Fixed Point Iteration
Fixed-point iteration x_{k+1}=g(x_k) converges to a fixed point x*=g(x*) of g, with local linear convergence when |g'(x*)|<1.
Purpose
To reformulate root-finding as a fixed-point problem and analyze convergence conditions.
Mechanism
Rewrite f(x)=0 as x=g(x). Banach fixed-point theorem: if g is a contraction mapping (|g'|≤L<1), the iteration converges to a unique fixed point. Convergence speed depends on the derivative of g at the fixed point.
Use cases
- Equation root finding
- Iterative method analysis
- Numerical methods