Re: Numerically solving a system of polynomial equations

On 2006-02-25, yaroslavvb@xxxxxxxxx <yaroslavvb@xxxxxxxxx> wrote:
This is a problem that often comes up in statistics --

Suppose we need to find positive solution of a system of n polynomial
equations in n variables, where none of the variables are raised to
more than 1st power, like the following

x+xy=m1
y+xy=m2

In practice people solve this using Newton's method or Conjugate
Gradient, but it may take many iterations. Are there any methods that
could use the special structure of the system to solve it quicker?

You could use the fact that the system is polynomial. There are special
agorithms to obtain numerical solutions for such system. For example,
PHCpack[1] by Jan Verschelde is a so-called homotopy-continuation solver
for polynomial systems. The main difference betweek using PHCpack
compared to guessing and using Newton's method is that it will help you
find all the solutions of the system as well as their multiplicities.

Hope this helps.

Igor

[1] http://www.math.uic.edu/~jan/download.html
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