Numerical methodsLesson 5 of 5 in the course Root finding

Fixed-Point Iteration: Convergence, Stability, and Acceleration

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

Fixed-point iteration turns the search for a solution into a repeated calculation. Starting from an initial guess, we apply a function again and again and observe whether the resulting sequence approaches a fixed point.

This produces the sequence x0,x1,x2,…x_0, x_1, x_2, \dots, where, for example, x1=f(x0)x_1=f(x_0) and x2=f(f(x0))x_2=f(f(x_0)). If the sequence converges to a limit xfixx_{\text{fix}} and ff is continuous at that limit, then taking limits in the iteration rule gives f(xfix)=xfixf(x_{\text{fix}})=x_{\text{fix}}.

All flashcards

What is a fixed point?
A value xfixx_{\text{fix}} that is unchanged by a function: f(xfix)=xfixf(x_{\text{fix}})=x_{\text{fix}}.
What is the fixed-point iteration rule?
Starting from x0x_0, repeatedly calculate xn+1=f(xn)x_{n+1}=f(x_n) to form a sequence of approximations.
Why is continuity relevant to fixed-point iteration?
If the iterates converge to xfixx_{\text{fix}} and ff is continuous there, the iteration equation implies f(xfix)=xfixf(x_{\text{fix}})=x_{\text{fix}}.
How does the Babylonian method approximate a square root?
For a>0a>0, it repeatedly replaces xx with the average of xx and a/xa/x: f(x)=12(a/x+x)f(x)=\frac{1}{2}(a/x+x). The iteration approaches a\sqrt{a} from a positive starting estimate.
What is linear convergence?
A convergence rate where the error is bounded by a constant times qnq^n for some 0<q<10<q<1, so the error decreases geometrically.

Quiz questions

  1. What equation defines a fixed point of a function ff?Answer: f(xfix)=xfixf(x_{\text{fix}})=x_{\text{fix}}
  2. Given an initial value x0x_0, what is the standard fixed-point iteration update?Answer: xn+1=f(xn)x_{n+1}=f(x_n)
  3. Why does continuity help show that a convergent iteration limit is a fixed point?Answer: Continuity lets the limit pass through ff, giving f(xfix)=xfixf(x_{\text{fix}})=x_{\text{fix}}.
  4. Which update function is used in the Babylonian method for approximating a\sqrt{a}?Answer: f(x)=12(ax+x)f(x)=\frac{1}{2}\left(\frac{a}{x}+x\right)
  5. What does an error bound proportional to qnq^n, where 0<q<10<q<1, indicate?Answer: The error decreases geometrically, which is linear convergence.

Based on the Wikipedia article Fixed-point iteration, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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