Elements of Numerical Analysis

Introduction

Let us suppose, there are given (n+1) pair of values (xi, yi), i = 0(1)n. This data may be an outcome of an experiment in which for different values of a variable x, the values of y are observed; thus no relation between variables x and y is known. Alternatively, it may be that the values of a known function y = f(x) are given for specific values of x. The abscissas xi, i = 0(1)n are called tabular points or nodal/pivotal points. Without loss of generality, we can assume x0 < x1 < x2 … < xn, i.e., the values of y are prescribed at (n+1) points in the interval [x0, xn]. Interpolation means to find the value of y for some intermediate value of x in (x0, xn). If x lies outside the interval (x0, xn), the process is called ‘extrapolation’.

The methods for interpolation may be put into two categories according to whether the tabular points are equidistant (equally spaced or equi-spaced or evenly/uniformly spaced) or they are not necessarily at equal interval; in other words, whether the interval xi − xi−1, i = 1(1)n is same throughout or not. In any case, it will be assumed that the behaviour of y w.r.t. x is smooth i:e. there are no sudden variations in the value of y.

The basis for an interpolation method is to approximate the data by some function y = F(x), say, which may satisfy all the data points or only some of them. The function is called interpolating function and the points on which the function F(x) is based are called interpolating points. Although there may be several functions interpolating the same data, we shall be confined to polynomial approximation in one form or another. Before describing various interpolation methods let us give some essential preliminaries which will be required in the development of the methods.