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== Recent changes should be reverted ==

I am concerned that after a flurry of changes the current version [[Special:Permalink/1219909644]] by @[[User:Fangong00|Fangong00]] is significantly less clear and less accessible to non-expert readers than the previous version [[Special:Permalink/1218106312]], as well as not really matching expected Wikipedia style.

In my opinion this whole collection of changes should be reverted, and any dramatic change along these lines should be first worked on as a draft (e.g. at [[User:Fangong00/Newton's method]] or [[Talk:Newton's method/draft rewrite]]) and then discussed before adoption here.

In my opinion, within the first few sections of this article we must focus on the most common and basic case of finding the zeros of a single-variable real-valued (or maybe also complex-valued) function, and everything about arbitrary multivariate functions, Banach spaces, etc. must be deferred to sections far down the article about more general contexts. The early sections must also avoid unnecessary jargon and must provide clear descriptive background rather than a wall of inscrutable technical details about this or that variant. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:12, 20 April 2024 (UTC)

:I disagree. The first screen need to tell visitors what Newton's iteration is and what convergence conditions are. 
:Wikepedia is not a teaching platform, it is for quick reference. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 13:57, 21 April 2024 (UTC)

Update: I tried reading the first few sections of Fangong00's version aloud to a friend, and we both found it almost impenetrable, so I decided to just revert the changes, pending discussion. Fangong00: I don't want to discourage you from trying to make improvements though. I'm sure there are things that could be expanded, reworked, etc. about the article as it was when you arrived. Feel free to make a rewrite draft, or to bring concrete criticisms of the existing article here for discussion. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:26, 20 April 2024 (UTC)

: I agree with jacobolus.  [[User:Bubba73|Bubba73]] <sup>[[User talk:Bubba73|You talkin' to me?]]</sup> 02:39, 21 April 2024 (UTC)
:I completely agree with jacobolus. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 02:54, 21 April 2024 (UTC)
::Also, after the page is back to a stable version, the Nash-Moser section (even though empty) should link to [[Nash–Moser theorem]]. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 02:58, 21 April 2024 (UTC)
:I agree with jacobolus. [[User:PatrickR2|PatrickR2]] ([[User talk:PatrickR2|talk]]) 06:10, 21 April 2024 (UTC)
:Please be specific. Which part is "impenetrable"? I can certainly try to improve it. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 13:05, 21 April 2024 (UTC)
::In your proposed version of the page, the section called [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1220039213#Simpson's_extension Simpson's extension] is pretty much identical in content to the existing section [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1220039213#Systems_of_equations k variables, k functions] together with the existing section [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1220039213#In_a_Banach_space in a Banach space]. It's not good to have content repeated twice in different locations on the same page. And I believe the existing sections are more clearly written with non-mathematician readers in mind. What do you think is wrong with these sections? [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 13:39, 21 April 2024 (UTC)
::Didactically, we want to understand the classical 1d situation before even mentioning other cases. —[[User:Kusma|Kusma]] ([[User talk:Kusma|talk]]) 13:39, 21 April 2024 (UTC)
:::As I understand, Wikepedia is a reference, not place to teach Newton's method. Users want to know quickly from the first screen what Newton's method is and what convergence conditions are. 
:::I do plan to delete the duplicate part. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 13:44, 21 April 2024 (UTC)
::::Wikipedia is an encyclopedia, namely <i>neither</i> a reference nor an instruction manual. The intention should not be to front-load the information that someone might be trying to look up. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 13:54, 21 April 2024 (UTC)
:::::I suppose people look up an encyclopedia for a quick reference. A page about Newton's method shouldn't give users an impression it is only a special case, as Newton-Raphson is. Hiding what Newton's iteration really is later shouldn't be the way. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 14:03, 21 April 2024 (UTC)

I think the changes should be restored the article, at least in some form.  The status quo article is pretty bad, and this added content is in a much better shape for improvement.  For example, local quadratic convergence is stated clearly, and various important generalizations to the elementary version are stated up front in a consistent style.  Thus the new article is more useful to people who want to use the method, as opposed to those just learning it for the first time.  I think a separate "Introduction" section should be written, which covers the one variable theory that one learns about in Calculus, followed by the new stuff pretty much as written, modulo stylistic changes.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 12:16, 21 April 2024 (UTC)

