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mcastillo356
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- TL;DR Summary
- Want to understand which kind of functions I can be sure that the method of fixed-point iteration is suitable
Hi PF
Not every function works when we try to compute the root with this method
The following theorem guarantees that the method of fixed point iteration will work for a particular class of functions
A fixed point theorem
Suppose that is defined on an interval and satisfies the following two conditions
(i) belongs to whenever belongs to ; and
(ii) there exists a constant with such that for every and in
Then has a unique fixed point in , that is, , and starting with any number in , the iterates
converge to
Hints for the proof
1- Condition (ii) of theorem implies that is continuous on . Use condition (i) to show that has a unique fixed point on . Apply the Intermediate-Value Theorem to on .
2- Use condition (ii) of theorem and mathematical induction to show that . Since , we know that as . This shows that
First: ¿Why bounded Newton quotient means continuity on the interval?.
Thanks
Not every function works when we try to compute the root with this method
The following theorem guarantees that the method of fixed point iteration will work for a particular class of functions
A fixed point theorem
Suppose that
(i)
(ii) there exists a constant
Then
converge to
Hints for the proof
1- Condition (ii) of theorem implies that
2- Use condition (ii) of theorem and mathematical induction to show that
First: ¿Why bounded Newton quotient means continuity on the interval?.
Thanks