Talk:Chebyshev nodes

Indices in Interpolation Error

Shouldn't the indices in the formula for the interpolation error run from i = 1 up to i = n, since x_0 does not exist? The final expression for the interpolation error will then become

$\|f-p\|_{\infty }\leq {\frac {1}{2^{n-1}n!}}\max _{\xi \in [-1,1]}f^{(n)}(\xi )$

Could be I'm mistaken, though. 62.194.131.196 (talk) 21:28, 9 January 2009 (UTC)[reply]

The problem seems to be that the definition of n changes in the article. In the first section, Definition, the index runs from i = 1 to i = n, while in the second section the index starts at i = 0. That's confusing, and thus I rewrote the second section. It now arrives at the formula you quote. Many thanks for your comment. -- Jitse Niesen (talk) 22:49, 9 January 2009 (UTC)[reply]

Inconsistency of Ordering of Nodes

This edit changed the affine transformation so that the resultant nodes were ordered in ascending order, that is, the first node corresponds to the node closest to b, and the last corresponds to the node closest to a. However, this is inconsistent with the formula $x_{k}=\cos \left({\frac {2k-1}{2n}}\pi \right){\mbox{ , }}k=1,\ldots ,n.$ given in the previous section, which has the first node closest to the value 1 (corresponding to a), not -1 (corresponding to b). The order of the nodes doesn't particularly matter, but the two formulas should be made consistent. The sign of the cosine term in one equation should be changed. — Preceding unsigned comment added by 170.140.147.203 (talk) 18:12, 7 May 2014 (UTC)[reply]

Roots of Chebyshev polynomial of the second kind

The article incorrectly states that the Chebyshev nodes of the second kind are roots of the Chebyshev polynomial of the second kind. This is not true because the Chebyshev nodes of the second kind include the endpoints, but the Chebyshev polynomial of the second kind takes on its extrema \pm (n+1) at the endpoints. Rather, the set of n Chebyshev nodes of the second kind are roots of (x^2-1)U_{n-2}(x). Grompvevo (talk) 01:55, 9 June 2024 (UTC)[reply]

I don't understand the point of recent changes

@Dyspophyr, I might revert most of your recent changes. They seem largely like disimprovements to me. The new text is choppier, full of unnecessary and distracting bold text, has confusing headings, moved the basic summary out of the lead section, and a lot of the changes seem arbitrary and not really supported by prevailing convention (e.g. "Chebyshev extrema" is a more common name than "Chebyshev extreme points", so it's not clear why you changed that). I don't really understand what your goal is here. –jacobolus (t) 08:44, 1 April 2025 (UTC)[reply]

Sorry to hear these concerns; please give me a little time to react. -- Dyspophyr (talk) 08:46, 1 April 2025 (UTC)[reply]

I concur that "extrema" is better than "extreme points", change made. -- Dyspophyr (talk) 08:48, 1 April 2025 (UTC)[reply]

My goals: correctness, clarity, brevity, in this order.

Bold text is used for alternative names of the main lemma, what could be wrong with that?

The previous text was confusing to me, as it did not explain that the zeroes of U_n are the extrema not of T_n but of T_{n-1}.

Previously, the definition was disrupted by trivia like the affine transform of the interval. Also, what was the point of mentioning open/closed intervals in the definition?

Please take these points into account, or address them here, when you try to further improve the article. Cheers, -- Dyspophyr (talk) 08:56, 1 April 2025 (UTC)[reply]

You took text written more or less using paragraphs of ordinary prose, and split it into a separate paragraph for each sentence. Now the text does not flow at all. I don't understand the logical order. I feel like I'm reading someone's first draft of a power-point bullet list. –jacobolus (t) 09:18, 1 April 2025 (UTC)[reply]

I took verbous prose, removed unnecessary or duplicate information, and was left with four remarks that do not belong to the definition but are nonetheless valuable enough to be preserved in four little paragraphs. Suggestions for a more logical order? -- Dyspophyr (talk) 09:30, 1 April 2025 (UTC)[reply]

As remarks 1-3 are all about describing the point sets, I am fine with merging them into one paragraph. -- Dyspophyr (talk) 09:33, 1 April 2025 (UTC)[reply]

I think there's just a problem of basic writing style. Your comments here follow a somewhat similar style of many tiny chopped up ideas, without any particular narrative flow or order, strung together like a list of bullet points. I find this style extremely difficult and tedious to read, much more taxing than ordinary paragraphs (which were not by any means "verbose"). It also doesn't match the conventional style of Wikipedia articles. Perhaps there are other contexts where this style is preferred though? (Maybe business emails, math lecture notes on a chalkboard, or personal journal entries?) –jacobolus (t) 09:43, 1 April 2025 (UTC)[reply]

