function [x,fail] = secant(f,x1,x2,eps,N)
%
%  Applies the secant method to the function f, starting
%  at the points x1 and x2.  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.
%
%  Note that the implementation below is inefficient, as it does
%  not store the function evaluations.
%
%  Example : 
%  >> [x,fail] = secant('cos',pi/4,2*pi/3,0.0001,100)
%
fprintf('running the secant method...\n');
for i = 1:N 
   fx2 = feval(f,x2);
   fx1 = feval(f,x1);
   dx =  fx2*(x2 - x1)/(fx2 - fx1);
   x1 = x2;
   x2 = x2 - dx;
   fx2 = feval(f,x2);
   fprintf('  x = %e, dx = %e, f(x) = %e, \n', x2, dx, fx2);
   if (abs(fx2) < eps) | (abs(dx) < eps)
      x = x2; fail = 0;
      fprintf('succeeded after %d steps\n', i);
      return;
   end
end
fprintf('failed requirements after %d steps\n', N);
x = x2; fail = 1;