function [] = chap_6_sec_6_prob_2()
%
% 
% Written by:
% -- 
% John L. Weatherwax                2007-04-23
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

close all;  clc; 

% perform all iterations (and tabulate chain links):
% 

N_INFTY = 15; 

% assign the initial conditions for our two by two example 
% and our phi function iterations 
% 
x0 = 10; y0 = 10; t0 = 0; 

% the "N" matrix (needed for the iterations):
% 
N = (1/12000)*[ 310 -20 ; 290 20 ]; 

% plot the iterates:
% 
fi=figure; plot( x0, y0, 'b.', 'MarkerSize', 15 ); hold on; grid on; 
%figure(fi); plot( x0, y0, 'gs', 'MarkerSize', 15 ); hold on; grid on; 
title( 'the iterates' ); 

% Iterate each scheme:
%
xn=x0; yn=y0; 
tn=t0; 
errs=zeros(N_INFTY,2); 
%fprintf('');
fprintf('ii=%3d: K=(%12.6f,%12.6f,%16s) T=(%12.6f,%16s)\n',0,xn,yn,'',tn,'');
for ii=1:N_INFTY,

  % iterate our modified Newton-Raphson method:
  fvn = [ xn^2 + yn^2 - 200; yn^3 + xn*yn - xn^3 ];
  nTmsf = N * fvn;
  xnp1 = xn - nTmsf(1);
  ynp1 = yn - nTmsf(2);
  %[ xnp1, ynp1 ] 
  %[ xn-xnp1, yn-ynp1 ]
  errs(ii,1) = norm( [ xn-xnp1, yn-ynp1 ] ); 
  xn=xnp1; yn=ynp1; 
  figure(fi); plot( xn, yn, 'x', 'MarkerSize', 10, 'MarkerEdgeColor', [ 0 1 0 ], 'MarkerFaceColor', [ 0 1 0 ] ); 

  % iterate our phi function:
  tnp1 = 1/6 + (tn^2)/6 + (tn^3)/300; 
  errs(ii,2) = norm( [ tn - tnp1 ] ); 
  tn=tnp1; 

  fprintf('ii=%3d: K=(%12.6f,%12.6f,%16.6g) T=(%12.6f,%16.6g)\n',ii,...
          xn,yn,errs(ii,1),tn,errs(ii,2));  

  %pause; 
end

% plot the error observing the various convergence: 
%
figure; semilogy( errs+eps, 'o' ); grid on; 
legend( gca, 'Mod Newton', 'Phi' ); 
%saveas( gcf, '../WriteUp/Pictures/chap_6_sec_6_prob_2_conv.eps' );