📂Numerical Analysis

Interpolation in Numerical Analysis

Definition ¹

For a given pair of data $(n+1)$ and $(x_{0}, y_{0}) , \cdots , (x_{n} , y_{n})$, the method or the function itself that satisfies $f (x_{i} ) = y_{i}$ while possessing some specific property is called interpolation.

Description

For example, consider the situation where there’s data available as shown above, but the middle part is missing. Of course, it’s best to have actual data, but if not, there might be a situation where we need to make predictions to fill in the gaps. In this sense, the term Interpolation is appropriate because it involves filling in the missing parts.

The simplest example of interpolation is Linear Interpolation, where points are connected by straight lines. This type of interpolation has the advantage of being intuitive; however, it is not possible to differentiate at each data point. Therefore, if a method is needed to smoothly connect the points as shown below, linear interpolation cannot be used. As such, interpolation isn’t limited to just one method, and it’s necessary to find the method that meets one’s needs or preferences.