nLab
Newton's method

Contents

Idea

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.

Definition

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.

References

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