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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, * )



     #include <sunperf.h>

     void shseqr(char job, char compz, int n, int ilo,  int  ihi,
               float  *h,  int  ldh,  float *wr, float *wi, float
               *sz, int ldz, int *info);

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  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  com-
               puted 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 sub-
               matrix 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 other-
               wise.

     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 ille-
               gal 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 suc-
               cessfully computed.

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