【作者】 殷巧玉；

【导师】 潘状元；

【作者基本信息】 哈尔滨理工大学 ， 应用数学， 2008， 硕士

【摘要】 对于牛顿型迭代格式等经典的算法,近年来经过很多学者的研究已经取得了丰硕的理论成果,包括收敛性定理、Kantorovich型定理和误差估计。局部收敛性定理事先假定了方程组有解存在,并且初始近似与解充分接近,则迭代序列收敛到方程组的解。然而对计算理论更为重要的是存在性、收敛性定理。在不知道解的情况下能够验证收敛条件,并且往往同时可以断定解的存在性乃至唯一性,因此对于各种迭代法建立存在性收敛性定理,始终是迭代法理论研究的中心课题之一。对于求解非线性方程组的Newton型分裂方法和离散Newton型分裂方法,Jochen W.Schmidt,Wolfgang Hoyer和Christian Haufe只给出了局部收敛性定理,并没有给出Kantorovich型存在性、收敛性定理,因此研究用分裂迭代格式求解非线性方程组,并给出Kantorovich型存在性收敛性定理,是对非线性方程组理论体系的完善,因此具有重要的理论意义。本文研究了用Newton型分裂方法求解非线性方程组,给出了Kantorovich型存在性、收敛性定理。全文共分四部分。第一章,在绪论部分主要阐述了国内外有关求解非线性方程组研究的发展概况,并介绍了本文的主要研究内容、课题背景和研究意义。第二章,给出了Newton型分裂方法的Kantorovich型定理。第三章,给出了离散Newton型分裂方法的Kantorovich型定理。第四章,给出了半离散Newton型分裂方法的Kantorovich型定理。完善了Newton型分裂方法的收敛性定理。更多还原

【Abstract】 Rich theoretical results of Newton-type iterative scheme and other classical algorithms have been made by many scholars in recent years,including convergence theorem,Kantorovich-type theorem and error estimate.The local convergence theorem assumpts that the solution of nonlinear system of equations exists,moreover the initial approximation approaches the solution sufficiently. But the existence and convergence theorem is more important to the theory of computation.We can verify the convergence conditions when we don’t know about the situation of the solution,and assert the existence and uniqueness of the solution.Therefore,establishing the existence and convergence theorems for all kinds of the iteration methods has always been one of the center of theoretical analysis in iteration method.Jochen W.Schmidt,Wolfgang Hoyer and Christian Haufe only gave the local convergence theorems of Newton-type decomposition methods and discrete Newton-type decomposition methods for solving nonlinear system of equations.Therefore,researching Newton-type decomposition methods for solving nonlinear system of equations and giving the Kantorovich-type existence and convergence theorem will improve and perfect the theoretical system of nonlinear system of equations.Therefore,it has very important theoretical significance.In this paper,the Newton-type decomposition methods have been applied in solving the nonlinear system of equations,moreover,Kantorovich-type theorems are given.There are four parts in this paper.In the first chapter,the solving nonlinear system of equations development at home and abroad,the main contents,background and significance are introduced in the preface.In the second chapter,the Kantorovich-type theorem for Newton-type decomposition methods is given.In the third chapter,the Kantorovich-type theorem for discrete Newton-type decomposition methods is given.In the fourth chapter,the Kantorovich-type theorem for semi-discrete Newton-type decomposition methods is given.The theory of convergence theorem of Newton-type decomposition methods has been perfected.更多还原