Is it possible to make NIntegrate faster?

[Leo Alekseyev]

Dear Mathgroup,

Recently I have been using NIntegrate fairly extensively. I am
dealing with an oscillatory integral that has a singularity.
NIntegrate is able to treat it reasonably well -- the only default I
had to change was increasing MaxRecursion. However, it is slow. 10
points of my integrand take about 40 seconds to evaluate. After I
ported my code to another system, this same integral took about a second using
the Gauss-Kronrod method (quadgk in the other system). Furthermore, by
increasing the absolute and relative tolerance values, I could improve
the speed without losing too much precision, so currently the
integrals evaluate in 0.4 seconds.

I have been playing with various NIntegrate parameters to try to
improve the speed, to no effect. My integrands are straightforward
(although long) algebraic expressions involving a few Bessel functions
and exponentials, wrapped inside a Module; all subexpressions use N[]
so that nothing should be symbolic... Ideally I hoped to find some
sort of a speed/accuracy tradeoff, but that hasn't happened.

I read the numerical integration tutorial in the docs, but am finding
it hard to figure out how to improve the efficiency of my integration.
I would expect Mathematica to get to at least within an order of
magnitude of the other system using the same integration strategy. The current
performance isn't satisfactory -- but neither is the solution of
porting perfectly good Mathematica code to the other system...

I would much appreciate any suggestions.

Thanks,
--Leo

[michael.p...@googlemail.com]

On 23 Nov, 11:51, Leo Alekseyev <dnqu...@gmail.com> wrote:
> Dear Mathgroup,
>
> Recently I have been using NIntegrate fairly extensively. I am
> dealing with an oscillatory integral that has a singularity.
> NIntegrate is able to treat it reasonably well -- the only default I
> had to change was increasing MaxRecursion. However, it is slow. 10
> points of my integrand take about 40 seconds to evaluate. After I

> ported my code to another system, this same integral took about a second =

using
> the Gauss-Kronrod method (quadgk in the other system). Furthermore, by
> increasing the absolute and relative tolerance values, I could improve
> the speed without losing too much precision, so currently the
> integrals evaluate in 0.4 seconds.
>
> I have been playing with various NIntegrate parameters to try to
> improve the speed, to no effect. My integrands are straightforward
> (although long) algebraic expressions involving a few Bessel functions
> and exponentials, wrapped inside a Module; all subexpressions use N[]
> so that nothing should be symbolic... Ideally I hoped to find some
> sort of a speed/accuracy tradeoff, but that hasn't happened.
>
> I read the numerical integration tutorial in the docs, but am finding
> it hard to figure out how to improve the efficiency of my integration.
> I would expect Mathematica to get to at least within an order of

> magnitude of the other system using the same integration strategy. The=

current
> performance isn't satisfactory -- but neither is the solution of
> porting perfectly good Mathematica code to the other system...
>
> I would much appreciate any suggestions.
>
> Thanks,
> --Leo

Hi

It's difficult to know what to suggest without seeing the Integrals.

Best Wishes,
Mike
www.walkingrandomly.com