Talk:Fixed-point iteration

What examples should be given in this article?

The Numerical analysis article gives one example of fixed point iteration: $x_{k+1}=(x_{k}^{2}-2)^{2}+x_{k}$ . This article can accommodate several examples and can illustrate them with graphics (maybe animations too?).

The example in the NA article can be complemented here by comparing it with the the superficially similar iteration: $x_{k+1}=(x_{k}^{2}-2)+x_{k}$ .

The Fixed point (mathematics) article has a nice graphic of an iteration of sin(x) that could be used to show that FPI can be used on many types of functions.

Examples of "what can go wrong" would also be nice.

Of course, examples are only part of the article.

--Jtir 14:13, 8 October 2006 (UTC)[reply]

I redirected this for now. Feel free to undo that if more example are added. Oleg Alexandrov (talk) 01:31, 9 October 2006 (UTC)[reply]

"fixed point iteration" is a technique in theoretical computer science

(copied from User talk:Oleg Alexandrov) --Jtir 06:17, 9 October 2006 (UTC)[reply]
Actually "fixed point iteration" is a technique in theoretical computer science: definition by recursion is regarded as solution of a fixed point problem g = F(g) and iterates of F converge to the fixed point. This technique has various flavors: an order theoretic one and a metric space one. --CSTAR 06:06, 9 October 2006 (UTC)[reply]

How are the iteration and the function related?

The third example has me wondering about the relation between the iteration and the function — why this function and not some other? Put another way, if someone gives you an iteration, can you say what function corresponds(?) to it?
This article is the place to explain the relationship between the two.

"... - this function is not continuous at x=0, and in fact has no fixed points."
I'm not sure what the dash in the third example means. Can it be replaced with the word "because"? --Jtir 18:05, 10 October 2006 (UTC)[reply]

an example comparing FPI and Newton's method?

I tried computing the example x_n+1 = sin x_n on a handheld calculator. I was down to about 0.2 when I got tired of pressing the sin button. The Further properties section could give an example comparing the number of iterations needed to reach some value using FPI with the number needed using Newton's method. --Jtir 18:25, 10 October 2006 (UTC)[reply]