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Using the image,  a) Is it possible to find an initial value x0, such that this iteration diverges? b) When the iteration converges, what is its convergence speed? ( linear, quadratic, rate constant?) Problem from lecture never solved

Using the image,  a) Is it possible to find an initial value x0, such that this iteration diverges? b) When the iteration converges, what is its convergence speed? ( linear, quadratic, rate constant?) Problem from lecture never solved

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

Using the image, 

a) Is it possible to find an initial value x0, such that this iteration diverges?

b) When the iteration converges, what is its convergence speed? ( linear, quadratic, rate constant?)

Problem from lecture never solved 

Consider the fixed-point-iteration, i+1 = 2xi 3 1 Xi
Transcribed Image Text:Consider the fixed-point-iteration, i+1 = 2xi 3 1 Xi
Expert Solution
Step 1: Prerequisite

Theorem 1:

The fixed point iteration method with iteration x subscript i plus 1 end subscript equals f open parentheses x subscript i close parentheses converges when open vertical bar f apostrophe left parenthesis x right parenthesis close vertical bar less than 1.


Theorem 2:

Let f be continuous on [a, b] and f apostrophe be continuous on (a, b). Furthermore, assume there exists k < 1 so that open vertical bar f apostrophe left parenthesis x right parenthesis close vertical bar less than kfor all x in (a, b).

  • If f apostrophe left parenthesis r right parenthesis not equal to 0, then the sequence converges linearly to the fixed point.
  • If f apostrophe left parenthesis r right parenthesis equals 0,  the sequence converges at least quadratically to the fixed point.
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