Talk:Fixed point (mathematics)/Archive 1

Archive 1

Eigenvectors

Fixed points seem to be deeply related to eigenvectors. I added "Eigenvector" to the "See also" section, but perhaps a mathematician could go into more detail? —Ben FrantzDale 23:25, 26 September 2007 (UTC)

Attractive Fixed Points

The statement about the |f'(x₀)|<1 implying an attractive fixed point needs either a proof or a reference to a proof. Azurefox (talk) 18:03, 15 August 2008 (UTC)

It is an easy consequence of the Banach fixed point theorem, using the topological definition of continuity and the mean value theorem.  --Lambiam 23:36, 19 August 2008 (UTC)

What about f(x)=exp(x), and x₀=-1/2? f'(x)=exp(x) -> f'(x₀)≈0.60653066<1 . But with this value f(f(f(...x))) quickly diverges. Maybe I'm not understanding this right...Goldencako 18:15, 4 October 2008 (UTC)

The statement requires that x₀ be a fixed point. If the derivative at a fixed point is < 1, then it is an attractive fixed point, which means that nearby values converge when you do f(f(f...(x))). But exp(x) has no fixed points, so the statement about derivatives doesn't say anything. Staecker (talk) 23:36, 4 October 2008 (UTC)

Convergence section moved to Convergence_(mathematics)

This article is about fixed point but rather abruptly introduced a mathematical definition of convergence, without informal explication in the text of how the two notions are connected. This was confusing, at least for me. I have moved this section in the article about the mathematical sense of Convergence, and hinted to it in the section "Applications". Sylvain. —Preceding unsigned comment added by 217.81.166.165 (talk) 21:26, 3 February 2010 (UTC)

EXAMPLE OF CALCULATED FIXED POINT ITERATION

The function g(x)= e^-x has a unique fixed point some where near x=0.6. To locate this fixed point more precisely we will now perform fixed point iteration with g as the iteration function and Po=0. The first 10iterations yielded;

P1 = g(Po) = 1.0000000000

P2 = g(P1) = 0.3678794412

P3 = g(P2) = 0.6922006276

P4 = g(P3) = 0.5004735006

P5 = g(P4) = 0.6062435351

P6 = g(P5) = 0.5453957860

P7 = g(P6) = 0.5796123355

P8 = g(P7) = 0.5601154614

P9 = g(P8) = 0.5711431151

P10= g(P9) = 0.5648793474

The sequence appears to be converging very slowly. It takes more than 20 iterations for Pn to agree with the exact fixed point at least 5 significance decimal digits.

— Preceding unsigned comment added by Nraymoss (talk • contribs) 03:15, 24 September 2012 (UTC)

Where is this supposed to go on the page?

What defficiency is it supposed to address?

What does the reader learn from reading it?

Your math formatting is missing.

I also cleaned out the stuff you left about Fixed-point arithmetic.

Mjmohio (talk) 19:26, 24 September 2012 (UTC)

Making the graph and equation of the lead section match

The article could be a bit clearer if the prominent equation in the lead was plotted at right, rather than differing equation and plot. Any thoughts on why the article is this way at the moment? Otherwise, will change. Scientific29 (talk) 06:52, 19 February 2014 (UTC)

attractive fixed set

Attractive fixed set redirects here, but what is this? - üser:Altenmann >t 16:07, 16 December 2015 (UTC)

The terminology "Parabolic"/"Elliptic"/"Hyperbolic" fixpoints

One can find that terminology using google or mathworld, but the explanation is extremely cryptic for me, so I can only guess what it means.

So for instance for f(x)= 2^x-1 with Taylorseries ux + (ux)^2/2!+(ux)^3/3! + ... (where u=log(2)) there is one fixpoint t0=0 because f(0)=0. It is "attracting" because at the first term "ux" the coefficient "u" has value |u|<1 .

But: is it "parabolic"?"Hyperbolic"?"Elliptic"?

What about the second fixpoint, t1=1? We have f(1)=1. Developed around that fixpoint the Taylorseries has representation wx + (wx)^2/2!+... where w=2u ~1.386 . At the first term "wx" the absolute value of the coefficient "w" is larger than 1. So this fixpoint is "repelling".

Again: is it "parabolic"? "Elliptic" ? "Hyperbolic"?

Can we have three simple examples which illustrate that three types of fixpoints?

