Simultaneous polynomial equations.

[Tableandchair]

Hi,

I'm looking for a numerical method to solve simultaneous polynomial equations that can be implemented in a computer program. I have included an example of a typical pair of equations that I may need to solve. In this case the two variables that I need to solve for are x and y, all other terms are known constants (with b equal to approximately 5).

Equation 1:

A1 * (k1 - x) * (k1 - 1 + x)^b + A2 * (k2 - y) * (k2 - 1 + y)^b = 0

Equation 2:

[b * (k1 - x)^2 + (k1 - 1 + x)^2] - [b * (k2 - y)^2 + (k2 - 1 + y)^2] = 0

It has been suggested that a Gaussian elimination method along with the Newton-Raphson method be used. Unfortunately, i have been scratching my head over this one for a couple of days now, but still have not found an answer.

Is anyone able to offer any thoughts/suggestions on this subject?

Thanks

 

[rrogers]

This is an old question; but for anybody who wanders by:
Try Grobner Basis's: A good introduction is
http://www.msri.org/web/msri/static-pages/-/node/244
Axiom and Maxima both implement the algorithm; together with some specialized Algebraic Geometry programs.
The Axiom and Maxima implementations can both wander off; but when they work they are great.
Outside of a small hole I wandered into, they do work and offer insight. When they wandered off they just kept chugging. If they gave an answer it was always right.
Basically the process is to take the simultaneous equations and algorithmically change them into a triangular equation set which is successively solvable. You have control of ordering of the triangular system; and to a certain extent the meaning of "simplicity".
Gaussian Elimination on Steroids.
Ray