Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
College Algorithms
College Modern Algebra

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search