:Thank you. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 13:03, 21 April 2024 (UTC)
:I don't object at all to the content being on the page somewhere. (As I mentioned on wikiproject:math, it almost all already was on the page, in a less detailed version.) That's with the exception of the rank-r Newton method; is it clear that it deserves any space on the page? [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 13:22, 21 April 2024 (UTC)
::I do agree that it is debatable on the inclusion of rank-r Newton's iteration since it is quite new. However, it is the first and ONLY iterative method that converges to positive dimensional solutions. It is published in Math Comp. Maybe it should be a one sentence item. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 13:51, 21 April 2024 (UTC)
:::I am extremely skeptical of that. Such iterative methods go back to at least [https://projecteuclid.org/journals/duke-mathematical-journal/volume-17/issue-2/Some-mapping-theorems/10.1215/S0012-7094-50-01713-3.short 1950]. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 14:35, 21 April 2024 (UTC)
::::I checked up that page. Are you sure there is an iterative method proposed in "Some mapping theorems" [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 14:50, 21 April 2024 (UTC)
:::::The iteration is described in the proof of Theorem 1, also with a modification in the proof of Theorem 4. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 14:56, 21 April 2024 (UTC)
::::::I'll look that up. Thank you. It would be a shame for a method to be buried in a proof. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 15:12, 21 April 2024 (UTC)
::::::By a quick read, it deos not seem to be an iteration converging to positive dimentional solutions. The condition K(x) mapping to the whole space disqualify the Jacobian that is rank-deficient at such solutions. So the Jacobian can't map to the whole space. If it works, the linear mapping K is unknown so it is not a practical algorithm. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 16:46, 21 April 2024 (UTC)
::::You may be skeptical. But I know a lot of numerical analysts who try compute nonisolated solutions. No one knows a single algorithm for that purpose so they all go extra steps to isolate the solutions first before rank-r Newton's iteration. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 17:28, 21 April 2024 (UTC)
:::::As for Graves' 1950 paper, it is about underdetermined systems, where the space of solutions has positive dimension. In that case it's often true that the linearization is surjective (i.e. K(x) maps to the whole space). It seems that rank deficiency would correspond instead to overdetermined systems, where the linearization would instead be injective and the solution space would have isolated points.
:::::Regardless, what is important is that Wikipedia needs to contain only verifiable information; is there any reliable source which says (implicitly or explicitly) that this 2023 paper is significant? [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 17:35, 21 April 2024 (UTC)
::::::That makes sense. Actually Newton's iteration (with inverse replaced by Moore-Penrose inverse) converges to positive dimensional solutions and it was proved by M. Chu in 1970s for underdetermined systems with surjective Jacobians. What I meant was that no known method for general rank-deficient Jacobians. 
::::::You have a great point. A 2023 paper may be too new to mention. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 17:47, 21 April 2024 (UTC)
:I disagree with the outlook of @[[User:Tito Omburo|Tito Omburo]] and @[[User:Fangong00|Fangong00]]
:* "Thus the new article is more useful to people who want to use the method, as opposed to those just learning it for the first time. "
:Anyone wanting to apply Newton's method has the motivation to read deeper into the article and to look up alternative sources. Everyone else should not be forced to into a deep dive to get the basic concept. I've used some of these techniques in the past and yet I would stop reading at the second section of the new draft.
:A later section such as "Formal concepts" (or any number of similar titles) with deeper content can have the same value for the in-depth reader. [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 14:49, 21 April 2024 (UTC)
:: "Anyone wanting to apply Newton's method has the motivation to read deeper into the article and to look up alternative sources."  Surely this is a problem with the order in which topics appear, not whether topics should be covered in the article ''at all''.  (And before anyone says that the article already discussed the new additions to some extent: yes, but not as thoroughly nor as well-organized.)  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 17:14, 21 April 2024 (UTC)
::Strongly disagree. A scientific practitioner may just want to try a known method and don't want/need to "read deeper into the article and to look up alternative sources". I know a lot of people, including myself, do that all the time. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 17:23, 21 April 2024 (UTC)
:::@[[User:Tito Omburo|Tito Omburo]] Adding new topics / expanding existing topics was not the problem. The problem was that the first 3 sections of the article were pushed out of reach of the majority of the intended audience. It's entirely fine to make a section halfway through the article about the multidimensional case, with as much detail as desired. In cases where both experts and novices can be expected to be interested in the same topic, it is essential that the top of the article remains accessible for novices, and that the rest remains clearly motivated, put into context, organized with some narrative flow, etc. (In cases where one section of such an article, focused on an advanced topic, has enough to say about it that it starts ballooning in size, it can be helpful to leave a summary and split it into a new article focused on that subtopic. That is probably unnecessary here though unless someone wants to keep expanding this much further.) –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:56, 21 April 2024 (UTC)
:::: This is pretty much what I've been saying all along, that an introduction should be at the beginning, followed by more advanced material.  Let me just ask: do you support adding this content if it is after the third section?   [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 00:13, 22 April 2024 (UTC)
:::::As far as I can tell nobody has any problem with expanding later sections about more advanced topics. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 00:44, 22 April 2024 (UTC)
:::::: The discussion is about wholesale ''reverting'' these changes, not moving them to a later section in a constructive fashion.  So yes: people taking this position do have a problem with improving later sections of the article.  Also [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&diff=prev&oldid=1220070545 this revert] exclusively affected primarily sections much later in the article.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 10:52, 22 April 2024 (UTC)
:::::::You are mischaracterizing what happened and misunderstanding people's intentions. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 11:20, 22 April 2024 (UTC)
--------------
Dear User:jacobolus, Bubba73, PatrickR2 and Gumshore2: 

Why do you not want a vistor know what Newton's method is on the first screen? This is the central issue and we need to focus on that, please.

