# Advantages of using cubic splines to expand functions

I've recently determined the deuteron's binding energy using cubic B-Splines to expand the system of coupled differential equations I obtained for my problem. This method of expanding functions using such basis was suggested by a professor of mine and I arrived to quite satisfying results.

However I would like to understand better the advantages of these particular splines.

For example, I've read something about the fact these splines have "local minimal support". What exactly is that? Does it relate to the fact that I've defined the splines piecewise?

• A spline of degree $2$ or less is in general less exact than a spline of degree $3$ , this is obvious. But taking in account too many nodes, the interpolating polynomial can heavily oscillate and moreover the determination of the interpolation polynomial can be numerically very instable.The best compromis is a (piecewise) cubic spline. – Peter Jun 19 at 18:41
• Another possibility to avoid the Runge-phenomenon is Chebycheff-interpolating. The nodes are not equidistant in this method. – Peter Jun 19 at 18:46
• @Peter Nonetheless, even by defining the cubic spline piecewise, it still amounts for some instability correct? – RicardoP Jun 19 at 18:59
• I am not an expert concerning the stability, but I can barely imagine an example where a cubic spline fails due to instability. But I guess there are cases pathological enough, maybe someone can point out such a case (if it exists at all). – Peter Jun 19 at 19:04
• @Peter I see! In my case particularly, I noticed that small variations of some physical parameters which were included in my spline definition would lead to great changes on my final results and I was wondering if that was something related to some inherent instability of the method. Thank you so much for the answers! – RicardoP Jun 19 at 21:18