nLab Newton's method

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 $f$.

Related concepts

• Puiseux series

References

• Wikipedia, Newton’s method