Hello,
I was curious about the issue of developing routines for the definite
integrals of piecewise smooth functions.
This is related to finding the zeroes of functions:
For instance,
h(x) = max(f(x),g(x)) = f(x) if f(x) - g(x) >=0
= g(x) if f(x) - g(x) < 0
So, if f(x) - g(x) had only finitely many zeroes, say
a_0,...,a_s
then one could simply right the integral as a sum after picking out
which of f or g gives the max.
After some search, I found the papers by Paul Wang, etc. which prove
that the problem of finding a zero (exactly) for polynomials
involving powers of x and sin(x) is recursively unsolvable.
I interpret this to mean that it is not useful to try to do symbolic
integration routines for max(f(x), g(x)) unless f(x) and g(x) are
polynomials. Hence, one shoud concentrate on numerical routines.
My questions is whether there are interesting mathematical problems
involved in this area, or whether the most important things to do
involve the implementation of known numerical routines.
Any comments?
TIA,
-sen
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| Sheldon E. Newhouse | e-mail: sen1 at math.msu.edu |
| Mathematics Department | |
| Michigan State University | telephone: 517-355-9684 |
| E. Lansing, MI 48824-1027 USA | FAX: 517-432-1562 |
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