Talk:Order of approximation

Chomsky
I removed the paragraph on Chomsky as I felt it inappropriate in an article whose main topic is the mathematical sense of 'order of approximation'. Zero sharp 05:03, 19 November 2006 (UTC)

Article needs work
While orders of approximation are used for data fitting, they are also used in theory work. The zeroeth order, first order, second order expansions are used when the relevant terms in the expansion series become significant enough to affect theoretical predictions. This should be made more clear. "First order" and "Second order" effects are often used to describe perturbations and deviations from linear models. We should have some text on this. --ScienceApologist 14:03, 2 December 2006 (UTC)

Is this article necessary?
I'm having a hard time understanding the purpose of this article. It has already been flagged for having no sources, the information in the article seems largely subjective, and if other people are like me then the information that they are actually looking for is in the Taylor Series article. 67.128.198.190 (talk) 20:32, 8 March 2013 (UTC)

This article is necessary because it explains something else than the Taylor Series article. The Taylor Series article does not make any references to orders of precision/approximation that are useable by the general audience.  --Jangirke (talk) 20:50, 21 March 2013 (UTC)

Two separate themes
The current article strikes me as trying to merge two separate issues into one: 1. the number of significant digits in the estimation of a quantity. 2. the degree of a polynomial fit. I suggest it would be far more pedagogical to treat these two issues separately. /216Kleopatra (talk) 21:15, 17 October 2013 (UTC)

Historic approximations
The discussion started here on the need for a new section with examples of "historic approximations", where order of approximation was very important for some reasons. Suggestions are invited, in addition to the three topics already mentioned. The descriptions should not be too long, as this section is should not substitute for full articles. — Preceding unsigned comment added by C. Trifle (talk • contribs) 15:43, 26 March 2016 (UTC)

Would a graph help?
User:Pacerier added "Unclear" tables to three units. User:Pacerier was very active on April Fools Day and on a day before making lots of contributions to different articles. In fact, the remark that the text may be unclear to readers could be added to just any text but I do not think the added criticism was just a practical joke. In my opinion, a graph to show the three examples would help. Also, this article should be extended by more examples (see above). C. Trifle (talk) 19:45, 2 April 2016 (UTC)


 * No, I am sorry. A graph will not help without a linguistic decision. The same phrase should not be used to mean two different things. For example, if the phrase "nth-order approximation" is linked to the meaning of the nth-power of ten and in the same text to the meaning of a polynomial of an n-th degree, then 10 to the power of 1 is confused with a straight line with a slope, i.e. a polynomial of degree 1. It is not possible to dictate which meaning writers outside Wikipedia should choose but I am afraid that without clearing this up on the level of this article nothing will help. Perhaps it will be easier to understand if one uses the notion of  significant figures (or digits) to mean the numerical accuracy?C. Trifle (talk) 23:03, 3 April 2016 (UTC)
 * I agree with you. As rewriting this article correctly appears to be difficult, I suggest to make it a disambiguation page, which could be reduced to:

Order of approximation may refer to
 * The number of significant figures of a numerical approximation
 * The degree of the Taylor polynomial that is used for approximating a function
 * IMO, the only problem, which would remain open by this change of the article, is where to place what physicists call "computing at order k". This is in fact the arithmetic of truncated Taylor series. This seems to be lacking in Wikipedia, and deserves either a specific article or a section in an existing article (which one?). D.Lazard (talk) 08:49, 4 April 2016 (UTC)


 * How about rearranging the examples first? I think there are more things that are lacking in Wikipedia, including the historic usage of the "order of approximation". So is rewriting it really so difficult? Why not put the Oth,1st and 2nd "order of magnitude" examples (with the town residents) into one section with the link(s) to the main article(s) and do the same with the remaining Taylor series examples with the three points, and then add the new sections? I agree that one can make it a disambiguation page and add the remains to some other articles, but it seems that this could result in a need for one or more new articles where these things would be placed scattered. The article has stayed here for so many years, maybe first try to give it one more chance? C. Trifle (talk) 12:59, 4 April 2016 (UTC)
 * IMO, rewriting this article as a single article is impossible, as this would imply to talk about two different thing in the same time. This is the reason for which a dab page is needed. However, as we are only two for discussing this, I'll start a WP:RfC. D.Lazard (talk) 14:19, 4 April 2016 (UTC)

