Given a set of points
for i = 0, 1, 2, ..., n, we want to find
a function (usually a polynomial) which passes through
all of the points. For example, Figure 1 shows
4 points and a polynomial which passes through them.
Figure 1. An interpolation of four points.
Often in engineering, you will only be able to sample
points (usually periodically) and interpolation is one
tool which can be used to estimate the values of points
between those points which were sampled.
We will look at three techniques for finding
- The Vandermonde method (easy to calculate, easy to generalize)
- Lagrange polynomials (easy to find by hand)
- Newton polynomials (efficient to implement when
using Horner's rule)
The Third Tool
Interpolation also forms the third tool which we will use in
developing other numerical techniques. After iteration, it
forms the next most common tool. It is not as powerful as
Taylor series, but is most useful when model information
is either unavailable or too difficult to use efficiently.
The first recorded attempts at quadratic interpolation begin with
the Persian physicist Al-Biruni (973-1048) who was one of the earliest adopters
of the scientific method; however, the first
use of finite differences began with Sir Thomas Harriot (1560-1621) whose other
legacy is the use of the symbols < and >. [Goldstine]
who used interpolation for navigation.