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Interpolation

6.1 Introduction

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

x0     x1     x2     ⋯     xn

f(x0)     f(x1)     f(x2)     ⋯     f(xn)

Then interpolation means to find an approximate value of f (x) for an x between two x-values in (x0, xn). This is done by finding a polynomial called an interpolating polynomial which agrees with the function at the nodal points xi (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

(xi, yi)     (i = 0, 1, 2, ⋯, n)        (6.1)

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

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