function [J, grad] = costFunctionReg(theta, X, y, lambda)
%COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization
%   J = COSTFUNCTIONREG(theta, X, y, lambda) computes the cost of using
%   theta as the parameter for regularized logistic regression and the
%   gradient of the cost w.r.t. to the parameters. 

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
%               You should set J to the cost.
%               Compute the partial derivatives and set grad to the partial
%               derivatives of the cost w.r.t. each parameter in theta



predictions =  sigmoid(X*theta);

leftPart = -y' * log(predictions);

rightPart = (1 - y') * log(1 - predictions);

thetaZero = theta;

thetaZero(1) = 0;

lambaCostPart = (lambda / (2 * m)) * sum(thetaZero .^ 2);

lambdaGradPart = lambda / m * thetaZero;

J = (1 / m) * (leftPart - rightPart) + lambaCostPart;

grad = ((1/m) * (X' * (predictions - y))) + lambdaGradPart;



% =============================================================

end
