Talk:Interpolation

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Crikey! This page is strangely marked-up.

The section on cosine interpolation includes the following text

This is a slight improvement over linear interpolation, the resulting function is continuously differentiable, but the differentiable is predictable since the interpolation still has only linear accuracy.

What does it mean that the differentiable is predictable? A reference for this method of interpolation would also be appreciated. -- Jitse Niesen 22:39, 23 May 2004 (UTC)

I removed the section on cosine interpolation. Searching for it on MathSciNet gives no hits, while I got hundreds for polynomial interpolation and spline interpolation. So it seems that cosine interpolation is not that important to be mentioned here. -- Jitse Niesen 22:13, 5 Jun 2004 (UTC)

There is such a thing as cosine interpolation, but in terms of its accuracy it is pretty much the same as linear interpolation but is more expensive to compute. Basically, it tries to avoid some of the artifacts created by linear interpolation by lending the function the continuous differentiability property of the cosine curve. My opinion is that the computational costliness involved is why you had difficulty finding it, cubic interpolation has a similar cost but is far more effective, so cosine interpolation has become a mathematical oddity with no real use. I believe what I have said has quite an overlap with the article. Apart from some diagrams and the equation to generate the function, I do not see what else could go into an article. However, I do think on principle even if this topic represents a mathematical evolutionary dead end there should be an article, for the clarification of a term that I assume from your research was difficult to find information on. Chadernook 23:30, 15 January 2007 (UTC)

Contents

[edit] I cant find the topic "Spatial interpolation"

I was looking for information and techniques for Spatial interpolation and was surprised to see wikipedia, not having this topic.

Can anyone look into this?

Thanks.

See the page on multivariate interpolation. --Berland 18:31, 23 June 2007 (UTC)

[edit] Fix the table

Could someone who understands tables in html fix the table in the Example section? The columns are so close together that if 2 appears in the left column and −3 in the right, then it looks as if "2 − 3" appears there. Michael Hardy 00:37, 14 Feb 2005 (UTC)

It should be better now. -- Jitse Niesen 17:21, 1 Mar 2005 (UTC)

[edit] Remark (moved from the article page)

But there are some cases where a polynom interpolate in end point. (anoninmous comment, moved from the article page by Oleg Alexandrov 19:44, 5 Jun 2005 (UTC))

[edit] Polynomial interpolation --> "infinitely differentiable"?

The section on polynomial interpolation states:

"the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation solves all the problems of linear interpolation."

This is referring to the remark that a linear interpolation cannot be differentiated at its endpoints. However, surely this is also the case for generalised polynomial interpolation?

Oli Filth 09:57, 15 October 2006 (UTC)

I may be wrong but at or beyond the endpoints of the interpolation I think it is still infinitely differentiable, but using the function here makes the process technically extrapolation? Chadernook 23:30, 15 January 2007 (UTC)

[edit] Lagrange polynomial

I find this page (and linked pages too, such as Linear interpolation, polynomial interpolation) to be too focussed on the practicalities of interpolation using computers, and too little on the mathematical theory of the subject. At the very least the foundations of the subject (Newton, Lagrange, Hermite, Bessel) should be covered, and references made to and analogies pointed out with the related subject of numerical integration.

I was thinking about Lagrange too! Could someone mention it in the right place? --Adoniscik (talk) 09:20, 17 February 2008 (UTC)

[edit] Piecewise constant interpolation

I made a graph similar to the existing ones (same dataset) to illustrate nearest neighbor interpolation. Should it be included alongside linear and spline-interpolation in this article? --Berland 18:54, 22 June 2007 (UTC)

That's a good idea. However, the vertical lines at x = 1/2 etc. where the interpolant is discontinuous are officially not part of the graph. I'd prefer it if you could get rid of them, or replace them by dashed lines, or something like that. -- Jitse Niesen (talk) 11:07, 23 June 2007 (UTC)

I totally agree on the vertical lines, I'll see what I can do with gnuplot. All these graphs should possibly also be converted to SVG. --Berland 14:46, 23 June 2007 (UTC)

Here is my go at it--Cronholm144 15:58, 23 June 2007 (UTC)
Umm... Berland, you kinda steamrolled my work. Look at the history of you svg upload--Cronholm144 16:08, 23 June 2007 (UTC)

Yes, I just noticed, and now I see I have broken the style you applied to all of them. However, I liked my style better, have a look at my gnuplot code. --Berland 16:10, 23 June 2007 (UTC)

I saw the code. If you export them all as svg's, then there can be some preservation of the style. Unfortunately all of my work will go to waste :( --Cronholm144 16:14, 23 June 2007 (UTC)

Jitse made the original png's, maybe he has an opinion. Anyway, it is a pity your images don't come with (gnuplot?) source code so that they are reproducible. --Berland 16:50, 23 June 2007 (UTC)

Nope, they are hand-drawn in inkscape, I use gnuplot for plotting curves, I have never used it for data interpolation. I agree about waiting for Jitse. It would be nice if you could tweak his code to export as an svg for comparison. I would do it myself, but I don't have quite the handle on the program that you seem to have(I only started using 5 days ago). Anyway, I am not attached to those images and I certainly can't compete with a graphing program. :)--Cronholm144 18:54, 23 June 2007 (UTC)
I ended up replacing your figures with svg made in Gnuplot. I eventually figured I had to use Gnuplot 4.2 for smooth lines in SVG. Also, I added a stub section on piecewise constant interpolation, please feel free to elaborate. --Berland 21:30, 25 June 2007 (UTC)

[edit] Spline interpolation before Polynomial interpolation

I moved the section on spline interpolation atop of polynomial interpolation, but an IP-editor reverted it. I think it is more natural that splines come first. Polynomial interpolation generalizes constant, linear and spline interpolation, but if often generates one function passing through all points, which is a different procedure than the other ones. Also, its usability, due to Runge's phenomenon, is much less, and I therefore clearly prioritize spline interpolation. --Berland 20:50, 26 June 2007 (UTC)

[edit] Microsphere interpolation?

Do we want a section on "Microsphere interpolation"? Is it original research? See http://en.wikipedia.org/wiki/Talk:Microsphere_projection. Andrew Moylan 14:53, 3 September 2007 (UTC)

I removed the section in this page. I think it violates WP:NOR, at least enough to warrant this removal. --Berland 16:53, 3 September 2007 (UTC)

[edit] Simplest form of interpolation?

On this page, the "simplest form" of interpolation is two things: midpoint and piecewise constant. Maybe this should be altered. --Axel 129.125.178.61 14:45, 9 October 2007 (UTC)