The Newton-Raphson is only a special case of Newton's method. It is a disservice to hide what Newton's method is in numerical analysis.  Fangong00
: As I mentioned on wikiproject:math, I don't think there would be any issue with modifying the lead section to highlight that there are different formulations of Newton's method, including the standard multidimensional version which you are calling "Simpson's extension." The lead section should also say that there are a number of different theorems which guarantee convergence of Newton's method to solutions. It isn't hard for readers to use the table of contents to click to where in the article they want to go in order to get details.
: Also, I hope you are aware of the [https://en.wikipedia.org/wiki/Wikipedia:Edit_warring#The_three-revert_rule three-revert rule]. I believe you are already in violation of it; if you continue to revert the page it is very possible that an admin will temporarily block you. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 14:42, 21 April 2024 (UTC)
: It might be okay to ''add'' a sentence or two to the end of the lead section, as well as expanding later sections. However, what you did was ''remove'' all of the readable introductory explanation from the top of the article, which made it very difficult for most readers to understand. I urge you to find a non-mathematician friend, print out the first 3–4 sections of your version of the article and the first 3–4 sections of the previous version, and ask them which one makes more sense. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 15:31, 21 April 2024 (UTC)
::I took your suggestion. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 16:03, 21 April 2024 (UTC)
:::@[[User:Fangong00|Fangong00]] Can you please work on your proposal in a draft outside of the article, e.g. in your user namespace, as a subpage of this talk page, or proposing particular sentences or paragraphs directly here in the talk page? Then you won't end up just getting reverted repeatedly.
:::Your updated changes to the lead section still make it worse, in my opinion. Perhaps you can recruit someone who both knows deeply about this topic and has strong expository writing skills to help keep changes legible. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 16:17, 21 April 2024 (UTC)
::::I found your last paragraph interesting, to say the least. So I'd like to ask: Do you know why the claim "preserves the quadratic convergence rate" in the section "Slow convergence for roots of multiplicity greater than 1" is actually false despite an impeccable proof? Few people know it is false and even fewer know why.
::::You would know it is false if you ever test it. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 00:38, 22 April 2024 (UTC)
:::::I can't figure out what point you are trying to make. The convergence rate to higher-multiplicity roots, a topic I am certainly no expert in, is irrelevant to our conversation here. (But if this particular claim is wrong, then it should by all means be fixed.) If your point is just that you (and probably several other participants in this discussion) know more about this topic than I do, I am happy to concede that. I am not by any means an expert in root-finding algorithms, nonlinear optimization, or whatever. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 00:49, 22 April 2024 (UTC)
::::::You know what point I am making, and I made it.  There is no shame for not knowing that. 
::::::The claim is practically false because, in numerical computation, tiny number divided by tiny number results in a random number generator, like m*f(x)/f'(x) near a multiple root where f(x)=f'(x)=0. The theorem has a hidden condition: f(x) and f'(x) can be exactly computed. The condition shouldn't be hidden because it can not be met in numerical analysis. The proof is impeccable with that hidden condition assumed, of cause. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 01:21, 22 April 2024 (UTC)
:::::::No I really do not understand what point you are trying to make. It seems like some kind of passive aggressive status thing. Please just be clear and direct and say what you mean. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 01:27, 22 April 2024 (UTC)
::::::::Do we agree no one is supposed to act like having some kind of status here? [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 02:10, 22 April 2024 (UTC)
:::::::::I cannot make heads or tails of this subthread. As far as I can tell it has nothing to do with the rest of the conversation. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:18, 22 April 2024 (UTC)
::::::::::It is interesting you avoided answering. Besides the point I already made, the status quo page you advocate is bad scientifically and in presentation, IMHO, maybe as bad as or worse than you think my corrections are. I'll list more errors when I get a chance. I do agree it is time to stop this subthread. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 03:10, 22 April 2024 (UTC)
:::::::::::I didn't reply because it seems off topic and I don't entirely understand what you mean by "no one is supposed to act like having some kind of status". I will say that (a) each of us is just an editor (and just a person for that matter), and Wikipedia pages' content is decided by consensus (cf. [[Wikipedia:Consensus]]) rather than credentials or reputation inside or outside of Wikipedia, (b) playing word games, trying to catch people with "gotchas", and speculating about other editors' motives or anything beyond what they directly did or said is usually a distracting waste of time, and (c) everyone should try to follow [[Wikipedia:Etiquette]].
:::::::::::{{tq|i=yes|status quo page you advocate}} – I did not ever "advocate" the previous page. All I said is that the previous version has essential features that should not be removed without consensus, and that any consensus for such substantial changes can only be achieved through discussion.
:::::::::::Please do make that list of errors, that sounds great. That was what I asked of you at the start: to (a) work on your draft in user space or as a subpage of this talk page instead of revert warring, and to (b) make a concrete list of criticisms / proposals here on this talk page. That other editors can concretely reply to those point by point. But please put it into a new talk page section instead of this deeply nested comment thread, and try not to address it personally toward any particular editor. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 05:04, 22 April 2024 (UTC)

