## splines

A description of how to construct higher order splines (splines_article.pdf), that are not b-splines (as far as I can see).
Published on arXiv as http://arxiv.org/abs/0905.3564.
They can be used for interpolation on uniform rectangular grids in any dimension, and they can be generalized to nonuniform rectangular grids (but I think the formulas would be huge).

I constructed them because I need very smooth interpolants --- continuity of as many derivatives as possible.

An article detailing the advantages of high order splines, to be published in the Journal of Computational Physics, can be found currently at http://dx.doi.org/10.1016/j.jcp.2009.10.046.

I've also attached my report on the implementation of high order splines, which contains mostly the same information as the arXiv article, but there's also a discussion on interpolation-solver "interaction".

I'm not sure I'm the first to have discovered this method, but I failed to find something similar on the internet when I needed it.

This file contains the exact expressions for a set of polynomials that is constructed for the splines. These expressions were useful to find a general formula for them.

This file and this file can be used with gnuplot for an example of interpolation.
Note that a weight function is used here, which is equivalent to using the formula in the article.
It's probably best if you know a little about gnuplot before using them; however, feel free to use the definitions of the beta polynomials in your code (and say where you got them from).

The file grid_splines.py does similar things, but it should be a bit easier to work with.

The file gs_example.tar.gz contains the source code for an example implementation of the grid splines (GPL license). You might extract some useful subroutines from it to reuse in your code if you are interested.