数值梯度 与 解析梯度

J(x1,x2)=x21+3x1∗x2

解析梯度：
function [value,grad] = simpleQuadraticFunction(x)
% this function accepts a 2D vector as input. 
% Its outputs are:
%   value: h(x1, x2) = x1^2 + 3*x1*x2
%   grad: A 2x1 vector that gives the partial derivatives of h with respect to x1 and x2 
value = x(1)^2 + 3*x(1)*x(2);


grad = zeros(2, 1);
grad(1)  = 2*x(1) + 3*x(2);
grad(2)  = 3*x(1);
end


数值梯度
function numgrad = computeNumericalGradient(J, theta)
% numgrad = computeNumericalGradient(J, theta)
% theta: a vector of parameters
% J: a function that outputs a real-number. Calling y = J(theta) will return the
% function value at theta. 


% Initialize numgrad with zeros
numgrad = zeros(size(theta));


% Implement numerical gradient checking, and return the result in numgrad.  


epsilon = 10^(-4);
n = size(theta, 1);
for i=1:n
    theta1 = theta;
    theta1(i) = theta1(i) + epsilon;
    theta2 = theta;
    theta2(i) = theta2(i) - epsilon;
    [J1, grad] = J(theta1);
    [J2, grad] = J(theta2);
    numgrad(i) = (J1-J2)/(2*epsilon);
end
end


x = [4; 10];
[value, grad] = simpleQuadraticFunction(x);


numgrad = computeNumericalGradient(@simpleQuadraticFunction, x);


% Visually examine the two gradient computations.   
disp([numgrad grad]);


% Evaluate the norm of the difference between two solutions.  
diff = norm(numgrad-grad)/norm(numgrad+grad);
disp(diff);