Solving Nonlinear Systems with Many Variables

Date: 10/25/2000 at 06:16:45
From: Carlos Lopez
Subject: Non-linear equations

I would like to know a numeric algorithm that solves systems of 
non-linear equations with up to 10 variables.

Date: 10/25/2000 at 13:18:04
From: Doctor Rob
Subject: Re: Non-linear equations

Thanks for writing to Ask Dr. Math, Carlos.

In general, this is a very hard problem.

If the equations are polynomial equations in the variables, there is 
the following method. Define a term order on the monomials in the 10 
variables that is consistent with multiplication. Write each equation 
in the form of a polynomial equal to zero. Take the polynomials you 
have, and find a completely reduced Groebner basis of the ideal they 
generate with respect to the term order chosen, using Buchberger's 
Algorithm or some variant of it. 

The solutions of the equations you got by setting all these basis 
polynomials equal to zero are the same as the solutions of the 
original system. One of the equations should contain just a single 
variable. Find its roots using Newton's Method. Substitute the answers 
back into the Groebner basis, one at a time. For each root found, you 
get a system of polynomial equations in one fewer variables.

Repeat the above process on each reduced system. After each round of 
this you have more and more systems in fewer and fewer variables to 
solve, with fewer and fewer equations. When you are done, you will 
have all the solutions.

This process is algorithmic, but it may consume large amounts of time 
and storage in the process.

If the equations involve more than polynomials in the variables, the 
problem is a good deal harder. A good book on numerical analysis will 
give you some information on how to proceed.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/