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Function Iteration was redirected to Function composition. Should this article redirect there as well (after any necessary merging)? - dcljr ( talk) 18:58, 11 November 2005 (UTC)
It appears that the articles Recurrence relation and Iterated function are about the same thing. Is there any objection to merging them? Which should be merged into which? Duoduoduo ( talk) 22:52, 28 May 2010 (UTC)
See discussion at talk:Recurrence relation. —Preceding unsigned comment added by Jowa fan ( talk • contribs) 06:19, 1 June 2010 (UTC)
Surely the expressions can be by far non rigurous, but can be someone to look my notes on [1] and improve that or comment about? — Preceding unsigned comment added by 80.25.164.213 ( talk) 20:06, 20 July 2012 (UTC)
From my time as student I develop some aproximation to this field. Maybe some ideas can be useful. [2] — Preceding unsigned comment added by 80.25.164.213 ( talk) 10:27, 20 August 2012 (UTC)
I think the article should say a bit more about the question of existence and uniqueness (or lack thereof) of fractional and continuous iterates. At the moment it simply says "In some instances, fractional iteration of a function can be defined", which is not tremendously illuminating. 86.160.216.252 ( talk) 13:31, 24 October 2012 (UTC)
There was a section in inverse function related to f 2, f k, and so, that I trimmed due to obvious WP:stay on topic concerns. Can somebody reuse this stuff here? Incnis Mrsi ( talk) 15:35, 3 July 2013 (UTC)
The section Some formulas for fractional iteration is very similar to my own unpublished results. My question is, can anyone verify the formulas are in a peer reviewed published work? Daniel Geisler ( talk) 20:21, 13 May 2014 (UTC)
It's been a week and nobody has provided any reason to think that the section on fractional iteration can be validated through being published. Ironically I don't disagree with the results, I just think they should be published first. Daniel Geisler ( talk) 11:22, 21 May 2014 (UTC)
I've rewritten the lead to make it make sense at least marginally. However, it struck me that the article seems to be more about the general notion of iterating functions (as a generaly activity, or subject of study) than about the specific subject of functions that are of the form f n. Therefore I think the name "Function iteration" would much better cover the contents than "Iterated function". Marc van Leeuwen ( talk) 08:42, 18 February 2015 (UTC)
When the article says, "Note: these two special cases of ax2 + bx + c are the only cases that have a closed-form solution. Choosing b = 2 = –a and b = 4 = –a, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table." Does it mean that we only know of two closed form solutions, or has it been proven that there are no others? — Preceding unsigned comment added by 75.243.141.93 ( talk) 21:20, 4 October 2015 (UTC)
I've discovered that not only fractional iteration is possible, but also complex iteration. It's not as intuitive as fractional iteration, but it does make sense once you have a new model for what iteration means. Probably the simplest explanation is with orbitals in the complex plain: iterating to a real values generates the complex orbitals as paths. Iterating to imaginary powers, meanwhile, generates paths perpendicular to the orbitals at every point. You need the entire vector field of orbitals to know the new path. For example, iterating a linear function (multiplying by a constant) creates orbitals which are rays. Iterating to imaginary powers creates paths with are concentric circles. After that, complex iteration can be done by composing real and imaginary values. For a more precise derivation, I had to invent a new type of derivative, it looks at the instantaneous change in value at a point as the function iterates. The formula for this derivative is: f*(x) = lim n->0 (fn(x) - x)/n Which is equivalent to the derivative of f's Abel function. The key about this derivative is it has the property that fn* = f x n . This enables iteration to imaginary powers to be calculated directly. Ultimately I think it is equivalent to taking the analytic extension of the Abel function, but it's a bit more intuitive than Taylor series witchcraft. I've done the calculations to arrive at Euler's identity using this method rather than the typical one, and graphically it makes much more sense: once it's understood how complex iteration works, it's just geometry to find the imaginary value e must be raised to in order to complete the arc from 1 to -1. — Preceding unsigned comment added by 216.49.181.128 ( talk) 15:34, 17 December 2015 (UTC)
In one of the subsections, it says "When n is not an integer, make use of the power formula y n = exp(n ln(y))". This will not work at all and makes no sense, n is not a power, it is the number of iterations. That line should be deleted.
There's no reply button, so I'll just have to edit here to say that I don't see anything in the Curtright, T.L. Evolution surfaces and Schröder functional methods. That article does not address what I asked, and instead of using facts and evidence, someone trolled by editing my question to say the article did, so I am reporting them for a moderator to look into.
I have not seen anyone actually bring up any logical contention with this change, so in two days (from my time zone) I will delete that sentence. Posted by User:Leakdope without signature, 12/2018.