(ec)
:  Fangong00  - introduce it simply.  Banach spaces and all of that doesn't belong in the introduction.  It doesn't even belong in the body of the article, until all of the main use of Newton's method has been covered.  [[User:Bubba73|Bubba73]] <sup>[[User talk:Bubba73|You talkin' to me?]]</sup> 16:04, 21 April 2024 (UTC)

::I disagree. Newton's method in Banach spaces is very important and is standard material. So I think it should be mentioned in the lead section, although I would put it closer to the end thereof, instead of in the very first sentence (where it is now).
::I think Fangong00's [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1220059681 new edit] is possible to work with and improve, although many of my previous objections stand. ("Simpson's extension" is not standard terminology, the sections on it and Gauss-Newton iteration are a little harder to read than the way they were previously expressed, and the rank-r Newton iteration is not notable enough to mention.) [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 16:19, 21 April 2024 (UTC)
:::: OK, then I think that the last sentence of the introduction could say that it can be generalized to Bsnach spaces.  Then the coverage of that should be near the end of the body.  [[User:Bubba73|Bubba73]] <sup>[[User talk:Bubba73|You talkin' to me?]]</sup> 22:48, 21 April 2024 (UTC)

:::@[[User:Gumshoe2|Gumshoe2]] What's this Banach thing? I just want to know about Newton's method. [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 16:43, 21 April 2024 (UTC)
::::The whole debate is about what visitors want to know. You may not want to know Newton's method on Banach spaces but there are certainly some visitors do, and may need it. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 16:49, 21 April 2024 (UTC)
:::::@[[User:Fangong00|Fangong00]] Fine, so put that detail done further in the article, not in the intro and first sections. [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 17:44, 21 April 2024 (UTC)
:::Just as with differentiation in general, it is best to first understand the one dimensional version, although the Banach space version is no different if viewed from the right angle. Best nit to dwell on it much in an introductory article. —[[User:Kusma|Kusma]] ([[User talk:Kusma|talk]]) 16:49, 21 April 2024 (UTC)
::::This is not a teaching platform. Someone trying to solve an equation in Banach space should get the information on the opening screen. I guess the original page was written by a teacher, not by a researcher or scientific computing practitioner. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 17:01, 21 April 2024 (UTC)
:::::Wikipedia is both a general and a specialist encyclopaedia. Articles should be targeted at the widest range of interested people. For something as relatively basic as Newton's method, that is undergraduates. —[[User:Kusma|Kusma]] ([[User talk:Kusma|talk]]) 17:05, 21 April 2024 (UTC)
::::::Mentioning its capabilities is exactly commodating "widest range". Your suggestion serves only undergraduates, who can skip the Banach part as they wish. 
::::::I don't understand why people want to hide the capacities of a great algorithm. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 17:13, 21 April 2024 (UTC)
::::::Why do people want to deny other visitors' access to crucial information simply because they don't need? [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 17:16, 21 April 2024 (UTC)
:::::::It's much clearer for a newcomer if the relevant part of the lead section says "[[multivariate function]]" instead of "[[Map (mathematics)|mapping]] between [[Vector space|vector spaces]] or  [[Banach spaces]]". The latter is being made gratuitously obscure for pretty much no benefit. Wikipedia has a huge problem of chasing readers away from technical articles because they find them inscrutable. To the extent possible we want to push back on that, even at the expense of occasional mild imprecision or inconvenience for advanced readers. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 19:10, 21 April 2024 (UTC)
::::::::To my personal taste, a lead section along the following lines would be ideal, with the hope that it's readable even if some of the words are unfamiliar. (I'd also prefer that even the single-variable formulas are pushed out of the lead section and into the first section.)
:::::::::Newton's method, also known as Newton's iteration, is an [[iterative method]] which uses the tangent lines of a single-variable function to iteratively define a sequence of numbers. Although there are exceptional cases, often this sequence is infinitely iterable and converges to a [[zero of a function|zero of the function]]. This can be studied empirically, although there are a number of mathematical theorems which provide conditions that guarantee the convergence of Newton's method to a zero.
:::::::::More generally, Newton's method has also been studied, both theoretically and empirically, in a number of multivariable contexts. In this context, the multivariable generalization of the tangent line known as the [[tangent plane]] is used to define the iteration. The multivariable Newton's method leverages the solvability of [[systems of linear equations]] to find solutions of nonlinear systems of equations. Computer implementations of this are useful in many practical problems of [[numerical analysis]] to solve complicated equations. This forms a powerful tool in the field of [[optimization]], where Newton's method is useful for finding [[critical point]]s. A number of variants of the standard Newton method have also been developed and are widely used in these applied contexts.
::::::::: In [[pure mathematics]], the multivariable Newton's method can naturally be put into the even broader context of [[Banach spaces]]. While the multivariable Newton method allows the points of the iteration to be [[tuples]] of numbers, this generalized Newton's method allows the points of the iteration to be, for example, functions. This flexibility is theoretically useful in parts of [[functional analysis]] and [[partial differential equation]]s.
