Hello Jimmy and Axel—how nice to meet you both again here on MaplePrimes!
As I wrote to Jimmy in a private email, we plan to include Geddes-Newton multiple integration in a future release of Maple, but that won't happen very soon. Meanwhile, we can develop short ad hoc Maple codes to apply the technique to Jimmy's problem.
Axel's change of variables is an important step in the right direction because it transforms the original integrand into a new integrand of the form f(x,y)*u(x)*v(y)
, where f(x,y)
is a symmetric function
; i.e., f(x,y) = f(y,x)
. The region of integration is transformed from one rectangle into another rectangle.
Now you want to expand the symmetric part f(x,y)
of the new integrand in a Geddes-Newton series on the smallest square which contains the new region of integration—the region of approximation
needs to be symmetric since the function is symmetric, and it clearly needs to include the region of integration. For this particular function, you can choose all of the splitting points on the diagonal of the square. This will generate a finite sum of terms of the form g[i](x)*h[i](y)
Next, multiply each term of the Geddes-Newton series by the weight function u(x)*v(y)
to obtain terms of the form g[i](x)*u(x)*h[i](y)*v(y)
. Now integrate termwise and separate the variables; i.e., integrate g[i](x)*u(x)
with respect to x
, then integrate h[i](y)*v(y)
with respect to y
, and multiply the results. Note that this reduces the 2-D integral of each term to a product of unnested
1-D integrals—that's the key idea!
This separation-of-variables method based on symmetric Geddes-Newton series expansions should easily accommodate high-precision integration for Jimmy's problem. I look forward to hearing about your results!
With best wishes,
Frederick W. Chapman, Postdoctoral Fellow, University of Waterloowww.fwchapman.infowww.geddes-series.info