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

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

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

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

## References

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