Solving Nonlinear Systems via Preprocessing and Interval Method
Yaohui Li, Xue Ji-wei · Journal of Sichuan University · 2004
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 designed 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.