function [x_star,success] = steffensen (x0, f, eps, maxitr)
% Accompanying program for the text
%
% Classical and Modern Numerical Analysis:
% Theory, Methods and Practice
% by Azmy S. Ackleh, Edward J. Allen,
% R. Baker Kearfott, and Padmanabhan Seshaiyer
%
% [x_star,success] = steffensen (x0, f, eps, maxitr)
% does iterations of Steffensen's method,
% using x0 as initial guess, and with function defined by y = f(x).
% For example, suppose an m-file xsqm2.m is available in Matlab's working
% directory, with the following contents:
% function [y] = xsqm2(x)
% y = x^2-2;
% return
% Then, issuing
% [x_star,success] = steffensen (1.5, 'xsqm2', 1e-10, 100)
% from Matlab's command window will cause an approximation to the square
% root of 2 to be stored in x_star.
% Iteration stops successfully if the difference between successive
% iterates is less than eps, and iteration
% stops unsuccessfully if maxitr iterations have been done
% without stopping successfully.
% On return:
% success = 1 if iteration stopped successfully, and
% success = 0 if iteration stopped unsuccessfully.
% x_star is set to the approximate solution to f(x) = 0
% if iteration stopped successfully, and x_star
% is set to x0 otherwise.
success = false;
diff=2*eps;
disp(' --------------------------------------------');
disp(' k x0 x1 ');
disp(' --------------------------------------------');
for k=1:maxitr
hlp= feval(f,x0);
x1=x0 - hlp*hlp/(feval(f,hlp+x0)-hlp);
diff=abs(x1-x0);
disp(sprintf(' %10.0f %15.9f %15.9f ', k, x0, x1));
x0=x1;
if (diff < eps) % Exit if error tolerance is achieved.
success = true;
break
end
end
x_star = x0;
if (~success)
disp('Warning: Maximum number of iterations reached');
end