Talk:Polynomial interpolation

message: Does anyone mind if I change the nodes from t to x ? I know t is used to indicate time but at the moment there is no reference to time on the page so I think it is clearer to use the standard variable x.MathMartin 13:34, 17 Sep 2004 (UTC)
replies: text: 1) I don't know, just try it out. 2) Yes, please add them, and also mention your textbook; a Russian reference is better than no reference. -- Jitse Niesen 13:19, 18 Oct 2004 (UTC)

text: Go for it. Jitse Niesen 22:13, 18 Jun 2005 (UTC)

message: What does "that is, it is prescribed there a whole k-jet" mean? That sounds ungramattical to me. Andrew (talk) 09:41, 1 July 2009 (UTC)
replies: text: It probably should read "that is, the whole k-jet is prescribed there". I doubt that introducing the concept of jets is useful though, so I removed that phrase. -- Jitse Niesen (talk) 14:54, 1 July 2009 (UTC)
thread_title:  Jet 

message: I think the Stone-Weierstrass theorem should at least be mentioned, as it addresses the polynomial interpolation of a curve. This article is about 
... the interpolation of a given data set by a polynomial
but later on in the article we state
Polynomials can be used to approximate more complicated curves
so we are considering not only data set interpolation but also curves fitting. If no one object to this, I will add it later --Marco4math (talk) 12:23, 26 February 2010 (UTC)
thread_title:  Stone-Weierstrass theorem 

message: I've found a cleared proof of the uniqueness using Vandermorde matrix, I'll replace the actual unclear proof 2 with a new one.
--Marco4math (talk) 23:25, 4 March 2010 (UTC)
replies: text: Do you have a citation to go with it? Anyway why do we need two proofs in the first place? The first proof looks like it adds to the article in an appreciable way but I think just removing proof 2 would be best. Dmcq (talk) 23:52, 4 March 2010 (UTC)
replies: text: I think two proofs contributes a more diverse range of thinking, even though the outcome is equivalent. Given that this is an encyclopedia it should contain all knowledge? Additionally in the sentence "...no matter what method we use to do our interpolation: direct, spline, lagrange etc., (assuming we can do all our calculations perfectly) we will always get the same polynomial." I don't think "spline" should be in this list as a spline will not be equivalent to an interpolated polynomial and the two proofs do not explicitly prove the uniqueness theorem for derivatives at points as well. Should this be changed?--Mrpalermo (talk) 04:29, 5 November 2013 (UTC)
thread_title:  Proof 2 

message: I was considering this problem on my own before looking it up, and came up with a solution that I believe is different from the one presented in the article. I asked a friend of mine what he thought, and his solution seems to be the one in the article. It was certainly different from mine.
The problem was, given a bunch of points, find the lowest-degree polynomial that passes through all of the given points. My friend's solution was this:
Suppose that there are (n + 1) points given. Then the polynomial that we are looking for is of the form
 anxn + an-1xn-1 + ... + a1x + a0 = y
We can plug in each (x,y) pair into the equation above to get a system of equations and solve the system for all an's.
My solution was this:
Suppose the points given have the x-values, say, of 5, 8, and 17. Then construct the polynomial as:
 y = a(x-5)(x-8) + b(x-5)(x-17) + c(x-8)(x-17)
Since the first term is the only non-zero term when x = 17, and we know the y-value when x = 17, it is trivial to solve for a. Same for b and c. Then, if you want, you can expand the equation to get a more normal-looking form.
Since even I managed to come up with this, I assume that someone else has. However, I don't want to put it on the page because it is original research. Does anyone else know if this method is already known? I think that it may be worth having on the page. It also requires no knowledge of matrices, and only simple algebra, so I think that it is more accessible to more people.
Thanks! hwalter42 (talk) 21:44, 7 January 2015 (UTC)
replies: text: you've just rediscovered the approach using Lagrange polynomials, which is given in the article. If you compare that with your method you'll see they are the same.--JohnBlackburnewordsdeeds 22:07, 7 January 2015 (UTC)
replies: text:  Ah, it was in a different form, so I didn't recognize it. Thank you for the fast response! hwalter42 (talk) 22:40, 7 January 2015 (UTC)
thread_title:  Alternate solution? 

message: Aviad.rubinstein (talk) 22:17, 26 January 2020 (UTC)
thread_title:  What's the difference between "polynomial interpolation theorem" and "unisolvence theorem"? 

message: I could not make sense of the illustration captioned "Geometric interpretation of Catmull–Rom cubic interpolation of the black point with uniformly spaced abscissae.[9]"
Is the image missing labels after the minus signs?
Is it really obvious what all those double-ended arrows are indicating?
Jmichael ll (talk) 19:00, 13 March 2022 (UTC)
replies: text: I agree thet the image is very confusing. As "Catmull–Rom interpolaton" is not mentioned elsewhere in the article, this image is definitely not relevant for this article, and I'll remove it. D.Lazard (talk) 20:10, 13 March 2022 (UTC)
thread_title:  graphic unclear