function showkepler ( )
%
%% showkepler() applies ODE23 and ODE45 to the Kepler ODE system.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, 'show_kepler ( )\n' );

  close ( 'all' );
  
  tInitial = 0;
  tFinal   =  2 * pi;
  tol = 0.0001;
  uInitial = [ .4; 0 ; 0 ; 2];
  trace = 0;
  
  [ t, u ] = ode23 ( @kepler, [ tInitial, tFinal ], uInitial, tol );  
  figure ( );
  plot ( u(:,1), u(:,3) );
  axis ( 'equal' );
  title(sprintf('ODE23 Steps = %5.0f, tol = %5.0e',length(t),tol));
  filename = 'showkepler_ode23.png';
  print ( '-dpng', filename );
  fprintf ( 1, '  Graphics saved as "%s"n' );

  [ t, u ] = ode45 ( @kepler, [ tInitial, tFinal ], uInitial, tol );
  figure ( );
  plot ( u(:,1), u(:,3) );
  axis ( 'equal' );
  title(sprintf ( 'ODE45 Steps = %5.0f, tol = %5.0e',length(t),tol));
  filename = 'showkepler_ode45.png';
  print ( '-dpng', filename );
  fprintf ( 1, '  Graphics saved as "%s"n' );

  tvals = linspace(tInitial,tFinal);
  xvals = spline(t,u(:,1),tvals);
  yvals = spline(t,u(:,3),tvals);
  figure ( );
  plot(xvals,yvals);
  axis ( 'equal' );
  title(sprintf('ODE45 Steps = %5.0f, tol = %5.0e',length(t),tol));
  xlabel('(With Spline Smoothing)');

  filename = 'showkepler_smoothed.png';
  print ( '-dpng', filename );
  fprintf ( 1, '  Graphics saved as "%s"n' );

  return
end