::::::::[[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 19:58, 21 April 2024 (UTC)
:::::::::I don't like the starting sentences. I would instead say something more like "Newton's method is an algorithm for numerically finding a solution of a nonlinear equation using a succession of linear approximations." I would recommend explicitly describing how the single-variable case works in the lead section, using both a picture and mathematical notation. We should explicitly describe the main idea: to find a root of a function, we make an initial guess, use knowledge of the function's value and derivative to construct a linear approximation near that guess, calculate the root of the linear approximation, then use that as our next guess. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 20:36, 21 April 2024 (UTC)
:::::::::One thing that I think would help a lot is an animated example image at the top of the article. We don't absolutely have to lead with formulas, but we should try to include a concrete explanation of what the main idea is behind Newton's method. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:57, 22 April 2024 (UTC)
::::::::::I agree, the starting sentences that I wrote there are pretty weak. Personally I'd like to avoid formulas in the lead section if possible, since they would have to be repeated pretty much immediately in the first section anyway. But animations would be very good, are they easy to make? [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 00:09, 23 April 2024 (UTC)
:::::"Someone trying to solve an equation in Banach space should get the information on the opening screen."
:::::(I assume that by "opening screen" you mean the beginning of the page.) There is absolutely no need for a page to be structured in this way. Why would it be so difficult for someone to use the table of contents to navigate perhaps very deep into the page to get what they need? [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 17:39, 21 April 2024 (UTC)
::::::Still don't get it. Why do you trying so hard to prevent one sentence in the opening graph mentioning the capabilities? Most visitors do not look at table of contents. [[User:Fangong00|Fangong00]] ([[User talk:Fangong00|talk]]) 17:55, 21 April 2024 (UTC)
:::::::I fully support the lead section giving a largely nontechnical summary of contents, as per [[Wikipedia:Manual of Style/Lead section]], and as such I support that the Banach space context should be mentioned there in a sentence. I am against deeper information or technical detail appearing at the beginning of the article; there is no problem with it being pushed towards the end. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 18:00, 21 April 2024 (UTC)
::::::::Yes. Sometimes the best way to balance the various requirements that a lead section ought to satisfy &mdash; both accessibility ''and'' summarizing the article that follows &mdash; is a brief synopsis of highly technical material towards the end. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 19:02, 21 April 2024 (UTC)
::::::::A few-sentence paragraph at the end of the lead section of clear prose explaining the importance of the multidimensional version of Newton's method would be fine ("Newton's method has been extended to multivariate functions, where it has several variants, and is widely used in application X, Y, and Z" or whatever).
::::::::I would leave the term "Banach spaces" out of the lead section entirely, especially when that isn't really discussed in any detail later on. Gratuitous jargon should be avoided, especially in the lead section. Readers interested in generalizations are not going to be confused by skimming down the page. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 19:05, 21 April 2024 (UTC)
:::::::: I would add to this proposed lede that the ideal convergence rate is extremely rapid, but that devising conditions to ensure ideal convergence is a difficult problem.  Chaos also deserves to be mentioned.  [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 11:06, 22 April 2024 (UTC)
:::::::::By "ensure ideal convergence," you mean that it's difficult to choose the initial point well? [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 00:10, 23 April 2024 (UTC)

== Questionable examples ==

I'm a little skeptical of two of the examples in [[Newton's Method#Failure analysis]] section.
# The example in the "discontinuous derivative" section. I agree with all of it except the key phrase "and Newton's method will diverge almost everywhere in any neighborhood of it," which seems by no means clearly established. The unboundedness of <math>f(x)/f'(x)</math> near the root seems to be, in principle, perfectly compatible with many/most Newton iterations converging to the root. A simulation with randomly chosen initial points seems to back this up.
# The second half of the "zero derivative" section. What is the meaning of "nearly double" in this context; does it mean that the root is close to a point where the derivative vanishes? Six also seems like a very moderate number of iterations to require. In all, I just don't know what this example is trying to illustrate.
Could anyone clarify or provide a relevant reference? If not, I think these examples should be removed.
[[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 00:06, 23 April 2024 (UTC)

:Also, although I don't have any issue with the "no second derivative" section, I haven't been able to find any standard textbook reference where this example or a similar one is given. Does anyone here know one? [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 00:12, 23 April 2024 (UTC)
:Since nobody has spoken up, I've removed these examples. I've also combined (and somewhat rewrote) the three different example sections into a single one, which I think is helpful for reading. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 14:19, 28 April 2024 (UTC)

