# Cubic Spline

- Author:
- Ryan Hirst

- Topic:
- Functions

In Numerical Analysis, a
We so often choose m=3 and S(x) composed of piecewise cubic polynomials, I will simply write "spline" to mean "cubic spline".
I continue to assume the points are evenly spaced.

*spline*S(x) of degree m is a function defined on an interval I = [a, b] which- coincides with a polynomial of degree m on each subinterval (at each tabular point),
- has continuous derivatives through order m-1

NOTES:

- The spline is peicewise continuous: it is less sensitive to small local changes such as discontinuities, escalating derivatives, and data errors. For example, the sample function above contains a term in the form g(x) = k |x-c|. Weierstrass' pretty theorem will fail: Polynomial[Pts] will not converge to f(x) as n increases. (Why?)
- The given conditions on S(x) lead to a tridiagonal matrix whose solution gives the coefficients of each subcurve (in red). For increased numerical stability, I have solved the system in advance, and used the resulting series to give the coefficients more directly.

**Tool:**Spline[{Points},n, y'(a), y'(b) ], for***equally spaced points***:http://www.geogebratube.org/material/show/id/244653