Gottfried--Gotti 09:37, 10 March 2016 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

I guess that you are confusing fixed points and stationary points. For a bivariate function, a stationary point may be called elliptic, hyperbolic or parabolic, depending on the number of real eigenvalues of the Hessian matrix: the point is elliptic if the eigenvalues are not real, and the stationary point is a local extremum. The point is hyperbolic if the eigenvalues are real and distinct, and the point is a saddle point. The point is parabolic if the eigenvalues are equal, and higher derivatives are needed to decide the shape of the surface in a neighborhood of the point. D.Lazard (talk) 22:16, 10 March 2016 (UTC)

No, see for instance the term "parabolic fixpoint" at mathworld and its completely cryptic description at http://mathworld.wolfram.com/ParabolicFixedPoint.html or the completely "freehandling" of "parabolic fixpoint", "hyperbolic tetration", "parabolic rational neutral fixpoint" or similar at http://tetration.org/Tetration/index.html --Gotti 12:34, 11 March 2016 (UTC) — Preceding unsigned comment added by Druseltal2005 (talk • contribs)

Assessment comment

The comment(s) below were originally left at Talk:Fixed point (mathematics)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs references. Geometry guy 21:35, 9 July 2007 (UTC) I think there's a typo under the section "Topological fixed point property". Two sentences in that section read: "The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The PPF is also preserved by any retraction." I'm assuming that the "PPF" should be a "FPP". Confluente 18:59, 16 October 2009 (UTC)

Last edited at 18:00, 16 October 2009 (UTC). Substituted at 02:06, 5 May 2016 (UTC)

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Convergence of a function

The § Applications section makes this claim:

The concept of fixed point can be used to define the convergence of a function.

But it neglects to indicate, even in general terms, how it can be so used. Accordingly, I've marked it with the {{vague}} template. Further, the one clue it gives to a method is the wikilink to "convergence of a function". However, that article, despite giving numerous definitions of the limit of a function, applies the verb "converge" solely to sequences, rather than functions; yet nowhere uses the phrase "fixed point". So how are readers supposed to satisfy themselves that the claim above is true?

The section is also singularly short on references, so I've flagged it with the {{refimprove section}} template. yoyo (talk) 11:29, 7 July 2018 (UTC)

Decluttered

It said, “Not all functions have fixed points: for example, if f is a function defined on the real numbers as f(x) = x + 1, then it has no fixed points, since x is never equal to x + 1 for any real number.” I shortened this to “Not all functions have fixed points: for example, f(x) = x + 1, has no fixed points, since x is never equal to x + 1 for any real number.” Editing out clutter. Okay?--Solomonfromfinland (talk) 01:28, 28 February 2021 (UTC)

Repulsive fixed point

"Repulsive fixed point links here", but there is not a single instance of the word "repulsive" in the article. — MFH:Talk 12:53, 28 March 2022 (UTC)

The more common term for this seems to be Repelling fixed point, which should probably also be made a redirect if the concept is described here.

A fixed-point a of a function f is said to be repelling if there is a neighbourhood U such that for any $x\in U$ such that $x\neq a$, there is an $n\in \mathbb {N}$ such that n-fold application of f takes x to a point outside of U.

Some textbooks more concerned with differentiable real functions reserve the moniker "repelling" for those functions and fixed points whose derivative at the fixed point is greater than one and use the term "weakly repelling" for the more general notion.

I would think we should include the former definition, since it matches our usage of the term attracting, but it does seem to me that the latter might be more common among textbook authors. Felix QW (talk) 16:20, 28 March 2022 (UTC)

References

Requested move 1 March 2022

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: No consensus: I don't see anything here that would suggest a consensus on what the name should be. I suggest further discussion of the article title.(non-admin closure) Spekkios (talk) 03:51, 8 April 2022 (UTC)

---

• Fixed point (mathematics) → Fixed point
• Fixed point → Fixed point (disambiguation)

– This requires to move first Fixed point to Fixed point (disambiguation). The reason, already discussed at talk:Fixed point (mathematics), is that all non-mathematical items listed in the dab page are far to be a primary topic D.Lazard (talk) 10:25, 27 February 2022 (UTC) — Relisting. Extraordinary Writ (talk) 23:03, 6 March 2022 (UTC)

This is a contested technical request (permalink). GeoffreyT2000 (talk) 01:33, 1 March 2022 (UTC)

