function [x,fail] = newton(f,df,x,eps,N)
%
%  Applies Newton's method to the function f, with derivative in df,
%  starting at the point x.  It produces a sequences of points.
%  Stops when the sequence reaches the point x where 
%  |f(x)| < eps or when two consecutive points in the sequence
%  have distance less than eps apart from each other.
%  Failure is reported (fail = 1) when the accuracy requirement
%  is not satisfied in N steps; otherwise fail = 0 on return.
%
%  Example : 
%  >> [x,fail] = newton('sin','cos',pi/4,1e-8,10)
%
fprintf('running the method of Newton...\n');
fx = feval(f,x);
for i = 1:N 
   dx = fx/feval(df,x); 
   x = x - dx;
   fx = feval(f,x);
   fprintf('  x = %e, dx = %e, f(x) = %e\n', x, dx, fx);
   if (abs(fx) < eps) | (abs(dx) < eps)
      fail = 0;
      fprintf('succeeded after %d steps\n', i);
      return;
   end
end
fprintf('failed requirements after %d steps\n', N);
fail = 1;