Several Accelerated Iterative Methods For Nonlinear Equations

In scientific computing, many equations derived from practical problems are alwaysthe nonlinear forms. So nonlinear problems play an important role in various fields.Therefore, how to quickly solve these nonlinear equations is a topic which is studiedby computational mathematics. The main content of the article is several acceleratediterative methods for nonlinear equations. The whole article contains of four chapters.In chapter 1, we introduce a number of numerical iterative methods for nonlinearequations.In chapter 2, we introduce the iterative methods which based on Adomian seriesmethod for nonlinear equation. As the series converges very fast, we only need to takethe first few series form solutions to approximate the root of the equations in practice.Di?erent order of convergence of the iterative schemes are derived from the di?erentnumbers in the items which are taken. For the coupled nonlinear equations form, wepropose iteration structure under another decomposition form of the nonlinear part ofthe equation, and we give a new method of fifth-order for solving nonlinear equation.Besides,the Adomian series method is promoted to high-dimensional case. Two fourth-order convergence of the iterative sequence for solving nonlinear equations are given.And the corresponding numerical examples to illustrate the e?ectiveness of the methodsat the end of the chapter.In chapter 3, we introduce the iterative methods based on numerical integration.Using the Newton - Cotes formula, a series of third-order convergence of iterative se-quences are presented. Starting from the inverse function, the corresponding iterativeschemes are given. Further, we extend to the case of solving nonlinear equations. Uti-lize a combination of techniques, we use the third-order convergence of the iterative sequence to replace the Newton step, when the Newton-Cotes quadrature coe?cientssatisfy certain conditions, the iterative methods for solving nonlinear equations can befourth-order convergence.In chapter 4,we get a super third-order convergence of iterative method by com-bineding with secant method and Newton method. Using the di?erence quotient toreplace derivative, we obtaine a seventh-order convergence of the three-step iterativemethod.