基于混合计算的非线性代数方程组的实根求解

Finding Real Solutions of Nonlinear Algebraic Equations Based on Symbolic-Numeric Methods

利用Grbner基理论将多项式方程组的求解化为有限维代数问题,并进一步化为单变元方程w(xi)的求根,然后利用区间方法求出每个单变元方程的根区间,最后使用区间分析从变元根区间的全排列中找出方程组的区间解.在区间分析求解中,提出并证明了区间解的性质定理,该方法易于并行化,不产生误差积累,且可以找到全部方程组的实解并达到任意精度.

By using Groebner basis, the paper transforms the multivariate nonlinear equations PS into univariate one ω（xi） over finite algebra. Then, the paper uses interval method to implement real root isolation for each ω（xi）. Finally, these real roots in different variants are combined to take as the possible solution of PS. We find the true solutions of PS from these combination by using interval analysis. In the paper, the theorem about properties of interval solution of PS is proposed and proved. The-advantage of our method is no error accumulation and easy to be parallelized. Besides these, it can find all real solutions with arbitrary precision for nonlinear algebraic equations.