Newton's method



What is called Newton’s method after Isaac Newton is an recursive procedure for computing approximations to zeros (“roots”) of differentiable functions with values in the real numbers.


Let f:f \colon \mathbb{R} \to \mathbb{R} be a differentiable function and for x 0x_0 \in \mathbb{R} a real number such that the first derivative f:f' \colon \mathbb{R} \to \mathbb{R} is non-vanishing at x 0x_0.

Let {x n|n}\{x_n \in \mathbb{R} | n \in \mathbb{N}\} be defined recursively by

x n+1x nf(x n)f(x n). x_{n+1} \coloneqq x_n - \frac{f(x_n)}{f'(x_n)} \,.

Under mild conditions, this sequence converges to a zero/root of ff.


Created on February 4, 2013 at 10:08:41. See the history of this page for a list of all contributions to it.