• This needs discussion, since if there is a primary meaning, it may be Fixed-point arithmetic. —⁠ ⁠BarrelProof (talk) 19:42, 28 February 2022 (UTC)
  • As arithmetic is a part of mathematics, the problem exists already with the current title: the disambiguating parenthesis in Fixed point (mathematics) means that this article is a main topic in mathematics. Moreover Fixed-point arithmetic is not about any sort of fixed points; it is about a specific arithmetic. So Fixed-point arithmetic cannot be a primary meaning for "fixed point". D.Lazard (talk) 22:26, 28 February 2022 (UTC)
    • I'm not following that explanation. I agree that the current title of Fixed point (mathematics) might have a problem, but that doesn't mean that these proposed moves are appropriate. You seem to acknowledge that Fixed-point arithmetic discusses a different topic than Fixed point (mathematics), but I fail to see how that leads to a conclusion that "fixed point" primarily refers to Fixed point (mathematics). I continue to think this is not an obviously unquestionable move request. —⁠ ⁠BarrelProof (talk) 22:59, 28 February 2022 (UTC)
      • To be clearer, Fixed-point arithmetic does not discuss any topic that is called a fixed point. In the article, "fixed-point", or "fixed point" (incorrectly, I think) is always used either as an adjective, or as an abbreviation of "fixed-point representation". So, Fixed-point arithmetic could be a primary topic for "fixed-point", but certainly not for "fixed point". D.Lazard (talk) 10:13, 1 March 2022 (UTC)
        • Although you and I know that Wikipedia tries to use nouns rather than adjectives for article titles, most readers don't know that. Many people think of "fixed point" as an alternative to "floating point". Per Uanfala below, when people encounter this page, the number of people that then look at Fixed point (mathematics) seems to be about twice the number that go to Fixed-point arithmetic, and some of them also go to Fixed-point iteration or other topics. This case seems to be somewhere near the borderline, although I personally suggest to move Fixed point (mathematics) to Fixed point of a function or Invariant point rather than to Fixed point, to reduce ambiguity, since there are are seven topics identified for Fixed point and the "mathematics" one is not overwhelmingly dominant. Note that we already have Graph of a function as an article title. —⁠ ⁠BarrelProof (talk) 06:07, 7 March 2022 (UTC)

• No one has reacted to my suggestion of Fixed point of a function. I thought that might be a good idea. To me it seems clear and natural. —⁠ ⁠BarrelProof (talk) 00:45, 19 March 2022 (UTC)

• Link to the outgoing traffic from the dab page: [1]. – Uanfala (talk) 01:31, 3 March 2022 (UTC)
• Comment (very weak support). Ugh. Don't really agree with proposal, but I really don't like the current title nor any of the alternatives proposed (not all functions have fixed points), and can't really think of an alternative myself right now (fixed point mapping? fixed point (function)? fixed point (mathematical function)? Bleh. Even worse). So I am tempted to support. But I will keep watching if a better alternative comes up. Walrasiad (talk) 01:41, 8 March 2022 (UTC)
• Comment. It it true that there are 7 items in the talk page, but only Fixed-point arithmetic and Benchmark (surveying) (the phrase "fixed point" or "fixed-point" does not appear in this article) are not subtopics of Fixed point (mathematics). More precisely, Fixed-point iteration, Fixed-point combinator, Renormalization group § Conformal symmetry and Recursive join use "fixed point" with the meaning of Fixed point (mathematics). So, we are almost in the case of WP:TWODABS, and the other links are here more for reader convenience that for really disambiguating: Except for Fixed-point arithmetic the phrase "fixed-point" or "fixed point" is never used alone the other articles, of, when used, it is with the meaning of Fixed point (mathematics). IMO, a consensus must take these facts into account. D.Lazard (talk) 09:56, 8 March 2022 (UTC)
• Move to Invariant point instead, as an alternative name that the subject is also commonly called in English reliable sources, albeit not as commonly as the preferred-but-ambiguous title (WP:NATURAL). I'm not an expert on the issue, but I'm familiar with the broader concept of Invariant (mathematics) so I'd more immediately recognize the topic than I would with the current title. No such user (talk) 12:32, 14 March 2022 (UTC)
• Support proposal. As D.Lazard points out, many of the other entries on the dabpage are subtopics/partial title matches. Fixed-point arithmetic is, in my view, the only other reasonable option, and is disambiguated since, being an adjective, it has a hyphen (WP:SMALLDETAILS).
  Oppose invariant point. To me that falls far afoul of WP:COMMONNAME. On the search function of the American Mathematical Society website, there are only 244 results for "invariant point", and 3846 (books)+4259 (journals up to 12/1990)+3174 (journals from 1/1991 to 12/2010)+2243 (journals since 1/2011)=13522 results for "fixed point" (the website has a maximum of 5000 results, so I had to split these into many disparate searches). That is to say, "fixed point" is over fifty times as common as "invariant point" within the mathematical literature (though it should be noted that, while the AMS does seem to have several international collections, it may naturally have an American bias). Google Scholar and JSTOR show a similar story (with the caveat that while they are international, they may include non-mathematical uses of "fixed point".) In short, IMO "invariant point" is far too rare for fixed points to be "commonly called" that for WP:NATURAL purposes. eviolite (talk) 00:15, 28 March 2022 (UTC)
• Support original move proposal and oppose invariant point per Eviolite's diligent research and sound reasoning. — Preceding unsigned comment added by Felix QW (talk • contribs) 08:48, 28 March 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Merge in "Least fixed point" article