And in my perception, it is not good style to start a discussion here by threatening to "revert most of your recent changes". -- Dyspophyr (talk) 10:01, 1 April 2025 (UTC)[reply]

Sorry, I'm not trying to be insulting. I just don't think these changes are, by and large, helpful. I find the article now significantly harder to read, without much clear benefit to the style or organization change. When reverting, I'll try to look carefully at each change you're making and fix specific problematic details you identified; overall though the previous version was better, so I'll probably end up with something close to that. I don't have time to do it right now, and to be honest I'm tempted to revert the changes wholesale (cf. WP:BRD), but I'm going to wait until I have a while to spend some careful attention, hopefully within a day or two. –jacobolus (t) 10:14, 1 April 2025 (UTC)[reply]

As WP:BRD says, bold revert is not encouraged. It has a huge risk of demotivating your opponent, or of resulting in dire conflict. It is utterly unnecessary in the present case, as I am available to discuss with you, and have already made changes to the article in response to your complaints.

When you try to restore the previous prose, then please take into account the following criticism that drove my edits. Sorry, again a list:

Keep the lead section short; don't provide details that will be fully duplicated in the Definitions section.
Don't mix definition and implications.
The definitions must not be tied to open/closed intervals.
The possiblility of an affine transformation is obvious, and should not be mixed with the definition.
It is essential information that the Chebyshev extrema pertain to T_{n-1}, not to T_n.
Chebyshev-Lobatto points redirects here. It is a widely used alternative name, and should therefore appear in bold.
"Note that the Chebyshev nodes of the second kind include the end points of the interval while the Chebyshev nodes of the first kind do not include the end points." - is overly verbous. Compare "While the second-kind nodes include the interval end points -1 and +1, the first-kind nodes do not." - which is not only much shorter, but even, for better readability, includes redundant information "-1 and +1".
"These formulas generate Chebyshev nodes" - duplicates information for no gain in clarity.

Perhaps, rather than risking forth-and-back edits, we should ask for a third opinion??

Best regards, Dyspophyr (talk) 11:49, 1 April 2025 (UTC)[reply]

I think the recent changes are a disimprovement, particularly moving summary content out of the lede and information about geometry from the definition section. To some extent, I share jacobolus' feeling that what was there flowed better, but I'm not sure I would go quite so far as to characterize it like as a transcription from lecture notes—just stylistically different. (Incidentally, I don't think what was there originally was perfect, just that on balance the recent edits do not move in the right direction.) Tito Omburo (talk) 12:07, 1 April 2025 (UTC)[reply]

It did "flow better". It just had redundancies, inaccuracies, and omissions. Isn't there a way to move forward without threat of total revert? -- Dyspophyr (talk) 12:22, 1 April 2025 (UTC)[reply]

For what it's worth, I also don't think the original text was amazing. This article could certainly use more help; beyond questions of prose style, there is a lot more we could say about this topic, we can add more sources, etc. –jacobolus (t) 16:23, 1 April 2025 (UTC)[reply]

Way more important, the article was mathematically wrong in suggesting that the set of roots of U_n equals the set of roots of T_n. By now, the math should be correct. Checks will be welcome. -- Dyspophyr (talk) 14:59, 1 April 2025 (UTC)[reply]

From what I can tell the article did not previously suggest that. It said: "The Chebyshev nodes of the second kind, also called the Chebyshev extrema, are the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind." This can of course be unpacked more precisely with technical details, but seems like a fine summary. It would be especially helpful for this to graphically plot an example of each type of polynomial, showing how the extrema of one align with the zeros of the other. (Edit: it should have said "extrema of a Chebyshev polynomial ...".) –jacobolus (t) 16:28, 1 April 2025 (UTC)[reply]

I added a possible figure, floating to the right here. As you can see the extrema of one type of polynomial correspond to the roots of the other. –jacobolus (t) 19:29, 1 April 2025 (UTC)[reply]

Nice figures, compliments. -- Dyspophyr (talk) 07:27, 2 April 2025 (UTC)[reply]

jacobolus, so you imposed your will, boldly and absolutely, regardless of all my efforts to compromise and to accomodate your wish for non-choppy prose. Be happy with the result. I'll just unwatch this page. -- Dyspophyr (talk) 11:16, 4 April 2025 (UTC)[reply]

I'm not quite sure what you mean. I tried to incorporate all of your feedback here, while also restoring features of the previous version that I thought were important. There is certainly room for further discussion about how this page should go. (And this article can be greatly expanded; it currently is only barely beyond a stub.) –jacobolus (t) 16:31, 4 April 2025 (UTC)[reply]