Question

I’m trying to define a cubic spline as a function in Mathematica 8 as I’ve got every P_{i} (which, of course, are polynomials of degree 3) for each interval [x_{i}, x_{i + 1}], i = 0, ..., n. What I want to do is to define s in the interval [x_{0}, x_{n + 1}] as s(x) = P_{i}(x) if x is in [x_{i}, x_{i+1}]. How can I do that as n varies? I was thinking of Piecewise but that didn’t work.

Answer 1

This does precisely what you ask, if I’m not mistaken. It’s a bit ugly though. There are better alternatives.

n = 5;
ClearAll[f];
f[x_] = Piecewise[Table[{x^k, (k - 1)/n < x <= k/n}, {k, 0, n}]]

f[1/2]

(* ==> 1/8 *)

If you want to make the result dependent on the current state of the global variable n (which I wouldn’t advocate) thne you can replace the Set (=) in the definition of f with SetDelayed (:=), but this implies re-evaluating the Table for every call of f. Not that bad for small values of n, but I don’t like it. Results in that case look like this:

n = 2; f[1/2]
n = 5; f[1/2]

(* ==>  1/2 

   ==>  1/8
*)