Jrheller1

CS1 error on Firehole Composites

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derivatives of Lagrange polynomials

To be honest, while I somewhat cleaned up the formulas about higher derivatives of lagrange polynomials that someone else added to Lagrange polynomial, I haven't ever thought too extensively about the subject. I reverted your change though, as from what I can tell all of the problems near nodes boil down to the inclusion of terms like $(x-x_{i})/(x-x_{i}),$ which shouldn't really be too big a problem to just simplify to 1 where applicable, in practical implementation. I don't think your proposed alternative suggestion of converting to monomial basis is a good general idea: it's likely to be numerically unstable and give poor practical results. If you want to improve this section, maybe you can find some good sources discussing the topic? (The current section is entirely unsourced and I'm not 100% convinced it is exactly correct.) I imagine some paper or another has gone into more detail about both abstract formulas and concrete numerical implementation. –jacobolus (t) 03:05, 17 January 2024 (UTC)

Those derivative formulas are getting very expensive for higher order derivatives. They will also always be inaccurate close to a node.

Power basis form is fine for many practical applications of Lagrange interpolation (such as the finite element method) where there are a fairly small number of nodes and the domain is the unit interval or the interval $[-1,1]$.

So your reversion is definitely not helpful for readers of this article. Jrheller1 (talk) 17:46, 18 January 2024 (UTC)

First of all, your addition is unsourced. Wikipedia should only make claims backed by sources. (See Wikipedia:Original Research and Wikipedia:Verifiability.) I'm fairly convinced sources could be found explicitly supporting the previous version, but if not it should be modified/removed. Secondly, the recommendation to convert to monomial basis still seems like a poor choice to me; Wikipedia should especially not make this kind of claim without a source.

However, I agree these derivative formulas get practically ridiculous.

In my opinion we should rewrite this section (I didn't initially add it), following Berrut & Trefethen (2004) who describe how to make a matrix for 1st or 2nd derivatives, with matrix multiplication taking values at the nodes to derivatives at the nodes. The derivatives at the nodes can then be thrown into the barycentric formula to compute derivatives at some arbitrary other point(s). My understanding is that this method gives pretty good results; maybe there's a paper analyzing the numerical stability etc. (Aside: It logically seems to me like the 2nd derivative matrix should just be the square of the 1st derivative matrix, and the matrix for an arbitrary other derivative should be some higher power. But I haven't played around with it.) –jacobolus (t) 18:08, 18 January 2024 (UTC)

"Elementary numerical analysis" by Conte and de Boor does not even discuss derivatives of Lagrange polynomials. So they are clearly assuming that people will convert to power basis form to compute the derivatives.

Have you implemented in computer code the derivative formulas given in this article and also conversion of Lagrange polynomials to power basis form and computation of derivatives using the power basis form? I have done this and it is clear that the power basis method is very accurate when the nodes are uniformly spaced points on the unit interval. This also shows that the formulas given in the article are correct at points not very close to a node. Jrheller1 (talk) 18:42, 18 January 2024 (UTC)