Talk:Divided differences

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[edit] Does this section belong in the mean value theorem article?

(Moved from my talk page by Oleg Alexandrov (talk) 17:17, 20 November 2005 (UTC))

Why don't you think this belongs with MVT? As I see it, DD appears to be more of a notation for MVT, rather than a separate concept. If you let "c" = LB<P<UB (while P is technically the variable of integration, since P is between LB and UB, at some place P will equal c, so essentially "LB<P<UB" is a more defining, descriptive expression for "c", with both of them meaning "a point between LB and UB providing the mean derivative"):

F'(c)=F'(LB\!<\!P\!<\!UB)=\sum_{TN=1}^{UT=\infty}\frac{F'(P_{TN})}{UT} \quad (LB=P_{1},\;UB=P_{UT}).\,\!

DD achieves the same thing, except that it is from the perspective of the average being of a divided minuscule difference:

\sum_{TN=1}^{UT=\infty}\frac{F'(P_{TN})}{UT}= \sum_{TN=1}^{UT=\infty}\frac{1}{UT}\frac{F(P_{TN})-F(P_{TN-1})}{P_{TN}-P_{TN-1}} \quad (P_{TN}=P_{TN-1}+iota).\,\!

 =F[P_1,P_{UT}]\,\!

I put it above "Mean value theorems for integration", because MVTfI appears to be an application of MVT.
Look at the Mathworld link: Two of its references have "A Mean-Value Property..." in their titles. If DD doesn't belong in MVT, where would it be better suited—I originally had it in Mean, but it wasn't too well received there, either! P=)
Would it be a candidate for its own, separate article? ~Kaimbridge~ 00:17, 20 November 2005 (UTC)

I guess I just found that section not well-written. Can that section be explained without referring to non-standard analysis, or is this theorem valid only for non-standard analysis? I find the notation F'(LB\!<\!P\!<\!UB) to be unusual, and I could not locate the mean value theorem in that section (supposedly there should be a statement like "there exists P, or x, such that ...."). But again, I don't know anything about nonstandard analysis, and maybe that explains it. Oleg Alexandrov (talk) 00:25, 20 November 2005 (UTC)

No, it's probably me! P=)
My formal math is only through high school algebra II (which I liked) and some "proof" type geometry (which was meaningless to me)—20 yr.s ago!—any and all calculus (and most trigonometry) I've learned on my own, over time (and mostly just the concrete, numerical tangibles, rather than the abstract, graphical analysis—thus most of my contributions are concretely numerical in nature). Thus, as I said before in terms of DD, as \frac{F(b)-F(a)}{b-a} appears to be common to both DD and MVT, and there doesn't seem to be much theory involved with DD (other than the derivative being a special case of DD), DD appears to be more a notation for MVT than a separate, distinct concept/theory. As for "LB<P<UB", that is pretty much arbitrary and essentially means "a<c<b": My forte/interest is geometric geodesy, which involves elliptic integrals, where a and b (and even c) have defined meanings.

Let's say you wanted to know the mean arcradius of an ellipse's perimeter, Pr: Using E2'(Lat) as the integrand for the elliptic integral of the 2nd kind, you could say Pr=a\frac{2}{\pi}\int_0^\frac{\pi}{2} \operatorname{E2}'(Lat)dLat or

Pr=a\times\operatorname{E2}[0,\frac{\pi}{2}]\mbox{ or }a\times\operatorname{E2}'(0\!<\!Lat\!<\!\frac{\pi}{2})
Now what if you wanted less than complete? Pr_{1,2}=a\times\operatorname{E2}[Lat_1,Lat_2]\mbox{ or }a\times\operatorname{E2}'(Lat_1\!<\!Lat\!<\!Lat_2)

What would a\times\operatorname{E2}'(c) mean? P=)
I don't know, maybe I should give it its own page (maybe expand it to include higher orders) then just let others come along and fix it up (maybe even slap on a math-stub)? ~Kaimbridge~ 15:01, 20 November 2005 (UTC)

OK, I forked it off to mean value theorem (divided differences). I guess you need to do more work on it, Some introduction would not hurt, the notation would need to be made standard, etc. I could help, but to start, I don't understand what the article wants to say. Oleg Alexandrov (talk) 17:04, 20 November 2005 (UTC)
I'm afraid it's not clear to me either what this article wants to say. You could call the fraction
\frac{f(b)-f(a)}{b-a}
on the right-hand side of the mean value theorem a divided difference, but, so what? By the way, divided differences are discussed a bit at finite difference, and this article hints that a theory of finite and divided differences does exist (indeed, there are several books called "The calculus of finite differences"). Perhaps this is the article that User:Kaimbridge is looking for? I suppose there one could mention that divided differences appear in the mean value theorem and in the definition of a Riemann integral.
The current article lacks focus. The notation is rather odd; for instance, why use f{x} instead of f(x) ? The references also do not match: neither nonstandard analysis nor the integral is mentioned in them, as far as I can see. -- Jitse Niesen (talk) 11:53, 21 November 2005 (UTC)
No—at least as far as I can tell—finite difference is wrong! Look at the two Mathworld articles: Finite Difference and Divided Difference, as well as an even better DD reference I will add. "Finite difference" appears to be just another name for the definite integral: F{UB} - F{LB}, whereas the divided difference (which is what I believe—at least in the first part of the article—finite difference actually means) is the finite difference divided by "UB - LB". As for the "F{}" notation—yea, okay, for this type of article (where special argument enclosures are involved), its probably better to stick with "F()"! P=)
Right now the article is just a loose fragment, as it was originally meant to be just a section...but that will change. One little nit to pick, though: Shouldn't this article be called "Divided difference", with "Mean value theorem (divided differences)" being a redirect? ~Kaimbridge~ 16:03, 21 November 2005 (UTC)
I guess what you mean is:
  • "finite difference" refers to f(b) - f(a), \,
  • "divided difference" refers to \frac{f(b)-f(a)}{b-a} \,.
Right! ~K~
This sounds quite logical, but, and this has confused me a lot, in practice "finite difference" often refers to the quotient; see for instance [1], [2], [3], as well as some numerical analysis books. The numerical method for solving partial differential equations is called finite difference method, though it used what you would call divided differences.
I think the current article should indeed be moved to "Divided difference". Perhaps it is even better to combine this article with finite difference; what do you think about that? -- Jitse Niesen (talk) 17:26, 21 November 2005 (UTC)
I don't know about those sources—I think, they too, are confusing "finite" with "divided".
I just added a formal theorem to the article (based on Herman's page). One question I have with that, though, is the Leibniz notation: Shouldn't \frac{D^{(n)}F(P)}{n!}\,\! actually be \frac{D^{(n)}F(P)}{DP}\,\!?
At this point I think it would probably be better to just mention finite difference, but keep it as a separate article (at least until we see how this article plays out). ~Kaimbridge~ 19:20, 21 November 2005 (UTC)
Well, it might be true that all the sources confusing finite and divided differences are wrong, but it is a fact that there are a lot of them. But we can leave that till later.
I moved the article to divided difference and made some changes so that it says what I think you mean. I hope I guessed correctly; let me know if there's anything you do not agree with.
I am not sure what you want to say about the integral. My best guess is that you can approximate the integral as a Riemann sum, and that this sum can be considered as a sum of divided differences. Does that come close? -- Jitse Niesen (talk) 19:59, 21 November 2005 (UTC)

