
computational simplification and accuracy
Posted:
Sep 23, 2011 11:52 AM


I was wondering if any CAS could even attempt to handle the following problem.
There is a naturally occurring probability distribution on the positive reals, with cumulative distribution function
F(x) = coth(x)  x*csch^2(x)
and density
f(x) = 2*csch^2(x) * (x*coth(x)  1).
Both of these have a removable singularity at 0. The point is that one can, and should, use a simple procedure computing ONE transcendental function, different for small and large values, to compute these without much loss of accuracy.
Now I have found such a way, and it is not difficult. Can any CAS accomplish this? I have shown mine below, to show how it can be done.
For x near 0, use coth(x) = 1/x + x*m(x). Then F(x) = x*(1  m(x)  (x*m(x))^2) and f(x) = 2*((1+x^2*m(x))^2x^2)^2*m(x).
For x large, exp(2*x) is the only transcendental needed. Calling exp(2*x) y, we have F(x) = 1 + 2*y/(1y)  4*x*y/(1y)^2, and f(x) = 8*(x*(1+2*y/(1y)) 1)*y/(1y)^2.
Possibly the algebra can be rearranged for some slight improvement. "Small" and "large" overlap.
 This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)4946054 FAX: (765)4940558

