function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
    %% GRADIENTDESCENT Performs gradient descent to learn theta
    %   theta = GRADIENTDESENT(X, y, theta, alpha, num_iters) updates theta by 
    %   taking num_iters gradient steps with learning rate alpha

    % Initialize some useful values
    n = length(y); % number of training examples
    J_history = zeros(num_iters, 1);

    for iter = 1 : num_iters

        % ====================== YOUR CODE HERE ======================
        % Instructions: Perform a single gradient step on the parameter vector
        %               theta. 
        %
        % Hint: While debugging, it can be useful to print out the values
        %       of the cost function (computeCost) and gradient here.
        %
        
        % compute the gradient
        grad = zeros(size(theta));
        error = X * theta - y;
        for j = 1 : length(grad)
            grad(j) = alpha * sum(error .* X(:, j)) / n;
        end
        
        % simultaneously update all theta entries
        theta = theta - grad;
        
        % Save the cost J in every iteration    
        J_history(iter) = computeCost(X, y, theta);
    end
end