RfC: Should we transform this article into a dab page?
Most posts in this talk page complain that the this article is unclear and confusing. It appears that one reason is that it mixes two different notions of order of approximation. It has been recently suggested to transform this article into a disambiguation page, whose content is given in the preceding section. As this article belongs to four different projects, the discussion between only two editors is no sufficient for such a dramatic change. D.Lazard (talk) 14:29, 4 April 2016 (UTC)
 * I am in favour of the motion. I think the following should also be included:
 * The order of approximation of a numerical procedure for solving differential equations (e.g. the Euler method is called a first-order method)
 * The number of terms in the Stirling's series that is used as an approximation for factorials
 * C. Trifle (talk) 23:37, 4 April 2016 (UTC)


 * Support per agreement with the previous section of this talk page; but why an RFC? It may impact multiple projects, but they can always revert later if they disagree (WP:BRD). RFCs are usually used when the issue is contentious and larger input is needed. I do not see that anyone disagreed, here. Tigraan (talk) 12:06, 6 April 2016 (UTC)


 * Shall we wait a week for more opinions? C. Trifle (talk) 09:04, 8 April 2016 (UTC)

What to do with the information for general reader? I have read some old edits since 2003. I think the intention was good but there was some danger of confusion from the beginning. One might understand that if you have some three points then you begin with zero significant digits and as a result you get a constant, after which you get one significant digit and a slope, and then two digits for a parabola with which most scientists are happy and here they usually end. IMO the problem is that there is defintely a need in Wikipedia for something that is not given in this article, but should be placed somewhere. I mean a bit more reliable piece for the general reader. (1) the phrase "order of approximation" is generally used in language in various contexts (2) there was some historic usage that does not match today's views. C. Trifle (talk) 09:04, 8 April 2016 (UTC)


 * Support These two concept should be un-slotted from each other and treated separately; the current state is confusing. The general information about scientific interpretation of first-, second-order etc. is valuable but would seem to fit well into Significant figures, I believe. However those parts need sourcing if they are to be retained - I can anecdotally agree with these interpretations but couldn't tell you whether they are generally accepted.-- Elmidae  (talk) 06:44, 10 April 2016 (UTC)


 * Strong support. I'm meant to know this stuff, but I find the article incomprehensible. I suspect that there's more than two different concepts all mixed together there. Maproom (talk) 06:39, 11 April 2016 (UTC)


 * Oppose - this RFC seems fuzzily suggesting what seems an inappropriate path for the named lacks.  Small nit in being fuzzy that RFC should put the intended language in the RFC, not point to prior discussions.  However, presuming it means to dab to Significant figures and Taylor polynomial, then bigger item is to reject as inappropriate.  Because instead of tackling 'unclear or confusing' by explanation or example, it instead would just ask the reader to jump to an article which does not use the term and figure out how such a term would apply to that situation ???  Significant figures closest approach to 'order of' is about the 'order of magnitude' for the sample size.  Taylor polynomial is only a redirect iteself to Taylor's theorem, and it's closest approach would be the finite order truncated at.  In both cases the reader would be left having to figure it out from there, which to make the 'unclear or confusing' worse, not better.     Markbassett (talk) 16:13, 11 April 2016 (UTC)


 * Oppose - an objetive of this article is to be didactic, to "Non-mathematical People"... The article covers an important concept (and jargon), that is in fact frequent in "scientific talkings" and formal and informal literature. This article is a "semantic bridge" for Curve fitting, Scientific modelling, Scale analysis (mathematics), False precision, Big O notation,  and, in second-order,  Significant figures, Generalization error, Taylor polynomial, and perhaps many others, to non-mathematical people. This RFC seems not offers an option to cover the gap that this article is covering. The best option is to enhance this article. --Krauss (talk) 09:20, 12 April 2016 (UTC)


