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Math Help - Which method for solving system of nonlinear polynomials equations?

  1. #1
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    Which method for solving system of nonlinear polynomials equations?

    Hello. I have a research project and need to solve the system of nonlinear polynomial equations like that:
    1.The polynomial equations have degrees 2, 4, 6, 8 respectively...........
    2.The system is positive-dimensional.

    For example:

    <br />
\left\{<br />
\begin{array}{rr}<br />
1+x_1-x_1 x_2 = 0\\<br />
1+x_1-3 x_1 x_2+\frac{3}{2} x_1 x_2^2-\frac{1}{6}x_1 x_2^3 = 0<br />
\end{array}<br />


    (Of course, my system is biger than that very much). I only need one of its solution, which satisfied condition x_i > 0. The system is very unstable, it can not be solved by any numerical method, such as Newton method, Homotopy method or so... and needs a "arbitrary precision arithmetic" environment to calculate.
    I use Maple, command Triangularize - package RegularChains - to transform this system to triangular form. That command gave me nice solution, but it only works for system up to 4 equations. For system consists of 6 equations, Maple used ~2Gigabytes of RAM and gave up.
    So, do you know how to solve this problem? Which method is efficient to my system? Thank you very much for your help
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  2. #2
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by Beet View Post
    Hello. I have a research project and need to solve the system of nonlinear polynomial equations like that:
    1.The polynomial equations have degrees 2, 4, 6, 8 respectively...........
    2.The system is positive-dimensional.

    For example:

    <br />
\left\{<br />
\begin{array}{rr}<br />
1+x_1-x_1 x_2 = 0\\<br />
1+x_1-3 x_1 x_2+\frac{3}{2} x_1 x_2^2-\frac{1}{6}x_1 x_2^3 = 0<br />
\end{array}<br />


    (Of course, my system is biger than that very much). I only need one of its solution, which satisfied condition x_i > 0. The system is very unstable, it can not be solved by any numerical method, such as Newton method, Homotopy method or so... and needs a "arbitrary precision arithmetic" environment to calculate.
    I use Maple, command Triangularize - package RegularChains - to transform this system to triangular form. That command gave me nice solution, but it only works for system up to 4 equations. For system consists of 6 equations, Maple used ~2Gigabytes of RAM and gave up.
    So, do you know how to solve this problem? Which method is efficient to my system? Thank you very much for your help
    In the particular case You have proposed the solution is relatively confortable. From the first equation You derive...

    \displaystyle x_{2}= 1+\frac{1}{x_{1}} (1)

    ... and if You insert (1) in second equation You obtain a third order equation in x_{1} which can be solved using a standard appproach...
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