Talk:Fixed point iteration

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[edit] 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 --uhvpirate 04:55, 12 April 2007 (UTC)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)

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

[edit] "fixed point iteration" is a technique in theoretical computer science

(copied from User talk:Oleg Alexandrov) --Jtir 06:17, 9 October 2006 (UTC)
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)

[edit] 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)

The function f and the iteration rule x_n \rightarrow x_{n+1} are related by xn + 1 = f(xn). The point of the third example is to show that the condition that f is continuous is a necessary condition; if f is not continuous then the iterations of f can converge to a value that is not a fixed point of f. Gandalf61 14:39, 11 October 2006 (UTC)

[edit] an example comparing FPI and Newton's method?

I tried computing the example xn+1 = sin xn 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)

The Newton iteration is a fixed point iteration. Loisel 22:37, 10 October 2006 (UTC)

I took that distinction from the article, which reads:
  • "Fixed point iteration is not always the best method of computing fixed points. For example, Newton's method ..."
--Jtir 22:47, 10 October 2006 (UTC)

Well obviously the article is missing a point. Loisel 00:24, 11 October 2006 (UTC)

You are right. Newton's iteration is a fixed point iteration. I fixed that in the article. Oleg Alexandrov (talk) 03:27, 11 October 2006 (UTC)
Sorry, I think I overwrote your text when I got an edit conflict. I had written up a bunch of stuff which seemed to moot your changes. Loisel 04:27, 11 October 2006 (UTC)

[edit] example of the Babylonian method

Thanks for adding the Babylonian method to the examples. I noticed that it is the only example that does not give the iteration. --Jtir 17:03, 12 October 2006 (UTC)

[edit] complex fixed points?

not sure if this is the right place to ask this but here goes anyway: obviously functions that don't cross f(x)=x shouldn't have fixed points, like ln(x). But you can use fixed point iteration on ln(x) to obtain an attractive complex fixed point. Does anyone have any information on complex fixed points?--uhvpirate 04:55, 12 April 2007 (UTC)

[edit] Toilet flushing

I had some diarrhea today and when i flushed i wondered: Can the question of whether the water spirals clockwise or counter-clockwise finally be answered by using a fixed-point problem? —Preceding unsigned comment added by 130.85.237.88 (talk) 11:55, 14 November 2007 (UTC)