Fixed point (mathematics) is at 5010 characters / 1.2k words, Least fixed point is at 10316 characters / 4.5k words, according to DYK Check. It seems foolish to have a sub-topic be more in-depth than the main article when the main article is not anywhere near the 50kb point to start thinking about splitting. So I think it makes sense to list all the types of fixed points on one page, instead of shoehorning greatest into LFP. @D.Lazard and Jochen Burghardt: what do you think? --Mathnerd314159 (talk) 22:01, 25 February 2022 (UTC)

I'm not sure that the articles are complete. For example, fixed point (mathematics) might be extended in future to discuss aspects related to bifurcation theory and fractals. (I'm not familar with this field of mathematics, so I can't estimate how much stuff, if any, could be added.) That is, the size relations might change. - As another aspect, the current split "fixed point" / "least fixed point" may be suboptimal, a better one may be "fixed points in computer science" (functions on Scott domains) / "fixed points in calculus" (functions on real/complex numbers); and maybe even other applications I'm unaware of. - Jochen Burghardt (talk) 22:43, 25 February 2022 (UTC)

I agree with Jochen. Here are some more comments.

Both articles are very incomplete. The current contents are compatible with a merge, but my impression is that, when the article will be correctly completed, a merge would not make sense anymore. So, for the moment, I oppose to the merge.

Fixed point (mathematics) is very short because it is incomplete. It should explain the subject of all items of the section "See also" that have "fixed-point" in their name. However, one of the item is related to computer science and is more related to the second article. Several other items may be grouped in a section that could be called "Fixed point of a group action" or "Invariants". The article should also explain that an iterative method is a method for solving a problem by searching a fixed point of some auxiliary function. Newton's method must also be mentioned, as being the best known iterative method. Also, most optimization methods are based on the search of fixed points. In summary, the title of the article suggest a much wider content than presently.

As far as I know, the concept of "least fixed point" has been elaborated for the need of computer science, and is used only there. So, after having improved the article, it could be worth to rename it Fixed point (computer science), and interlinking the two articles. D.Lazard (talk) 09:48, 26 February 2022 (UTC)

Oppose As has been mentioned above, the subject of the least fixed point article is more used in computer science and finite model theory. However, I think keeping the mathematical and the computer science aspects of least fixed point theory together is a good thing (they intersect in Fixed-point logic), so I would prefer the status quo over the changes suggested by D. Lazard and J. Burgardt above. Felix QW (talk) 20:35, 26 February 2022 (UTC)

Oppose This is not a "subtopic" of Fixed point (mathematics); rather it is a topic that relies on the concept named by the title of the latter article. Michael Hardy (talk) 22:06, 23 April 2022 (UTC)

Ok, so my idea of how one would split the topics mentioned so far and the ones already in the articles:

Fixed point	Fixed point (computer science)	Fixed-point iteration
Fixed point of a function Fixed points in calculus (real/complex numbers) Optimization (main article Fixed-point iteration) Fixed point of a group action Cycles and fixed points Fixed-point subgroup Fixed-point subring Topological fixed point Fixed-point theorem Fixed point logic Applications	Fixed points of Scott domains Prefixedpoint / postfixedpoint Least fixed point Optimal (greatest) fixed point Fixed-point combinator	Attracting fixed point Iterative methods Newton's method Chaos game (fractals) Bifurcation theory

I'm thinking we would move Fixed point (mathematics) to Fixed point and the current Fixed point to Fixed point (disambiguation), because the mathematical notion of fixed point is the simplest and most common, AFAICT.

But still, there are only 10 topics in Fixed point and 5 in the CS article. So the sections would have to be more than 3.3k characters on average to make the merged math-CS article too large, which seems unlikely since most of them will be summaries of linked articles. I find your arguments that the articles will expand beyond mergability to be unconvincing. This topic outline is just as readable as the split articles:

Combined article outline

* Fixed point of a function Fixed points in calculus (real/complex numbers) Optimization (main article Fixed-point iteration) Fixed point of a group action Cycles and fixed points Fixed-point subgroup Fixed-point subring Topological fixed point Fixed-point theorem Fixed points of Scott domains Prefixedpoint / postfixedpoint Least fixed point Optimal (greatest) fixed point Fixed-point combinator Fixed point logic Applications

--Mathnerd314159 (talk) 02:49, 27 February 2022 (UTC)

I agree with moving Fixed point (mathematics) to Fixed point, since the dab page does not contain any non-mathematical meaning that can be considered as a primary topic. I'll try to make the move myself, and if I cannot, make a move request.

I agree globally with the outline (some details may have to be changed at writing time). Go on boldly.

About the proposed merge: It remains unclear whether Least fixed point should remain a separate article with a {{main article}} tag in the main article, or should become a redirect. This would be better to decide this later. D.Lazard (talk) 09:39, 27 February 2022 (UTC)