function [] = chap_1_prob_1(nIter)
% CHAP_1_PROB_1 - 
% 
% Written by:
% -- 
% John L. Weatherwax                2005-06-02
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

xA = [ 0:0.1:1.0 ]; 

% Taylor series:
%
disp( 'Taylor series...' ); 
tic; 
for xi=1:length(xA)
  x = xA(xi); 
  aatan = 0; 
  for n=1:nIter,
    aatan = aatan - ((-1)^n)*(x^(2*n+1))/(2*n+1);
  end
  fprintf( 'x = %f; atan = %f; aatan = %f; abs acc = %f; rel acc = %f\n', xA(xi), ...
	   atan(xA(xi)), aatan, ...
	   abs(aatan-atan(xA(xi))), abs(aatan-atan(xA(xi)))/abs(atan(xA(xi))+eps) );
end
tm = toc; fprintf( 'Approx. computation time = %f\n', tm/length(xA) ); 
disp( '------------------------------------------------------------------------' ); 


% Continued fraction (loop to create a string to pass to "eval"):
%
disp( 'Continued fraction...' );
tic;
for xi=1:length(xA),
  x = xA(xi); 
  str = '1/(';
  for n=1:nIter,
    term = [ num2str(2*n-1),'+(',num2str(n),'*x)^2' ];
    str = [ str, term ];
    if( n~=nIter )
      str  = [ str, '/(' ];
    end
  end
  % Append enough closing parenthesis:
  for n=1:nIter, str = [ str, ')' ]; end
  % Perform the eval we have worked so hard for:
  aatan = x * eval( str ); 
  fprintf( 'x = %f; atan = %f; aatan = %f; abs acc = %f; rel acc = %f\n', xA(xi), ...
	   atan(xA(xi)), aatan, ...
	   abs(aatan-atan(xA(xi))), abs(aatan-atan(xA(xi)))/abs(atan(xA(xi))+eps) );
end
tm = toc; fprintf( 'Approx. computation time = %f\n', tm/length(xA) ); 
disp( '------------------------------------------------------------------------' ); 


% Rational function approx: 
%
disp( 'Rational function approx...' ); 
tic;
for xi=1:length(xA),
  % Initial conditions:
  P = [ 1 3 ]; 
  Q = [ 1 3+xA(xi)^2 ];
  for n=2:nIter,
    P = [ P, (2*n+1)*P(end) + (n^2)*xA(xi)*P(end-1) ];
    Q = [ Q, (2*n+1)*Q(end) + (n^2)*xA(xi)*Q(end-1) ];
  end
  aatan = xA(xi)*(P(end)/Q(end));
  fprintf( 'x = %f; atan = %f; aatan = %f; abs acc = %f; rel acc = %f\n', xA(xi), ...
	   atan(xA(xi)), aatan, ...
	   abs(aatan-atan(xA(xi))), abs(aatan-atan(xA(xi)))/abs(atan(xA(xi))+eps) ); 
end
tm = toc; fprintf( 'Approx. computation time = %f\n', tm/length(xA) ); 
disp( '------------------------------------------------------------------------' );