Numerical Analysis in Splines 📂Numerical Analysis

Numerical Analysis in Splines

Buildup

Interpolation is not about restoring the exact function but finding a similar yet more manageable function. Of course, it would be ideal if we could find one that is explicit and easy to calculate, but the universe is not so simple.

Depending on the problem, it might be necessary to quickly solve simple parts and meticulously solve complex parts, and continuity might not even be guaranteed. In this way, finding different interpolations for each segment, rather than the entire range, is called Piecewise Interpolation.

Of course, just dividing it into pieces doesn’t automatically make it better, among these we still need a usable interpolation.

Definition ¹

Let’s say we divide the interval $[a,b]$ into nodes points like $a \le x_{0} < x_{1} < \cdots < x_{n} < \cdots x_{N} \le b$. The following properties possessed by $s$ are called a $m$th degree Spline:

(P1): At each $[x_{i-1} , x_{i}]$, $\deg s < m$
(P2): For $0 \le r \le m-2$, $s^{(r)} (x)$ is continuous over $[a,b]$

Explanation

A spline is a polynomial interpolation with good properties regarding differentiability. In reality, some conditions are added to its use. Being a piecewise interpolation, it is better than typical polynomial interpolation, especially when the given data is messy.