# Interpolation

Given a function \(f\) and a set of values \(x_i\) an interpolation is a curve that passes through the points \(\left(x_i, f\left(x_i\right)\right)\) and is used to approximate the function within the intervals defined by adjacent pairs of the values, typically because only a sample of the function is known.

One of the simplest approaches is linear interpolation in which the adjacent points are connected by straight lines

One of the simplest approaches is linear interpolation in which the adjacent points are connected by straight lines

\[
f(x) \approx f(x_i) + \frac{x-x_i}{x_{i+1}-x_i}\left(f\left(x_{i+1}\right)-f\left(x_i\right)\right)
\]

for \(x\) in \(\left[x_i, x_{i+1}\right]\).