---
tag: K6ZF91051B
kind: concept
name: Fixed-point iteration
msc: [47J25, 65H05, 40A05]
sources:
  - url: https://en.wikipedia.org/wiki/Fixed-point_iteration
same_as:
  wikidata: Q1030759
  wikipedia: Fixed-point iteration
---

## Summary

Fixed-point iteration computes a solution of $f(x) = x$ by applying $f$ again and again to a starting point.
If the resulting sequence converges and $f$ is continuous, its limit is a fixed point of $f$.
The [[Banach fixed point theorem|0VK6VEH345]] guarantees convergence for a contraction on a complete metric space.
Many numerical methods, such as [[Newton's method|07EXVNF9VP]], are fixed-point iterations.

## Explanations

### L1 human

Pick a number, apply a function to it, apply the function to the result, and keep going.
If the numbers settle down, they settle at a point that the function does not move.
Try it with the cosine key of a calculator in radians: starting from $1$, the values approach about $0.739085$, the only real solution of $\cos x = x$.
Iteration is a simple way to solve an equation when one can rewrite it as $x = f(x)$.