function [a, b, k, n] = min_bisection(fun_expr, alpha, beta, epsilon, lambda)
% Bisection method for finding the local minimum of a function.
%
% fun_expr: (symbolic expression over x) The symbolic expression of the
%           objective function
% alpha:    (number) The starting point of the interval in which we seek
%           for minimum
% beta:     (number) The ending point of the interval in which we seek
%           for minimum
% epsilon:  (number) The epsilon value (distance from midpoint)
% lambda:   (number) The lambda value (accuracy)
%
% return:
%  a:       (vector) Starting points of the interval for each iteration
%  b:       (vector) Ending points of the interval for each iteration
%  k:       (number) The number of iterations
%  n:       (number) The calls of objective function fun_expr
%



% Error checking
if alpha > beta || 2*epsilon >= lambda || lambda <= 0
    error ('Input criteria not met')
end

% Init
a = alpha;
b = beta;
n = 0;
fun = matlabFunction(fun_expr);

% wrapper call count function
function r = count_fun(x)
    n = n + 1;
    r = fun(x);
end

k=1;
while b(k) - a(k) > lambda
    % bisect [a,b]
    mid = (a(k) + b(k)) / 2;
    x_1 = mid - epsilon;
    x_2 = mid + epsilon;
    
    % set new search interval
    k = k + 1;
    if count_fun(x_1) < count_fun(x_2)
        a(k) = a(k-1);
        b(k) = x_2;
    else
        a(k) = x_1;
        b(k) = b(k-1);
    end
end

end