Newton-Type Decomposition Methods For Solving Nonlinear System Of Equations

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.