基于预处理和区间计算的非线性方程组实根求解(英文)

Solving Nonlinear Systems via Preprocessing and Interval Method

提出了利用混合方法进行多变元非线性方程组实根求解的算法。该方法与符号计算方法相比,最大优点是不需要将非线性方程组三角化,并且可以求出指定区间内达到任意精度的全部实根。在求解过程中,首先采用区间压缩、因式分解和去除重因子等方法对非线性方程组进行预处理。然后,采用区间二分法对给定的区间矢量进行二分并判断每个区间是否有解。如果区间内无解,将该区间舍弃;否则使用带有符号预处理的区间Gauss Seidel方法进一步对区间缩小。当根区间达到所要求精度时则输出该区间;反之,重复上述过程继续进行二分和迭代计算。在算法中,由于采用了区间二分法和区间扩展除法,可以对根可能存在的区间进行判断从而求出多变元非线性方程组的全部实根。另外,通过实例对该算法的求根情况和效率进行例证。最后,指出了进行实根求解下一步所要解决的问题。该方法可有效解决工程实践中的一些较为复杂的非线性问题。

This paper presents a hybrid method for finding real solution of nonlinear equations with arbitrary precision. In contrast with symbolic computation, the system of nonlinear equations don' t need to be triangularized. In the procedure of computation, we combine the methods, including contraction of the initial interval by analysis, factorization and squarefree decomposition, to preprocess the nonlinear systems firstly. Then, interval dichotomy is used to bisect the de-signed interval vector. After this, it is examined whether there is zero point in each sub-interval vector. If these is no solution in sub-interval vector, the sub-interval vector is abandoned, or else we use multivariate Newton Gauss-Seidel method with symbolic preconditioner to refine this sub-interval vector. It is one solution if each interval in interval vector is not greater than the tolerance. Or else, the above procedure is repeated till the error is less than the tolerance. In the algorithm, as interval dichotomy and extended interval division are used, it is certain that all sub-interval boxes can be examined to guarantee all real roots of system of nonlinear equation can be attained. Its performance is shown in solving examples from various applications. Finally, it is pointed out that there is some related works to be researched further. This method can solve some complex problem in practice effectively.