The function is interpolated by a polynomial through Chebyshev points of the first kind on

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

The function f(x)=14+x2f(x) = \dfrac{1}{4+x^2} is interpolated by a polynomial through n+1n+1 Chebyshev points of the first kind on [−1,1][-1,1]. A convergence study reports that at n=10n = 10 the maximum interpolation error is 2.54×10−82.54\times10^{-8}. ((a)) Locate the singularities of ff nearest the interval [−1,1][-1,1] and find the Bernstein ellipse parameter ρB\rho_{\rm B} for this function. ((b)) Predict the digits of accuracy gained per extra node, and use it to predict the maximum error at n=20n = 20.

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