== Proof of quadratic convergence for Newton's iterative method ==

Does anyone think it's necessary to give a full proof of this as in [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1221206566#Proof_of_quadratic_convergence_for_Newton's_iterative_method the present version]? It's possible to summarize it reasonably and convincingly in about two lines, and there are plenty of standard books to reference where the full proof appears. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 14:28, 28 April 2024 (UTC)

:I agree. Also, the preceding subsection is also confusing because of too much technical details. Imo, one must reduce the proofs and explanations to functions of [[differentiability class|class]] C2, with the simple mention that there are generalizations to slighty larger classes of functions. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:12, 28 April 2024 (UTC)

== "Poor initial estimate" ==

The following paragraph was added [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&diff=707315222&oldid=703802972 in 2016]:
:A large error in the initial estimate can contribute to non-convergence of the algorithm. To overcome this problem one can often linearise the function that is being optimized using calculus, logs, differentials, or even using evolutionary algorithms, such as the Stochastic Funnel Algorithm.  Good initial estimates lie close to the final globally optimal parameter estimate.  In Nonlinear Regression the SSE equation is only "close to" parabolic in the region of the final parameter estimates.  Initial estimates found here will allow the Newton-Raphson method to quickly converge. It is only here that the Hessian of the SSE is positive and the first derivative of the SSE is close to zero.
It is both very unclear and, based on the username of the editor, seems to be added as an advertisement for [https://researchoutput.csu.edu.au/en/publications/using-a-stochastic-funnel-to-find-nlr-starting-values their own research]. For now I've removed it, possibly someone can salvage quality material from it. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 15:09, 28 April 2024 (UTC)

:Remove this paragraph: if the initial estimate induce non-convergence or convergence to a wrong local optimum, the best method if to choose another initial estimate. It is unclear whether this paragraph is about the choice of a better initial estimate. On the opposite, is seems to suggest to replace the function with an approximation of it, which is a very bad idea, as it may change the optimum dramatically. Moreover, the choice a good initial estimate depends on the nature of the problem, and I doubt that there exist ''general'' methods for that, that are better than a random choice. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:48, 28 April 2024 (UTC)

== Computable numbers ==

I'm removing the following paragraph:
:When Newton's method can be applied to a transcendental equation, and converges to a solution of the equation, this implies that the solution is a [[computable number]] that is exactly represented by the pair formed by an initial approximation and an [[algorithm]] for increasing the accuracy of any approximation.
As stated, it seems definitely wrong. If {{mvar|c}} is an uncomputable number, then the root of {{math|''x'' - ''c''}} is {{mvar|c}}, and Newton's method initialized anywhere converges to the root (and even does so in a single step). [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 15:43, 28 April 2024 (UTC)

:The paragraph becomes true if "transcentdental equation" is replaced with "equation defined by a computable function". This is the simplest way to prove that [[algebraic number]]s are computable. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:38, 28 April 2024 (UTC)
::I don't know much on this topic, but the wikipage [[computable function]] suggests that the domain and range are only natural numbers, or tuples thereof. Is this the wrong context?
::But it does make sense to me that Newton's method, along with other root-finding algorithms, can prove that algebraic numbers are computable. Is there a standard reference for this? I haven't been able to find it. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 19:15, 28 April 2024 (UTC)
:::{{to|Gumshoe2}}Sorry, by trying to give sense to the removed paragraph, I did (wrong) original research. For computability with real numbers, one must look on the work by [[Stephen Smale]] and his followers. Also, for applying Newton's method to a transcendental equation, one requires to work with an increasing approximation of the equation and the involved derivatives. This seems to not be an easy task to define this propertly. This has probably been done in [[constructive mathematics]], but I do not know this subject well.
:::By the way, in section {{alink|Multiplicative inverses of numbers and power series}}. it must be said that it is the quadratic convergence that insures that the computation of a multiplicative inverse has the same bit complexity as multiplication, even if fast multiplication is used (beacause of the quadratic convergence and the property of geometric series, all iterations but the last one take together no more time than the last iteration). This is also Newton's method that is used to show that matrix inversion has the same complexity as matrix multiplication, independently of the used multiplication algortihm. (As usual, I an unable to give references, but this should be found in most textbooks on complexity theory). [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 20:38, 30 April 2024 (UTC)

== Systems of equations article draft ==

A draft article now exist for solving systems of equations, "Newton's method for systems of nonlinear equations".  If approved, it should replace the existing sub-heading on the subject.  While waiting for review, I would be helpful if interested parties were to weigh in with additional edits to the draft on subjects such as mathematical proofs, convergence theorems, etc.  It will make a more credible draft and help with the review process. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 01:50, 10 May 2024 (UTC)

