Consider on , interpolated at Chebyshev points of the first kind. The nearest…

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

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

Problem

Consider

f(x) = \dfrac{1}{x^2+9}

[-1,1]

, interpolated at

n+1

Chebyshev points of the first kind. The nearest singularities of

f

in the complex plane are the poles at

x = \pm 3i

. For a function with poles at

\pm ic

and the interval

[-1,1]

, the Bernstein ellipse parameter is

\rho_{\rm B} = c + \sqrt{c^2+1}

, and Chebyshev interpolation gains

\log_{10}\rho_{\rm B}

decimal digits of accuracy per additional node. ((a)) Compute

\rho_{\rm B}

and the number of digits gained per node for this

f

. ((b)) The maximum interpolation error at

n=4

(5 nodes) is measured to be

7.5\times10^{-6}

. Predict the maximum error at

n=10

(11 nodes).