Spline Interpolation

Abstract

Given the set of n + 1 data points,

$$\displaystyle{ \begin{array}{l|l|l|l|l} x_{0} & x_{1} & x_{2} & \ldots & x_{n} \\ \hline y_{0} & y_{1} & y_{0} & \ldots & y_{n} \end{array},\text{ with }x_{0} < x_{1} < \cdots < x_{n}, }$$

we have seen how we can obtain a polynomial function of degree (at most) n, $p(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0}$ that interpolates these points. That is:

$$\displaystyle{ p(x_{0}) = y_{0},p(x_{1}) = y_{1},\ldots,p(x_{n}) = y_{n}. }$$

This, however, has a major drawback: the polynomial can have a very high degree (up to n) and hence, the interpolating function can oscillate too much. The oscillation may be quite wild even when all the y-values of the data set given are essentially constant.

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-16739-8_8