:Where is the draft available? I'm a little skeptical that it needs its own article, but certainly the present coverage should be expanded. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 02:32, 10 May 2024 (UTC)
::If someone wants to write one, I think this subject definitely ''can'' support its own article. There is a ton to say about it, enough to overflow 1–2 sections here. A new article would IMO be a much better solution than trying to generalize the first several sections of this article to cover all cases up front, but depending on the amount drafted, it could also remain a section. (I'm not sure whether a wordy explicit title would work better, or something more concise like [[Multivariate Newton's method]].) –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:51, 10 May 2024 (UTC)
:::I think there could be other solutions -- like you, I don't think the first several sections of this article should be generalized in place. I think it should be possible to more or less keep the present section layout on this page. But it all depends on the particular content in question.
:::As for titles, one issue is that, with the exception of freshman calculus texts, the "multivariate" Newton's method is the standard context that Newton's method is presented and used in. I think giving it a special page title like "Multivariate Newton's method" would make it appear unnecessarily niche. (Also see only 47 hits on Google Scholar for "multivariate Newton's method", 30 for "multivariable Newton's method") [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 03:06, 10 May 2024 (UTC)
::::If the consensus is to leave the Multivariate content in the current article, I am good with that.  However, my thinking is that the Multivariate content will grow, and the existing article focuses heavily on single variable.  The new draft article still needs an applications section, which I can contribute to, mathematical sections to prove out the details which math experts can contribute to, etc.  Since, as you point out, half the draft article is already in the existing section, I can place the remainder in the existing section, too, at least until the new article issue is finalized.   I am somewhat concerned that the remaining content, being primarily based on life experiences, may not meet verification standards, but we can try it.  Any further thoughts on this subject?
::::As for the name, "Multivariate", the draft currently uses terminology from old text books.  The theory still works, but naming conventions do change over time.  I will update the draft to conform to current naming conventions. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 15:58, 10 May 2024 (UTC)
:::::To be clear: I have no deep insight or particular care about what title/terminology to use. I was just throwing out an idea of a more concise title. But a longer one can also be fine. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 16:47, 10 May 2024 (UTC)
::::::Thank you for the suggestion.  I liked it.  I was unable to change the title in the draft, so I made a talk entry for it. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 00:42, 11 May 2024 (UTC)
::Seems to be at [[Draft:Newton's method for systems of nonlinear equations]]. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:58, 10 May 2024 (UTC)
:::I can comment later on the content, but I think that can all be well absorbed into this page. (The first half is already here.) [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 03:10, 10 May 2024 (UTC)

== Example is not referenced. ==

The section under Multidimensional methods > Example has no sources. A recent [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1226078611 edit] by an IP user changed one number without any edit summary. Without any way to verify the content, I [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1226095490 reverted] the edit. 

My change was [https://en.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=1226299270 reverted] by @[[User:Netshine2|Netshine2]] claiming that the IP user was correct. That's great, and I'm glad the content may now be correct. However we still have no way to verify it.

I think the content has marginal notability and absent a reference it may be a small kind of original research. Is there a case to keep this content? [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 19:50, 29 May 2024 (UTC)

:The references for the example would be the link to "Jacobian" matrix in the above section, and the multidimensional definition of Newton's method in the section above that.  It is assumed that the derivatives in the example are covered by the reference to "Jacobian", since the Jacobian matrix is a matrix of partial derivatives.  
:The presence of the example serves to clarify the definitions in the section above.  Users not familiar with Newton's method as it applies to actually doing something with it can easily see how the defining equations above work to obtain a valid converged solution.  The equations are validated by keeping them simple and breaking them down into small enough pieces such that anyone skilled in Mathematics can validate the converging processes each step at a time. I do not believe the term "original research" really applies here because the example is simple to create and reproduce by anyone skilled in mathematics.  The only challenge would be from user's not-so-skilled in mathematics, which is what the example is intended for.  I have no objection to replacing the example with an existing published simple example, provided it is simple enough for users to follow, and complex enough for users to grasp the usefulness of Newton's method for multidimensional applications, but I have to point out that published example can contain typos, too. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 20:21, 29 May 2024 (UTC)
::Thanks! (please us the reply button on Talk pages). I made a minor edit to the reference.
::{{done}} [[User:Johnjbarton|Johnjbarton]] ([[User talk:Johnjbarton|talk]]) 00:10, 30 May 2024 (UTC)
:::Looks Good!  I'll have to remember the footnotes for future reference.  Looks like we did good job of making the Wikipedia page better. [[Special:Contributions/24.121.207.148|24.121.207.148]] ([[User talk:24.121.207.148|talk]]) 00:27, 30 May 2024 (UTC)

:I did some additional research and was able to find a published reference that used a similar example to ours, but it used 3 equations instead of our 2.  For simplicity, I would prefer that our example keep the number of equations down to 2.  Note that even if we were to use the reference example as-is, it still needs to be validated for typos and accuracy, which I very capable of doing.  Please look the reference example, and let us know if you believe we should use it.  The reference is easier to follow if you create a free account for yourself with the reference site. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]])

== Is "Convergence dependent on initialization" right? ==

<s>By my calculations the result should be:
:<math> \frac{ - x_n^2 }{ 2 - x_n - 2 x_n^2 } \approx x_n \cdot \frac{ - x_n }{ 2 } </math>,

rather than x_n^3 .</s> And the second paragraph does not add anything useful. <s>So I would suggest removing the whole section.</s> [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 19:43, 6 May 2025 (UTC)

