function [x_vals, f_vals, k] = newton(f, grad_f, hessian_f, x0, tol, max_iter, mode)
    % f:         Objective function
    % grad_f:    Gradient of the function
    % hessian_f: Hessian of the function
    % x0:        Initial point [x0, y0]
    % tol:       Tolerance for stopping criterion
    % max_iter:  Maximum number of iterations
    
    % x_vals: Vector with the (x,y) values until minimum
    % f_vals: Vector with f(x,y) values until minimum
    % k:      Number of iterations


    if strcmp(mode, 'armijo') == 1
        gamma_f = @(f, grad_f, x0) gamma_armijo(f, grad_f, x0);
    elseif strcmp(mode, 'minimized') == 1
        gamma_f = @(f, grad_f, x0) gamma_minimized(f, grad_f, x0);
    else % mode == 'fixed'
        gamma_f = @(f, grad_f, x0) gamma_fixed(f, grad_f, x0);
    end

    x_vals = x0;  % Store iterations
    f_vals = f(x0(1), x0(2));
    
    for k = 1:max_iter
        grad = grad_f(x0(1), x0(2));
        
        % Check for convergence
        if norm(grad) < tol
            break;
        end
        hess = hessian_f(x0(1), x0(2));
        
         % Solve for search direction using Newton's step
        dk = - inv(hess) * grad;

        % Calculate gamma
        gamma = gamma_f(f, grad_f, x0);
        
        x_next = x0 + gamma * dk'; % Update step
        f_next = f(x_next(1), x_next(2));
        
        x0 = x_next; % Update point
        x_vals = [x_vals; x_next]; % Store values
        f_vals = [f_vals; f_next]; % Store function values
    end
end