[R] 2-D numerical integration over odd region

Frede Aakmann Tøgersen FredeA.Togersen at agrsci.dk
Wed Oct 24 08:47:49 CEST 2007

Well if you define

F(x,y) = I_{x > g(y)} f(x,y),

where I is an indicator function giving a 0 if x < g(y) and a 1 of x > g(y) then

\int_a^b \int_{g(y)}^Inf  f(x,y) dx dy = \int_a^b \int_{-Inf}^Inf  F(x,y) dx dy

There are several things you can do to make the integrals easier to be evaluated. The first thing I would is to make a substitution in x to map the interval (-Inf, Inf) to e.g. (0, 1). Then you only need to integrate over (0,1) in the inner integral. I think you can read about that in Abramowitz, Milton and Stegun, Irene A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ( or is it Gradshteyn, I.S. and Ryzhik, I.M. Tables of Integrals, Series, and Products??). Perhaps you or other can provide other references.

Med venlig hilsen
Frede Aakmann Tøgersen

> -----Oprindelig meddelelse-----
> Fra: r-help-bounces at r-project.org
> [mailto:r-help-bounces at r-project.org] På vegne af Chris Rhoads
> Sendt: 23. oktober 2007 22:58
> Til: r-help at r-project.org
> Emne: [R] 2-D numerical integration over odd region
>
> Hello all,
>
> I'm hoping to find a way to evaluate the following sort of
> integral in R.
>
> \int_a^b \int_{g(y)}^Inf  f(x,y) dx dy.
>
> The integral has no closed form and so must be evaluated
> numerically.  The "adapt" package provides for
> multidimensional integration but does not appear to allow the
> limits of integration to be a function.  I need to evaluate a
> number of integrals of this sort and I need the evaluations
> to be fairly precise (something like 10^-6 or 10^-7 would be
> sufficient) so I'd prefer to avoid time-consuming MCMC methods.
>
> Any ideas?
>
> Thanks,
>