Mathematica program to obtain a bounding function for a set of data points
- To: mathgroup at smc.vnet.net
- Subject: [mg42811] Mathematica program to obtain a bounding function for a set of data points
- From: gilmar.rodriguez at nwfwmd.state.fl.us (Gilmar Rodríguez Pierluissi)
- Date: Wed, 30 Jul 2003 04:07:39 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Dear friends:
I'm looking for a Mathematica program to obtain a bounding function
for a set of data points. What I have in mind is more in tune with
the "big-O", and "little-o" paradigms, i.e. when one function
seriously dominates another:
|f(n)|<=K*|g(n)| for all n, and some positive constant K.
For example:
If PrimePi[n]={x: x is Prime, x<=n}, then
PrimePi[n]=(li[n] + big-O(x*Exp[-K*Sqrt[Log[n]]])),
where li[n]=Integral[1/Log[t], {t,2,n}], for some positive constant K.
i.e. I see this program as a useful tool to explore bounding functions
for two-summands prime partitions, and other "bounding function-type"
problems.
You will find another (visual) example of a bounding function at:
http://www.gilmarlily.net/goldbach/mgdrotbf.htm
I usually have to spend a great deal of time using Mathematica's
Nonlinearfit program, and using some tactical points from the data set
that I'm interpolating, in order to build the bounding function g(n).
This is an indirect approach that I use for lack of a better method or
program. This Nonlinear fit approach to find the bounding function
works; but unfortunately it takes a considerable amount of trial and
error. It would be nice to have a program that finds the bounding
function directly, and gives the actual equation of the function, in a
manner similar to Nonlinerfit. Mathematica's convex hull program is
an example of a program that does a similar (but, different) sort of
specialized trick (in the case of the convex hull; finding a bounding
polygon around a set of points) that I'm talking about.
Your assistance with this matter will be greatly appreciated!