Introduction

In many practical situations, for a function $y = f(x),$ which either may not be explicitly specified or may be difficult to handle, we often have a tabulated data $(x_i,y_i),$ where $y_i = f(x_i),$ and $\; x_i < x_{i+1}$ for $i=0,1,2,\ldots,N.$ In such cases, it may be required to represent or replace the given function by a simpler function, which coincides with the values of $f$ at the $N+1$ tabular points $x_i.$ This process is known as INTERPOLATION. Interpolation is also used to estimate the value of the function at the non tabular points. Here, we shall consider only those functions which are sufficiently smooth, i.e., they are differentiable sufficient number of times. Many of the interpolation methods, where the tabular points are equally spaced, use difference operators. Hence, in the following we introduce various difference operators and study their properties before looking at the interpolation methods.

We shall assume here that the TABULAR POINTS $x_0, x_1,x_2, \ldots,x_N$ are equally spaced, i.e., $\; x_k-x_{k-1}=h$ for each $k=1,2,\ldots,N.$ The real number $h$ is called the STEP LENGTH. This gives us $x_k= x_0+kh.$ Further, $y_k=f(x_k)$ gives the value of the function $y=f(x)$ at the $k^{\mbox{th}}$ tabular point. The points $y_1, y_2, \ldots, y_N$ are known as NODES or NODAL VALUES.