 * Oppose, largely agree with Krauss above. The article as-is does serve an often overlooked purpose of presenting an informal concept in math that relates to many conceptually related formal methods. Students often get confused by these things, because they are not usually addressed in textbooks (E.g. The idea that both a truncated Taylor series and Stirling series both can have an "order of approximation," even though they are very different things). I think the problems raised by OP and in the RfC can and should be addressed by expanding the "see also" section, and working in better high-level descriptions and links to the related techniques throughout the article. SemanticMantis (talk) 14:57, 13 April 2016 (UTC)
 * Good points.-- Elmidae  (talk) 15:30, 13 April 2016 (UTC)


 * Comment, The current text mixes the series [approximation by constant value; linear fit; quadratic fit] with [0 significant figures; 1 significant figure; multiple significant figures]. These are two completely disjunct concepts and need to be separated. I don't know whether this would be best done by a disambiguation to two separate articles or by separating them in the text, though. --Slashme (talk) 07:56, 19 April 2016 (UTC)
 * Comment At a first approximation, it is hard to disagree with any of the foregoing remarks. The article is poorly structured and unclearly worded. It deals with an important class of concepts and needs serious attention. I agree in particular with the repeated remarks that it inappropriately conflates distinct concepts. One difficulty is that although some concepts are indeed distinct, I do not believe that putting them into separate articles would be appropriate; the nature of their differences as well as their substance needs clarification. The article should put them into proper perspective and distinction. JonRichfield (talk) 06:48, 26 April 2016 (UTC)


 * Support The current version is incomprehensible and will likely mislead readers on what order of approximation is. As noted above, order of approximation represents different concepts that need to be made explicit. Tale.Spin (talk) 17:32, 26 April 2016 (UTC)


 * Oppose - The current version is comprehensible to me. I also don't see ambiguity in what is being described here. I thought someone might have significantly improved the article since this RfC was started but apparently that's not the case . ~Kvng (talk) 23:57, 3 May 2016 (UTC)

Important article, only need work
The "main/reference article" about the subject, Scale analysis (mathematics), is "didactically horrible"... A merge will not solve the problem.

Articles to use as "reference article" are Big O notation and Curve fitting: the order of approximation is a jargon about "scientific modeling of reality". See curve fitting: can use similar illustrations, and here (the order of approximation article) can also add more generic illustrations, not only linear curves, but any other "fit to model" process (as "progressively more refined approximations" process)... see Scientific modelling.

--Krauss (talk) 08:55, 12 April 2016 (UTC)
 * Your two posts cite a lot of related articles. I may add Approximation, which is also "didactically horrible". I interpret your two posts as the lack of a broad-concept article about approximation, which should be useful and understandable for the layman. I agree that such an article is lacking. Nevertheless, the subject of such an article is much wider than "order of approximation", and writing it under this title would make difficult to find it, for a reader, which is not specifically focused on "order of approximation". If such an article would exist, it would be natural to redirect Order of approximation to it. But writing it is not easy and would take some time, and we are faced to the problem of what we should do now. I agree the the possible redirects from a disambiguation page are not well satisfactory. This could be improved by adding to the dab page a section "see also" redirecting to the articles you cite. Personally, I think that this is the best short-time solution. D.Lazard (talk) 10:23, 12 April 2016 (UTC)


 * Hi, thanks. You're right about "... take some time ... what we should do now"... Hum... Can I suggest something simple but not so immediate? I have experience to do/help and see good results, but is to do step-by-step and with some collaboration &mdash; perhaps only you and me, perhaps more people, and no illusion about time for perfect work. We can  "connect" some articles by a semantic-core box, let's do it here?  (... and after people read/change/collab oriented by the box ... we merge articles). --Krauss (talk) 00:05, 13 April 2016 (UTC)
 * The O in this box should definitely not be calligraphic. This is not the usual notation, which is just a regular upper case O. Sapphorain (talk) 07:13, 13 April 2016 (UTC)

