2011/10/13 Raymond Toy <toy.raymond at gmail.com>
> On 10/12/11 5:00 PM, Edwin Woollett wrote:
> > On Oct. 12, 2011, I wrote
> > -------------------
> >> My idea is that a method with less function evaluations is
> >> having an easier time of converging, and the result is
> >> then more likely to be more accurate. But this is of
> >> course an experimental question.
> > ----------------------------------
> > On second thought, I agree that I should use est abs error size
> > as the criterion. Especially since the writers of quadpack took
> > the trouble to hazard a guess as to that value.
> >
> Here is an example where the number of evaluations is not the correct
> criterion:
>
> (%i58) alpha:9;
> (%o58) 9
> (%i59) quad_qag(4^(-alpha)/((s - %pi/4)^2 + 16^(-alpha)),s,0,1,3);
> (%o59) [3.14157002086875,1.621492651486314e-8,1271,0]
> (%i60) quad_qags(4^(-alpha)/((s - %pi/4)^2 + 16^(-alpha)),s,0,1);
> (%o60) [3.141570020867747,2.996182815899798e-8,1113,0]
>
> Using the number of evaluations, quad_qags would be chosen, but, using
> the error, quad_qag should be chosen. And, it is, in fact closer to
> the true value of the integral.
>
> This integral is one of the tests from the quadpack book. The integrand
> has a vary narrow spike at %pi/4. For alpha > 10, quad_qags fails to
> converge. quad_qag converges until alpha > 18.
>
> And I just remembered that src/numerical/slatec/quadpack.lisp has the
> test examples from the book so you can run the tests yourself. Look at
> the comments at the end of the file.
>
> I'm sure we can cook up various integrals to demonstrate all kinds of
> bad things. :-)
>
> Ray
>
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