````Leakdope```` — Preceding unsigned comment added by Leakdope ( talk • contribs) 04:09, 12 December 2018 (UTC)
Please see my comments at Talk:Tetration#Moving_towards_a_verifiable_article which is relevant to both articles. Daniel Geisler ( talk) 18:53, 30 April 2019 (UTC)
Sorry, but I strongly disagree with the statement that fractional iteration is a mature functional conjugacy field. Papers are being published on extending tetration to the real and complex numbers solely on Abel's Functional Equation which is wrong because it is only valid for once the coordinate system is shifted to a fixed point then must be true. I am in the process of writing something up to send to these authors and the editors of the journals they published in.
If you review the older material from the beginnings of the Tetration Forum you will see them arguing that fixed points were not important, while my work starts with fixed points. So this work is inconsistent with Schroeder's Über iterirte Functionen, considered the first paper on dynamics.
Even though I try and track people and papers, both for myself and the community of mathematicians I communicate with. Maybe our different views of functional conjugacy are based on you being connected to people and papers I am unaware of. I'd appreciate any information you wish to share, either publically or privately. Thank you.
Daniel Geisler ( talk) 20:45, 1 May 2019 (UTC)
I'm sorry then, I see that we will not be agreeing. This is not an appropriate place for me to benefit people. I'm fine with letting history judge the merits of our work here. So best wishes to the folks here.
Daniel Geisler ( talk) 07:49, 2 May 2019 (UTC)
by the omission of the top illustration, which is overwhelmingly confusing.
It is confusing simply because it tries to convey about ten times as much information as one illustration can convey.
Often — and in this case — less is more.
If the illustration were replaced by a far simpler one, having almost no text, that would be a good thing. 2601:200:C000:1A0:5FB:D9A8:4BAF:D605 ( talk) 19:34, 24 November 2022 (UTC)
This article contains multiple passages with bad, confusing, and/or misleading writing. The top illustration is at least as confusing as any other bad illustration in Wikipedia, but even more so. Significant important relevant topics, like Koenigs's theorem as well as the central topic of embedding certain functions into a continuous flow, are left entirely unmentioned. And the passage about iterating f(x) = (√2)x is a train wreck. - 2601:200:c082:2ea0:a874:6184:ede8:53c1 11:56, 21 May 2023
if you use the f^0(x) axiom,as well as the exponentioation property of iterated functions you can conclude that the half itarate of x aka the identy map is itself. 100.2.153.196 ( talk) 01:24, 20 November 2023 (UTC)
I claim that for every ... we have ... . To prove it, let ... be ... . Then, we have ... = ... = ... by ... and ..., respectively. Hence, ... by ... . etc. ... Hence, we are done." - Jochen Burghardt ( talk) 10:13, 22 November 2023 (UTC)
This is the
talk page for discussing improvements to the
Iterated function article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||
|
Function Iteration was redirected to Function composition. Should this article redirect there as well (after any necessary merging)? - dcljr ( talk) 18:58, 11 November 2005 (UTC)
It appears that the articles Recurrence relation and Iterated function are about the same thing. Is there any objection to merging them? Which should be merged into which? Duoduoduo ( talk) 22:52, 28 May 2010 (UTC)
See discussion at talk:Recurrence relation. —Preceding unsigned comment added by Jowa fan ( talk • contribs) 06:19, 1 June 2010 (UTC)
Surely the expressions can be by far non rigurous, but can be someone to look my notes on [1] and improve that or comment about? — Preceding unsigned comment added by 80.25.164.213 ( talk) 20:06, 20 July 2012 (UTC)
From my time as student I develop some aproximation to this field. Maybe some ideas can be useful. [2] — Preceding unsigned comment added by 80.25.164.213 ( talk) 10:27, 20 August 2012 (UTC)
I think the article should say a bit more about the question of existence and uniqueness (or lack thereof) of fractional and continuous iterates. At the moment it simply says "In some instances, fractional iteration of a function can be defined", which is not tremendously illuminating. 86.160.216.252 ( talk) 13:31, 24 October 2012 (UTC)
There was a section in inverse function related to f 2, f k, and so, that I trimmed due to obvious WP:stay on topic concerns. Can somebody reuse this stuff here? Incnis Mrsi ( talk) 15:35, 3 July 2013 (UTC)
The section Some formulas for fractional iteration is very similar to my own unpublished results. My question is, can anyone verify the formulas are in a peer reviewed published work? Daniel Geisler ( talk) 20:21, 13 May 2014 (UTC)
It's been a week and nobody has provided any reason to think that the section on fractional iteration can be validated through being published. Ironically I don't disagree with the results, I just think they should be published first. Daniel Geisler ( talk) 11:22, 21 May 2014 (UTC)
I've rewritten the lead to make it make sense at least marginally. However, it struck me that the article seems to be more about the general notion of iterating functions (as a generaly activity, or subject of study) than about the specific subject of functions that are of the form f n. Therefore I think the name "Function iteration" would much better cover the contents than "Iterated function". Marc van Leeuwen ( talk) 08:42, 18 February 2015 (UTC)