[edit] Non-standard analysis?

My point in noting non-standard analysis was just to define the derivative with infinitesimality/iota, instead of a limit. Since divided difference involves a concrete interval, it just seemed relevant and convenient in comparing a derivative and dd, especially when the higher order forms are introduced. ~Kaimbridge~ 19:20, 21 November 2005 (UTC)

I don't think nonstandard analysis in necessary. This is just a standard calculus theorem, and mentioning nonstandard analysis makes things more difficult to understand, not simpler.
I think the whole point of this article is that the usual mean value theorem can be considered a statement about the first order divided differences, and this article aims to expand that to higher order divided differences. If I got it right, non-standard calculus is not needed. 20:31, 21 November 2005 (UTC)

[edit] Merging this to divided differences

This article, divided difference, needs to be merged into divided differences. Actually, there is nothing to merge. Maybe the external links, and the result

f[p_0,p_1,p_2,\ldots,p_n] = \frac{f^{(n)}(p)}{n!} . \,\!

which is related to Peano's form at divided differences. If you look there, that Peano form looks very similar to the formula above. I can't say if the above follows from the Peano form as I don't know anything about B-splines which show up in there. Comments? Oleg Alexandrov (talk) 02:53, 24 November 2005 (UTC)

Given that I started this whole ruckus, I'd have to say absolutely (just delete it all and make it a redirect—one can look up a pre-redirect version, if they wish)!
When I first attempted this article (actually just a section for MVT), I did a Wiki search for "divided difference" and didn't see anything in the results—I thought if there was an established page, it would be #1, "Relevancy: 100.0%"—though I didn't hit the You searched for "divided difference" or the "index" links (where I see, now, they both exist). The established DDs article appears beyond my understanding (while some of the notation seems off, the primary author seems competent, so it's probably me: E.g., shouldn't: [y_0,y_1] = \frac{y_1-y_0}{x_1-x_0} be [x_0,x_1] = \frac{y_1-y_0}{x_1-x_0}?), so I'm not even going to touch it. P=)
Searching around further, I think the article I'm actually looking to create is "difference quotient" (right now it's a redirect to Derivative, with a section on "Newton's difference quotient"): See The Difference Quotient.
\frac{\Delta F(P)}{\Delta P}=\frac{F(P+\Delta P)-F(P)}{\Delta P}=\frac{F(P_n)-F(P_0)}{P_n-P_0}
\Delta P = iota: \frac{\Delta F(P)}{\Delta P}=\frac{dF(P)}{dP}=F'(P)
\Delta P > iota: \frac{\Delta F(P)}{\Delta P}=\frac{DF(P)}{DP}=F[P_0,P_n]
From this, reference and relation to MVT and the definite integral can be made, as well as provide higher order formulation. ~Kaimbridge~ 16:35, 24 November 2005 (UTC)
You asked E.g., shouldn't: [y_0,y_1] = \frac{y_1-y_0}{x_1-x_0} be [x_0,x_1] = \frac{y_1-y_0}{x_1-x_0}?). The answer is no. It is assumed that the grid x_0, x_1, ..., x_n is fixed, and you take finite difference of various functions y on it. So it is more important to emphasize the y and not the x. But of course, the most comprehensive notation would be [y0,y1;x0,x1], but I don't think it is worth it. Oleg Alexandrov (talk) 18:52, 24 November 2005 (UTC)
I did the merge to divided differences. Oleg Alexandrov (talk) 23:38, 25 November 2005 (UTC)
Sorry, I have undone the merge, because I want to explicitly refer to the generalized mean value theorem from an article about means, and I don't want that the reader must search for the part of the article that is relevant. I think, that the generalization of the mean value theorem is interesting as such and look forward for more content. HenningThielemann 12:52, 7 November 2006 (UTC)

[edit] Math layout error?

In the examples section the first line stating [y_0] = y_0 is typeset differently than the rest. I see nothing wrong with the source. I wonder is this because of my browser? Or is it a wikicode error? Sustik 09:17, 12 January 2007 (UTC)