Contraction Mapping Principle
Definition
Banach Fixed Point Theorem
Proof Outline
- Cauchy sequence: , so converges, making Cauchy
- Convergence: Complete space
- Fixed point:
- Uniqueness: If , then , so
Example
Solve :
Let . Then on .
Starting from : , , ,
Applications to Differential Equations
Applications to Linear Systems
Convergence Rate
Linear convergence: error decreases by factor each step.
Generalizations
Applications
Banach's Lemma
This Neumann series converges and gives the inverse.
Error Bounds
| Iteration | Upper Bound |
|---|---|