:The derivation of the authors work is extremely tedious and easily prone to algebraic errors, at least for me.  However, I used a computer program to compute the derivative digitally and plug the derivative result into the the rest of the equation, and was able to validate the author's result for x-F(x)/F'(x) for F(x)=x(1+x^2)^-.5 to be -x^3, as stated in the paragraph.  Rather than bore the readers with all the derivation details that arrive at the solution, I think what is needed here is a footnote that goes through the derivation details. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 22:26, 6 May 2025 (UTC)
:Your calculations aren't right: if <math>f(x)=\frac{x}{(1+x^2)^{1/2}}</math> then
::<math>f'(x)=\frac{1}{(1+x^2)^{3/2}}</math>
:so
::<math>x-\frac{f(x)}{f'(x)}=x-x(1+x^2)=-x^3.</math>
:I don't support adding the details of calculation to the article, in a footnote or otherwise, since it's a routine calculation and not of any encyclopedic value.
:I think the second paragraph is good as an example of extreme sensitivity to initial conditions, which is an important aspect of Newton iteration. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 01:18, 7 May 2025 (UTC)
::In that case, I would recommend removing the "[dubious – discuss]" note in the paragraph, but bear in mind the note appears to indicate that some readers may not be able to validate the expression as efficiently as we can, and may find it to be suspect. [[Special:Contributions/24.121.207.148|24.121.207.148]] ([[User talk:Netshine2|talk]]) 03:03, 7 May 2025 (UTC)
:::That tag was only added by JRSpriggs, there's no reason to think that any other readers are suspicious about it. I'll remove the tag and still think it would not be good to add these calculations to the page, footnote or otherwise. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 04:33, 7 May 2025 (UTC)

::Since 
:::<math> f (x) = x \cdot ( 1 - x^2 )^{ -0.5 } </math>.
::Using the product rule, the rule for differentiating powers, and the chain rule, we get:
:::<math> f' (x) = 1 \cdot ( 1 - x^2 )^{ -0.5 } + x \cdot -0.5 \cdot ( 1 - x^2 )^{ -0.5 - 1 } \cdot ( 0 - 2x ) 
 = ( 1 - x^2 )^{ -0.5 } + x \cdot -0.5 \cdot ( 1 - x^2 )^{ -1.5 } \cdot -2x 
 = ( 1 - x^2 )^{ -0.5 } + x^2 \cdot ( 1 - x^2 )^{ -1.5 } 
 = ( 1 - x^2 + x^2 ) \cdot ( 1 - x^2 )^{ -1.5 } 
 = ( 1 - x^2 )^{ -1.5 } </math>
::Then
:::<math> x - \frac{ f (x) }{ f' (x) } = x - \frac{ x \cdot ( 1 - x^2 )^{ -0.5 } }{ ( 1 - x^2 )^{ -1.5 } } = x - ( x \cdot ( 1 - x^2 )^{ -0.5 } \cdot ( 1 - x^2 )^{ +1.5 } ) 
 = x - ( x \cdot ( 1 - x^2 )^{ -0.5 + 1.5 } ) = x - ( x \cdot ( 1 - x^2 )^{ +1.0 } ) = x - ( x \cdot ( 1 - x^2 ) ) = x - ( x - x^3 ) = x^3 </math>.
::So I was mistaken and the cube is OK after all. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 04:12, 7 May 2025 (UTC)

:::So, what is your opinion about placing your nice calculations in a footnote, given they are now complete and accurate?  It may help improve the credibility of the paragraph and it may save other readers some time and headache. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 04:19, 7 May 2025 (UTC)

::::I am sorry that I was forced by circumstances to log-off immediately after my previous post and was thus unable to revert my edit which put the "dubious" template into the section. I am happy to see that someone else removed it. If you want to put my newer calculation into a footnote, that would be fine with me. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 14:51, 7 May 2025 (UTC)
:::::I think a derivation footnote would be a good addition for the more novice readers and would be happy to install it.  However, I think we need more consensus, and do not yet have Gumshoe2 support.  Gunshoe2, are you OK with placing JRSpriggs's nice derivation work into a foot note?  Once we get your buy-off, I am happy to do the work. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 15:01, 7 May 2025 (UTC)
::::::No, I think it's way too much detail for an elementary computation. I've added one intermediate expression to the section, I think any more calculation detail than that would be inappropriate. [[User:Gumshoe2|Gumshoe2]] ([[User talk:Gumshoe2|talk]]) 18:09, 7 May 2025 (UTC)
:::::::I think your middle step looks good.  It will help give readers more confidence that the simple end result is valid, and will help those who attempt to derive the full solution. [[User:Netshine2|Netshine2]] ([[User talk:Netshine2|talk]]) 18:16, 7 May 2025 (UTC)