The function is smooth on but changes rapidly near the origin

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)=\tanh 5x=\sinh 5x/\cosh 5x

is smooth on

[-1,1]

but changes rapidly near the origin. It is interpolated at

n+1

Chebyshev points of the second kind. ((a)) Find the singularities of

f

in the complex plane, using

\cosh(iy)=\cos y

for real

y

, and identify the one that fixes the rate of convergence. ((b)) Find the Bernstein ellipse parameter

\rho_{\rm B}

and the digits of accuracy gained per node. ((c)) The maximum error is measured to be

4.7\times10^{-3}

n=20

. Predict the

n

needed for an error of

10^{-10}

. ((d)) A student argues that, since

\tanh 5x

is infinitely differentiable on the real line, the error must fall faster than any geometric rate. Say what is wrong with this.

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