Interpolation is important concept in numerical analysis. Quite often functions may not be available explicitly but only the values of the function at a set of points, called nodes, tabular points or pivotal points. Then finding the value of the function at any  non-tabular point, is called interpolation.

 
Definition :

Suppose that the function f (x) is known at  (N+1)  points (x₀, f₀), (x₁, f₁), . . . , (x_N, f_N)  where the pivotal points x_i spread out over the interval  [a,b]  satisfy  a  =  x₀  <  x₁ <   . . .   <  x_N   =   b and  f_i = f(x_i) then finding the value of the function  at  any  non  tabular  point x_s  (x₀ < x_s < x_N)  is called interpolation.

Interpolation is done by approximating the required function using simpler functions such as, polynomials. Polynomial approximations assume the data as exact at the (N+1) tabular points and generate an N^th degree polynomial passing through these (N+1) points. However, if the given data has some errors then these errors also will reflect in the corresponding approximated function. More accurate approximations can be done using Splines and Chebicheve,  Legender and Hermite polynomials but polynomials of degree N or less passing through (N+1) points are easy to develop and useful in understanding numerical differenciation and numerical integral. Hence the present chapter is devoted to developing and using polynomial interpolation formulae to the required functions.
 

Linear Interpolation :

Consider the data ( x₀, f₀),  (x₁, f₁). Problem is to find a function  f(x)  which passes through these two data points. Since there are only two data points available,   the maximum  degree  of  the  unique polynomial which passes through these points is one. Let us assume that f(x) = ax + b  is the straight line passing through the two points then

Solving for a and b gives

This is the required curve passing through the given two points.  Now one can substitute the value of   x Î (x₀, x₁)  in the equation to find f(x).   Similarly  the  same can be extended to find an unique N^th degree  polynomial  which  passes  through  (N+1)  data  points.   The  main  difficulty  in  the   above procedure is the lengthy calculations involved to arrive at the function f(x) which is a function of given data points for higher degree approximations.    For example let us consider the case with three data points. Let the points be

and the second degree polynomial approximation is  f (x) @ P₂(x) = ax² + bx + c   then we have

Solving these three equations for a, b and c gives

The corresponding second degree polynomial approximation for the given data is f (x)  = ax² + bx + c

So  finding  the coefficients  of  the  polynomial  is not an easy task for polynomials of degree greater or equal to two if we follow the usual algebric means to solve the system of equations. The alternative is to simplify the calculations by making use of Vandermonde's determinants or by generating difference formulae. So in the  following sections  we  discuss  Newton's  difference  formulae  which  depend  on  difference  operator  and Lagrange's  method  which  depends  on Vandermonde's determinant.