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Introduction to Numerical Analysis I

Introduction to Numerical Analysis I. Fixed Point Iteration. MATH/CMPSC 455. Fixed-Point Iteration. This can generate convergent or divergent sequences! Our interest lies mainly in the following case ：. Definition: The real number is a fixed point of the function if. Give .

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Introduction to Numerical Analysis I

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  1. Introduction to Numerical Analysis I Fixed Point Iteration MATH/CMPSC 455

  2. Fixed-Point Iteration This can generate convergent or divergent sequences! Our interestlies mainly in the following case： Definition: The real number is a fixed point of the function if

  3. Give No Converge? Yes Out put

  4. Root: Fixed Point: Example: Which one is better?

  5. Question: How can we guarantee F has a fixed point? Definition (Contractive Mapping): A mapping (or function) is said to be contractive if Theorem (Contractive Mapping Theorem): A contractive function has a unique fixed point. Moreover, the fixed point is the limit of the sequence obtained by the Fixed-Point Iteration.

  6. Example: Example: Example:

  7. Error Analysis Definition (Linear Convergence):Let denote the error at step n of a iterative method. The method is said to be obey linear convergence with rate S, if Theorem 1.6 (Linear Convergence of FPI): Assume that is continuously differentiable, that , and Then Fixed-Point Iteration converges linearly with rate to the fixed point for initial guesses sufficiently close to the fixed point.

  8. Example: Explain why the FPI converges for the following function: Example: Find the fixed points Example: Calculate by using FPI

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