Editing Newton's method in optimization (section)

==Higher dimensions==
The above [[iteration|iterative scheme]] can be generalized to <math> d>1</math> dimensions by replacing the derivative with the [[gradient]] (different authors use different notation for the gradient, including <math>f'(x) = \nabla f(x) = g_f(x)\in \mathbb{R}^d</math>), and the [[Multiplicative inverse|reciprocal]] of the second derivative with the [[Invertible matrix|inverse]] of the [[Hessian matrix]] (different authors use different notation for the Hessian, including <math>f''(x) = \nabla^2 f(x) = H_f(x)\in \mathbb{R}^{d\times d}</math>).  One thus obtains the iterative scheme

:<math>x_{k+1} = x_k - [f''(x_k)]^{-1} f'(x_k), \qquad  k \ge 0.</math>

Often Newton's method is modified to include a small [[Learning rate|step size]] <math> 0 < \gamma \le 1 </math> instead of <math>\gamma=1</math>:
:<math>x_{k+1} = x_k - \gamma [f''(x_k)]^{-1} f' (x_k).</math>
This is often done to ensure that the [[Wolfe conditions]], or much simpler and efficient [[Backtracking line search|Armijo's condition]], are satisfied at each step of the method. For step sizes other than 1, the method is often referred to as the relaxed or damped Newton's method.