Pseudo-Singular Integrals
- To: mathgroup at yoda.physics.unc.edu (Mathematica User's Group)
- Subject: Pseudo-Singular Integrals
- From: Keith Clay <clay at galileo.phys.washington.edu>
- Date: Fri, 20 May 1994 12:00:37 -0700 (PDT)
This is more of a problem of numerical analysis in genereal than of
anything specific to MMA, but I was wondering if anyone has specific
tools for MMA may to help me deal with it.
I have succesfully used Mathematica to analytically integrate out 3
of the variables in some very messy 5 dimensional expressions. The
results are functions of two variables (w and y) which must be integrated
over y from 0 to 1. These expressions are typically 100 kB of ASCII
code, so translation to another language is difficult and would not
necessarily solve the problem. Since the expressions as functions
of y cannot be integrated in closed form, the use of MMA's NIntegrate
is the clear choice.
The problem is that these functions are very nearly singular. A fictitious
but representative (although overly simplified) example might look like:
-PolyLog[2,-1/y] (1 - y) y 1 - y
----------------- + Log[---------] PolyLog[2, -(-----)] + (Less Singular)
4 w + y y
(w + y)
If I NIntegrate this with w set equal to 1, I have no problem. But if I
probe the region where w->0 (try w=10^-n for n = 4,5,6,7...), NIntegrate
grinds for hours and then fails miserably. The reason is clearly that
the first term (roughly Log[y]^2/y^4 for 1 >> y > w ) appears numerically
divergent until one probes the region y << w.
Is there any way to help MMA with this? I am looking for three or more
significant figures. Thus far, the use of clever variable changes has
not done the trick, even when they enlarge the region where y < w, presumably
because the Jacobians then become very nearly singular.
Thanks is advance for any help you can give me.
Sincerely,
Keith
------------------------------------------------------------------------
Keith Clay Department of Physics, FM-15
(clay at galileo.phys.washington.edu) University of Washington
( -or- clay at phys.washington.edu ) Seattle, WA 98195