Interpolation
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), . . . , (xN, fN) where the pivotal points xi spread out over the interval [a,b] satisfy a = x0 < x1 < . . . < xN = b and fi = f(xi) then finding the value of the function at any non tabular point xs (x0 < xs < xN) 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 Nth 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.
The concept of interpolation is to select a function P(x) from a given class of functions in such a way that the graph of y = P(x) passes through the given data points (xi, yi), i = 1, 2, . . . , n. The points may arise as measurements in a physical problem, or they may be obtained from a known function. The interpolating function is usually chosen from a restricted class of functions, with polynomials being the most commonly used class.
Theorem 1 (Weierstrass Approximation Theorem) :
If f(x) is a continuous function on [a, b] and ε > 0 is given, then there exists a polynomial P(x), defined on [a, b], with the property that
Theorem 2 : Given n 1 distinct points x0, x1, . . . , xn and n 1 ordinates y0, y1, . . . , yn, there is a polynomial P(x) of degree n that interpolates to yi at xi, i = 0, 1, 2, . . . , n. This polynomial P(x) is unique among the set of all polynomials of degree at most n.
Proof: Consider the following linear system of n 1 equations and n 1 unknowns
The unknowns are (a0, a1, . . . , an). The proof follows from the fact that this system has a unique solution.