Theory

In this page we briefly describe each interpolation method and how the error estimation is computed. For usage of the methods, go to the Interpolation methods section.

Linear interpolation

Given a function $f(x)$ defined on a domain $X \in \mathbb{R}$, and given two points $(x_0,\, y_0)$ and $(x_1,\, y_1)$ where $x_i \in X$ and $y_i \equiv f(x_i)$, we can approximate the function in the interval $[x_0,\, x_1]$ with the following second order polynomial:

\[ p(x) = f(x_0) + \dfrac{f(x_1) - f(x0)}{x_1 - x_0} (x - x_0)\, .\]

The error of the approximation is defined as

\[ R_T(x) = f(x) - p(x)\, .\]

If $f$ has a continuous second derivative, i.e., is a function of class $C^2$, then error is bounded by

\[ \left|R_T\right| \leq \dfrac{(x_1 - x_0)^2}{8} \max_{x_0\, \leq\, x\, \leq\, x_1} \left|f^{\prime\prime}(x)\right|\, .\]

In addition, we can further divide $f(x)$ in several subintervals, delimited by $\lbrace x_i \rbrace_{i=0}^{n}$ and apply a linear interpolation to each of the subintervals.

If the function does not have a continuous second derivative, we cannot use the previous equation to estimate the error bound. However, if the discontinuities are removable discontinuities or jump discontinuities at different points $d_i$, we can locally apply the linear interpolation at the open intervals $(d_{i-1},\, d_i)$.