Chapter 4 - The interpolating polynomial

Abstract

The Taylor polynomial is designed to approximate to a given function f very well at one point. However, there is a much simpler type of polynomial approximation in which the agreement with f is not all focused at one point, but is spread over a number of points. This is called the interpolating polynomial. Since p1 is a linear function of x, the evaluation of p1(x) is called the linear interpolation. The main purpose of constructing p1 is to evaluate p1(x) for a value of x between the points x0 and x1, and to use this as an approximation to f(x)—is known as interpolation. Insight into linear interpolation can also be gained by studying tables of standard mathematical functions. In these mathematical tables, functions are tabulated at equal intervals. The interpolating polynomial can be evaluated very efficiently by a scheme known as the Neville-Aitken algorithm.