---
title: "fixed-point iteration"
type: concept
source: https://www.jemoka.com/posts/kbhfixed_point_iteration/
confidence: high
status: active
---

Fixed point iteration is a method for finding a fixed point of a function, which is a value that remains unchanged when the function is applied to it (i.e., f(x) = x). The method works by repeatedly applying the function to an initial guess:
Start with an initial approximation x₀
Compute successive iterations: xₙ₊₁ = f(xₙ)
Continue until convergence (|xₙ₊₁ - xₙ| &lt; ε) or maximum iterations
The method converges if the function is a contraction mapping in the neighborhood of the fixed point (|f&rsquo;(x)| &lt; 1).
