Spike integral

Nick Hale, October 2010

We demonstrate the adaptive capabilities of Chebfun by integrating the 'spike function'

f = @(x) sech(10*(x-0.2)).^2 + sech(100*(x-0.4)).^4 + ...
         sech(1000*(x-0.6)).^6 + sech(1000*(x-0.8)).^8;

(which appears as F21F in [1]) over [0 1].

We create a Chebfun representation and plot the function, increasing 'minsamples' so that the spikes are not missed by an overly coarse initial sample:

ff = chebfun(f,[0 1], 'splitting','on','minsamples',129);
plot(ff,'b','linewidth',1.6,'numpts',1e4)
title('Spike function','FontSize',16)

Now we compute the integral. In order to estimate the time for this computation, we create the chebfun again without plotting it.

tic
ff = chebfun(f,[0 1], 'splitting','on','minsamples',129);
sum(ff)

ans =
   0.211717021214835

Time for creating this chebfun and integrating it:

toc

Elapsed time is 0.357464 seconds.

References:

[1] D. K. Kahaner, "Comparison of numerical quadrature formulas", in J. R. Rice, ed., Mathematical Software, Academic Press, 1971, 229-259.