Talk:Newton's method

Topic heading changes

Hey there, the heading "Generalizations --> Nonlinear systems of equations" is inappropriate for a number of reasons.

Firstly, Newton's method is (almost by definition) a method for finding the roots of non-linear equations (since finding the roots of linear equations does not involve the derivative / iteration). Hence, the term "non-linear" is redundant in the heading.

I think a heading of just "systems of equations" would be OK except that the body of the text specifically describes a single vector-valued function F : ℝ^k → ℝ^k. Further, I think it is more intuitive to think of things in this way, rather than as a system of equations, since the univariate N-R method finds the root(s) of a single real-valued function, and thus the natural extension is to find the "vectorised" roots of a single function that is now vector-valued.

--Statisticsmonkey (talk) 05:41, 27 August 2021 (UTC)

 Fixed. I have also changed other headings accordingly. D.Lazard (talk) 07:12, 27 August 2021 (UTC)

Choice of styles

The Manual of Styles states that "[w]here more than one style or format is acceptable under MoS, one should be used consistently within an article". There is currently no consistency of mathematical style in this article as there is a lot of both the latex math style as well as the HTML style (see Manual of Style : Mathematics for reference). I think the latex style is much better and I propose that we standardise the math style accordingly by replacing any HTML math style with latex math style.

There were many discussions among mathematics editors about this subject, and the present consensus seems the following: Avoid sanserif html (abcdef). If html is used for a formula, it must be enclosed inside {{math}} or {{mvar}}. If a formula contains special characters, such as $\mathbb {R} ,{\sqrt {~}},\circ$, latex is preferred, because of a bad rendering on some browsers. For displayed formulas, latex is preferred. Otherwise, the used style must be kept unless if the article section is edited for other reasons (one reason is that, changing the style induces errors frequently). Moreover, many editors prefer {{mvar}} for isolated variables.

It seems that this article follows these rules (I have not checked this in details), so the style of the formulas must not be changed per MOS:VAR. D.Lazard (talk) 07:42, 11 September 2021 (UTC)

Meaning of “sufficiently close” is unclear

I was reading through this article just now and am confused by the three conditions under "The term sufficiently close in this context means the following:". The first condition is “Taylor approximation is accurate enough such that we can ignore higher order terms”; this invokes an entirely different unanswered question of what it means for the Taylor approximation to be “accurate enough.” Is this condition equivalent to the other two conditions described by formulas, or do all three conditions have to hold independently for “sufficiently close” to be true? rdl381 (talk) 14:52, 25 October 2022 (UTC)

I rewrote the conditions to make them correct, simpler, and clearer. How is it now? Are you satisfied? JRSpriggs (talk) 21:04, 25 October 2022 (UTC)

Yes, I think it’s clear now. Thanks! Is it correct to merge conditions 1 and 3 as follows?

1. $f'(x)\neq 0$ for all $x\in I$, where $I$ is the interval $[\alpha -r,\alpha +r]$ for some $r$ such that $x_{0}\in I$

This way $r$ isn’t used before its meaning is explained. — Preceding unsigned comment added by Rdl381 (talk • contribs) 15:55, 27 October 2022 (UTC)

Since all the conditions are needed, I do not think that the order is particularly significant. I think it would be better to focus on getting x₀ closer to α rather than increasing r to try to include it. We do not want M to get so large that the last condition cannot be met. JRSpriggs (talk) 21:53, 27 October 2022 (UTC)

Recent changes should be reverted

I am concerned that after a flurry of changes the current version Special:Permalink/1219909644 by @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. –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. 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. –jacobolus (t) 18:26, 20 April 2024 (UTC)

I agree with jacobolus. Bubba73 ^{You talkin' to me?} 02:39, 21 April 2024 (UTC)

I completely agree with jacobolus. 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. Gumshoe2 (talk) 02:58, 21 April 2024 (UTC)

I agree with jacobolus. PatrickR2 (talk) 06:10, 21 April 2024 (UTC)

Please be specific. Which part is "impenetrable"? I can certainly try to improve it. Fangong00 (talk) 13:05, 21 April 2024 (UTC)

In your proposed version of the page, the section called Simpson's extension is pretty much identical in content to the existing section k variables, k functions together with the existing section 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? Gumshoe2 (talk) 13:39, 21 April 2024 (UTC)

Didactically, we want to understand the classical 1d situation before even mentioning other cases. —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. Fangong00 (talk) 13:44, 21 April 2024 (UTC)

Wikipedia is an encyclopedia, namely neither a reference nor an instruction manual. The intention should not be to front-load the information that someone might be trying to look up. 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. 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. Tito Omburo (talk) 12:16, 21 April 2024 (UTC)

Thank you. 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? 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. Fangong00 (talk) 13:51, 21 April 2024 (UTC)

I am extremely skeptical of that. Such iterative methods go back to at least 1950. 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" 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. 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. 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. 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. 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? 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. Fangong00 (talk) 17:47, 21 April 2024 (UTC)

I disagree with the outlook of @Tito Omburo and @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. 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.) 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. Fangong00 (talk) 17:23, 21 April 2024 (UTC)

@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.) –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? 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. –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 this revert exclusively affected primarily sections much later in the article. Tito Omburo (talk) 10:52, 22 April 2024 (UTC)

You are mischaracterizing what happened and misunderstanding people's intentions. –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 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. 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. –jacobolus (t) 15:31, 21 April 2024 (UTC)

I took your suggestion. Fangong00 (talk) 16:03, 21 April 2024 (UTC)

@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. –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. 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. –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. 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. –jacobolus (t) 01:27, 22 April 2024 (UTC)

Do we agree no one is supposed to act like having some kind of status here? 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. –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. 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.

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. –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. Bubba73 ^{You talkin' to me?} 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 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.) 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. Bubba73 ^{You talkin' to me?} 22:48, 21 April 2024 (UTC)

@Gumshoe2 What's this Banach thing? I just want to know about Newton's method. 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. Fangong00 (talk) 16:49, 21 April 2024 (UTC)

@Fangong00 Fine, so put that detail done further in the article, not in the intro and first sections. 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. —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. 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. —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. 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? 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 "mapping between 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. –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 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 points. 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 equations.

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. –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. –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? 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? 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. 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. 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 — both accessibility and summarizing the article that follows — is a brief synopsis of highly technical material towards the end. XOR'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. –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. 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? 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.

1. 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 $f(x)/f'(x)$ 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.
2. 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. 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? Gumshoe2 (talk) 00:12, 23 April 2024 (UTC)