Fixed-point iteration: xₙ₊₁ = g(xₙ). Converges if |g'(x*)| < 1 (contraction mapping). Rate: linear if g'(x*)≠0, quadratic if g'(x*)=0. Error: |xₙ−x*| ≤ Lⁿ/(1−L)|x₁−x₀| where L = max|g'|.
How to Use
Select a function:
- g(x): Iteration function
- x₀: Initial guess
- Output: Fixed point x*
Convergence
Banach fixed-point theorem: if g maps [a,b]→[a,b] and |g'(x)| ≤ L < 1 on [a,b], then there exists a unique fixed point and iteration converges for any x₀ in [a,b].
Applications
- Root finding: f(x)=0 → x = x−f(x) or x = x−f(x)/f'(x)
- Newton's method IS fixed-point with g(x) = x−f(x)/f'(x)
- Picard iteration for ODEs
Step-by-Step Instructions
- 1Select g(x).
- 2Enter x₀.
- 3Run iterations.
- 4Check convergence.
- 5Get fixed point.