function [] = chap_6_sec_6_prob_3()
%
% 
% 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 = 0; y0 = 0; t0 = 0; 

% the negative "M^{-1}" matrix (needed for the iterations):
% 
mInv = (1/1000)*[ 28 9 ; 4 37 ]; 

% 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; 
t1n=t0; t2n=t0; 
errs=zeros(N_INFTY,3); 
%fprintf('');
fprintf('ii=%3d: K=(%12.6f,%12.6f,%16s) T1=(%12.6f,%16s) T2=(%12.6f,%16s)\n',0,xn,yn,'',t1n,'',t2n,'');
for ii=1:N_INFTY,

  % iterate our modified Newton-Raphson method:
  fvn = [ xn^5 + yn^5 + 25; (xn^3)*(yn^3) + 18 ];
  nTmsf = mInv * fvn;
  xnp1 = nTmsf(1); 
  ynp1 = 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 first phi function:
  t1np1 = 0.862 + (280/5000)*(t1n^5) + (222/6000)*(t1n^6);
  errs(ii,2) = norm( [ t1n - t1np1 ] ); 
  t1n=t1np1; 

  % iterate our second phi function:
  t2np1 = 0.862 + (280/5000)*(t2n^5) + (54/6000)*(t2n^6);
  errs(ii,3) = norm( [ t2n - t2np1 ] ); 
  t2n=t2np1; 

  fprintf('ii=%3d: K=(%12.6f,%12.6f,%16.6g) T1=(%12.6f,%16.6g) T2=(%12.6f,%16.6g)\n',ii,...
          xn,yn,errs(ii,1),t1n,errs(ii,2),t2n,errs(ii,3));  

  %pause; 
end

% plot the error observing the various convergence: 
%
figure; semilogy( errs+eps, 'o' ); grid on; 
legend( gca, 'Mod Newton', 'Phi fn 1', 'Phi fn 2' ); 
%saveas( gcf, '../WriteUp/Pictures/chap_6_sec_6_prob_2_conv.eps' );