Mathematical Methods

6

Interpolation

6.1 Introduction

Let the values of a function y = f (x) be given at different x values; thus,

 

x₀     x₁     x₂     ⋯     x_n

 

f(x₀)     f(x₁)     f(x₂)     ⋯     f(x_n)

Then interpolation means to find an approximate value of f (x) for an x between two x-values in (x₀, x_n). This is done by finding a polynomial called an interpolating polynomial which agrees with the function at the nodal points x_i (i = 0, 1, 2, ⋯, n). In finding a suitable interpolating polynomial we need to have a formula for errors in polynomial approximation.

6.1.1 Formula for Errors in Polynomial Interpolation

Let y = f (x) be a function defined at the (n + 1) points

 

(x_i, y_i)     (i = 0, 1, 2, ⋯, n)        (6.1)

Suppose f (x) is continuous and differentiable ...