Draw us a picture
The lack of clarity in the examples could be reduced by drawing a simple graph for each example.&bull; &bull; &bull; Peter (Southwood) (talk): 05:50, 19 April 2016 (UTC)


 * Very good point. If no-one's done it by the weekend, I'll make some SVG graphs. --Slashme (talk) 08:00, 19 April 2016 (UTC)


 * True, but this opens an interesting concept that I did not notice in the article: fallacies of approximation. Two that occur to me are:


 * Successive inappropriate approximations to a limit, such as the proof that the diagonal of a unit square equals 2 instead of root 2. This also lends itself to graphical comparisons with valid approximation of circle circumference by inscribed polygons. I see that there is a convenient graphic in Wikimedia at File:Cutcircle2.svg|thumb|Cutcircle2, but I don't see one for the diagonal fallacy. Yet.
 * Crossing a chasm in two jumps JonRichfield (talk) 07:22, 26 April 2016 (UTC)

Introduction
I have changed the introduction a bit and added the references section and links to dictionaries and inside Wikipedia. There seemed to be some confusion about the usage of phrases with and without "order", and also the meaning of "precision" and "accuracy". It is waiting in my Sandbox. Is it better or worse? How to improve it? C. Trifle (talk) 14:04, 2 June 2016 (UTC)


 * I worked hard to make the introduction more readable. The text is still here. Now I'm going on holiday. If you have any comments, please write below. C. Trifle (talk) 16:36, 3 June 2016 (UTC)
 * I have replaced the lead by the one written by in its sandbox. D.Lazard (talk) 12:30, 27 November 2018 (UTC)
 * The reference to "zeroth approximation" is very confusing, . It seems out of place. Tale.Spin (talk) 23:13, 6 October 2019 (UTC)
 * Thank you for this remark, . Actually, it is not mine. It has been with this article for 16 years, since the first entry by at 03:43, 22 October 2003. Would you prefer a different spelling? For example, would a "zero-order approximation" sound better? Both forms seem to be used though. Or is there something else that you find repulsive about it?--C. Trifle (talk) 21:50, 13 October 2019 (UTC)
 * Wow, I had zero recollection of having originally started this article. Look at how far it's come!  :)  It looks like I started it in contrast to separate articles on first and second order approximations, and the first edit someone else made on it  was to merge in those other two articles. , it's worth noting that the phrasing here is "zeroth order approximation" to keep consistent with "first order approximation" and "second order approximation".  Saying "zero order approximation" would be like saying "one order approximation" and "two order approximation", so I don't think we should change it.  Yes "zero order approximation" is used, but it's not grammatically consistent.  We could put in that it's an alternative way of saying it though since it's relatively common.  zandperl (talk) 23:55, 16 November 2019 (UTC)
 * Agree : "one order" and "two order" does not sound acceptable to me. How about "order zero approximation" used together with "order one" and "order two approximation"? It is much less common but it happens. Does it not sound confusing to you, ?
 * In the section about the "zeroth order approximation" I would end the sentence after "y-values" as follows:

...is an approximate fit to the data, obtained by simply averaging the x-values and the y-values. (Stop here.) After that, it is necessary to show when we need to ...derive a multiplicative function for that average... here or in the next section because that seems to fit better to introducing the first order approximation. --C. Trifle (talk) 00:53, 26 November 2019 (UTC)

We need to begin with a clear and simple example.
I looked at the following articles: Order of approximation, Taylor's theorem, Taylor series, and Big O notation. They all user the word "order" without defining it. Links to this page would help. I propose to begin this discussion with the use of a Taylor series to approximate a simple function. I would like to place this soon after the following sentence:

"The formal usage of order of approximation corresponds to the omission of some terms of the series used in the expansion (usually the higher terms)."

I will remove the comment "(usually the higher order terms)" and instead give an example where the higher order terms are omitted. Then I will Taylor expand the inverse of (1+x) (exponential function), and identify the terms. Guy vandegrift (talk) 04:11, 10 June 2024 (UTC)