE PRECISION 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, 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.