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/
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