【作者】 周茵；

【导师】 曾金平；

【作者基本信息】 湖南大学 ， 计算数学， 2004， 硕士

【摘要】 非线性方程组 F(x)=0 (*)在物理、力学、工程等问题中有广泛的应用背景。随着科学技术的进步以及计算工具的不断更新，它的算法研究获得进一步的深入发展。特别是并行计算技术的出现和发展，使得求解非线性方程组的数值计算方法的研究获得新的进展，得到了一系列研究成果。 近些年来兴起的多重分裂算法，具有很好的并行结构，从而是易于进行并行计算的一类数值计算方法。在并行计算机发展的今天，此类算法更显现出发展的空间。对于线性微分方程边值问题的多重分裂算法，其收敛性理论以及收敛速度的分析都得到了较为系统的研究。而对于非线性微分方程边值问题的多重分裂算法，无论就算法的结构而言，还是就理论分析方面而言，都还存在着许多有待研究的课题。 本文首先介绍将多重分裂算法与科学及工程计算中广为应用的一种区域分解技术—Schwarz算法相结合，得到了一类多重分裂加性Schwarz算法以及两水平的多重分裂加性Schwarz算法用于求解非线性方程组 F(x)=Ax-G(x)-b=0， (**)其中非奇异，为一非线性映射。这类算法具有很好的并行性能，因而特别适用于并行计算。其次，考虑到非线性多重分裂算法用于求解非线性方程组(**)的局部收敛性定理已有许多，这里我们着重考虑对于一类较为特殊的非线性方程组的全局收敛性和单侧收敛性。并对算法做了改进，用m步Newton法来代替求得每个非线性多重分裂子问题的近似解，并给出相应的收敛性结论。最后，对文中所提到的算法均列出数值算例，证实了算法的有效性。更多还原

【Abstract】 The system of nonlinear equationsF(x) = 0 (*)has been widely applied in physics, mechanics and engineering. With the development of science and computing technique, many algorithms for solving the system of nonlinear equations have been constructed and analyzed. Especially with the development of parallel computing technique, corresponding numerical and theoretic-researches have made great progress.Multisplitting methods, which have been widely studied in recently years, are a class of numerical computing methods having a very good parallel structure. As a result, they are easily to be used to calculate in parallel. With the development of parallel technique, the study of these methods has been greatly stimulated. As for the boundary-value problems of linear differential equations, multisplitting methods have been extensionally studied. The theories concerned with the convergence as well as the convergent rate analysis have been established systematically. On the other hand, as for the boundary-value problems for nonlinear differential equations, both the algorithmic construction and the theoretical analysis of multisplitting methods are needed. Researches in the fields are relatively few.In this paper, we firstly present a multisplitting additive Schwarz algorithm and a two-level multisplitting additive Schwarz algorithm for solving a system of nonlinear equationsF(x) = Ax - G(x) - b = 0, (**)where A R is a nonsingular matrix, 6 R is a given vector and G is a given nonlinear mapping. This kind of algorithms, combining multisplitting and additive Schwarz algorithm, are easily to be applied to compute in parallel. The convergence and convergent rate is analyzed.Secondly, considering there are many local convergence theories of nonlinear multisplitting methods, we emphatically point out the global convergence and unilateral convergence of solving a kind of relatively special nonlinear system of equations. We also consider the convergence of a nonlinear multisplitting method, in which the solution of the subproblems are approximated by the iterative solution such as m-steps Newton’s iterative solution.IIIn the end. numerical examples based on the algorithms presented in this paper are given and the results show these algorithms are effective.更多还原