The Iterative Methods For Nonlinear Equations And Their Theoretical Analysis

One of the most important problems in scientific study is nonlinear problem, and one of thecomparatively research direction in nonlinear problem is to find the solutions of nonlinear equations.It is the key problem in scientific and engineering computations field. For it is hardly difcult toobtain accurate solutions by direct methods except a few special cases, so, at present,the key issuewhich need to be solved is to considerably practical significance to obtain approximate solutionswithin the error between the accurate solutions and approximate solutions is tolerated. Numericalmethods for nonlinear equations had got people’s attention and fast development in1960s. Recentlyit has becomes a hot topic again along with the development of large, high-speed, high-precisioncomputers, and also obtained a lot of high efcient iterative methods.Based on summarizing the results of previous studies, we obtain some efcient iterative methodsand discuss the convergence of some iterative methods in Banach space. The main subjects in thisdissertation are listed as follows:1. Summarizing the results(1) Construction of iterative methods. Based on the iterative methods which developed in recentdecades, we summarize and induce five skills, such as linearization, integral interpolation, Adomiandecomposition, Taylor expansion and multi-steps composition etc. In the same time, we also pointout the advantages and shortages of these skills.(2) Theory of iterative methods. The convergence of iterative methods is the most fundamentalproblem in nonlinear analysis. Semilocal convergence, which does not depend on the accurate so-lution, is the most widely used in application. The Kantorovich theorem of Newton’s method wasgiven by Kantorovich early in1948, which is famous in Banach space. Convergence analysis withweak conditions is a hot research problem as the Kantorovich condition is strong and hardly to beobtained. We summarize a variety of weak conditions and their proof methods: recurrence relationsand majorizing sequence in this dissertation. Finally, we gives the type and expression of majorizingfunctions which used in the proof.2. Three high efcient iterative methods(1) An improved secant-like method is proposed in this dissertation. We propose a new type ofhybrid technique proposed by Kanwar, and then we present an improved iterative method for solv-ing nonlinear equations without derivatives. Analysis of the convergence shows that the asymptoticconvergence order of this method is (1+√5)/2. The practical utility is demonstrated by numericalresults.(2) A new method for solving nonlinear equations is presented. We attempt to improve theorder of the classical King-Werner method, and then we present a new iterative method for solving nonlinear equations. Analysis of the convergence shows that the asymptotic convergence order of thismethod is1+√3which is equal to the method defined by Wang et al. proposed in[?]. Per iterationthe methods require two evaluations of the function and one of its first derivative and therefore theefciency, in term of function evaluations, of the new methods is equal to√31.618≈1.17398, whichis better than Newton’s method and the method proposed in[?]. Finally, some numerical examplesare also given.(3) We suggest and analyze a new iterative method for solving nonlinear equation by rewritingthe given nonlinear equation as a coupled system of equations and using the Taylor series. Analysisof the convergence shows that the asymptotic convergence order of this method is four. Per iterationthe methods require two evaluations of the function, one of its first derivative and one of its secondderivative and it is shown that this iterative method is more efcient than Newton’s methods and He’smethod [Ji-Huan He, AMC,135(2003)81-84.] by some numerical results.3. Semilocal convergence of three iterative methods in Banach spaceIn this dissertation, we study the semilocal convergence of three iterative methods for nonlinearequations in Banach spaces. Using recurrence relations, we establish the semilocal convergence ofharmonic mean Newton’s method and two fifth-order convergence mthods under Lipschitz conditionrespectively. Three existence-uniqueness theorem are given to establish the R-order of these methodsand their priori error bounds. Finally, some numerical applications are presented to demonstrate ourapproachs.