When the article says, "Note: these two special cases of ax2 + bx + c are the only cases that have a closed-form solution. Choosing b = 2 = –a and b = 4 = –a, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table." Does it mean that we only know of two closed form solutions, or has it been proven that there are no others? — Preceding unsigned comment added by 75.243.141.93 ( talk) 21:20, 4 October 2015 (UTC)
I've discovered that not only fractional iteration is possible, but also complex iteration. It's not as intuitive as fractional iteration, but it does make sense once you have a new model for what iteration means. Probably the simplest explanation is with orbitals in the complex plain: iterating to a real values generates the complex orbitals as paths. Iterating to imaginary powers, meanwhile, generates paths perpendicular to the orbitals at every point. You need the entire vector field of orbitals to know the new path. For example, iterating a linear function (multiplying by a constant) creates orbitals which are rays. Iterating to imaginary powers creates paths with are concentric circles. After that, complex iteration can be done by composing real and imaginary values. For a more precise derivation, I had to invent a new type of derivative, it looks at the instantaneous change in value at a point as the function iterates. The formula for this derivative is: f*(x) = lim n->0 (fn(x) - x)/n Which is equivalent to the derivative of f's Abel function. The key about this derivative is it has the property that fn* = f x n . This enables iteration to imaginary powers to be calculated directly. Ultimately I think it is equivalent to taking the analytic extension of the Abel function, but it's a bit more intuitive than Taylor series witchcraft. I've done the calculations to arrive at Euler's identity using this method rather than the typical one, and graphically it makes much more sense: once it's understood how complex iteration works, it's just geometry to find the imaginary value e must be raised to in order to complete the arc from 1 to -1. — Preceding unsigned comment added by 216.49.181.128 ( talk) 15:34, 17 December 2015 (UTC)
In one of the subsections, it says "When n is not an integer, make use of the power formula y n = exp(n ln(y))". This will not work at all and makes no sense, n is not a power, it is the number of iterations. That line should be deleted.
There's no reply button, so I'll just have to edit here to say that I don't see anything in the Curtright, T.L. Evolution surfaces and Schröder functional methods. That article does not address what I asked, and instead of using facts and evidence, someone trolled by editing my question to say the article did, so I am reporting them for a moderator to look into.
I have not seen anyone actually bring up any logical contention with this change, so in two days (from my time zone) I will delete that sentence. Posted by User:Leakdope without signature, 12/2018.
````Leakdope```` — Preceding unsigned comment added by Leakdope ( talk • contribs) 04:09, 12 December 2018 (UTC)
Please see my comments at Talk:Tetration#Moving_towards_a_verifiable_article which is relevant to both articles. Daniel Geisler ( talk) 18:53, 30 April 2019 (UTC)
Sorry, but I strongly disagree with the statement that fractional iteration is a mature functional conjugacy field. Papers are being published on extending tetration to the real and complex numbers solely on Abel's Functional Equation which is wrong because it is only valid for once the coordinate system is shifted to a fixed point then must be true. I am in the process of writing something up to send to these authors and the editors of the journals they published in.
If you review the older material from the beginnings of the Tetration Forum you will see them arguing that fixed points were not important, while my work starts with fixed points. So this work is inconsistent with Schroeder's Über iterirte Functionen, considered the first paper on dynamics.
Even though I try and track people and papers, both for myself and the community of mathematicians I communicate with. Maybe our different views of functional conjugacy are based on you being connected to people and papers I am unaware of. I'd appreciate any information you wish to share, either publically or privately. Thank you.
Daniel Geisler ( talk) 20:45, 1 May 2019 (UTC)
I'm sorry then, I see that we will not be agreeing. This is not an appropriate place for me to benefit people. I'm fine with letting history judge the merits of our work here. So best wishes to the folks here.
Daniel Geisler ( talk) 07:49, 2 May 2019 (UTC)
by the omission of the top illustration, which is overwhelmingly confusing.
It is confusing simply because it tries to convey about ten times as much information as one illustration can convey.
Often — and in this case — less is more.
If the illustration were replaced by a far simpler one, having almost no text, that would be a good thing. 2601:200:C000:1A0:5FB:D9A8:4BAF:D605 ( talk) 19:34, 24 November 2022 (UTC)
This article contains multiple passages with bad, confusing, and/or misleading writing. The top illustration is at least as confusing as any other bad illustration in Wikipedia, but even more so. Significant important relevant topics, like Koenigs's theorem as well as the central topic of embedding certain functions into a continuous flow, are left entirely unmentioned. And the passage about iterating f(x) = (√2)x is a train wreck. - 2601:200:c082:2ea0:a874:6184:ede8:53c1 11:56, 21 May 2023
if you use the f^0(x) axiom,as well as the exponentioation property of iterated functions you can conclude that the half itarate of x aka the identy map is itself. 100.2.153.196 ( talk) 01:24, 20 November 2023 (UTC)
I claim that for every ... we have ... . To prove it, let ... be ... . Then, we have ... = ... = ... by ... and ..., respectively. Hence, ... by ... . etc. ... Hence, we are done." - Jochen Burghardt ( talk) 10:13, 22 November 2023 (UTC)