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Some Iterative Methods For Solving Nonlinear Equations And Analysis Convergence

Posted on:2012-06-28Degree:DoctorType:Dissertation
Country:ChinaCandidate:T B LiuFull Text:PDF
GTID:1110330368478902Subject:Applied Mathematics
Abstract/Summary:PDF Full Text Request
Everything that exists in the ever-changing and colorful nature are nonlinear systems, linear system is a ideal simplification of nonlinear system. In 1950s the research of nonlinear science almost involving the all fields of natural sciences and social sciences, and we gradually come to understand the world and the reason of all things. Finding the approximate solution of nonlinear equations is an important research branch of nonlinear science, and plays a very important position in every sciences fields, so in a long time, the study of methods for solving nonlinear equations has always caused the scientists and engineers interest and attention. This paper consists of five chapters.The first chapter is the introduction, it is divided into four parts, in the first part, we summarize the research significance of the nonlinear problems. Nonlinear science almost related to the all fields of natural science and social science, and the methods of solving nonlinear equations play an important role in all science fields. The second part describes the background and status of iterative methods. The most classical iterative method is Newton method, numerical workers made a lot of high-order iterative methods base on Newton method. The third part introduces the theory and definition of iterative methods. The fourth part introduces the main results of this paper. In the second chapter, we present several Iterative methods based on the characteristics of the different curves.We use the characteristics and a single point information of the derivatives to construct high-order conver-gent iterative methods. In the first section, we consider approximating the function at a point by a cubic polynomial, and impose the tangency con-ditions, we get a approximate estimate of the derivative, and then modify the existing iterative methods by the approximate estimate, then we ob-tain a three-parameter fourth-order family of iterative methods for solving the nonlinear equations. We can obtain some different iterative methods by taking different parameter values,and includes, some existing iterative methods and some new methods, It is observed that the methods do not de-pend on the second derivatives in computing process. In the second section, we discuss a family of predictor-corrector iterative methods, the methods do not only have the third-order convergent, but also do not compute the second derivatives. In the third section, we use approximating the function at a point by a cubic polynomial, and impose the concavity and convex-ity of functions, which is important condition, we also get a approximate estimate of the Second-order derivative, and then modify the existing itera-tive methods, we can obtain a one-parameter fifth-order family of iterative methods for solving the nonlinear equations, we have some iterative meth-ods by taking the parameter values. In this chapter, we are concerned with the construction of the iterative methods based on the characteristics of the different curves, especially by using approximating the function at a point by a cubic polynomial and a single point information of the derivatives. As a result, we present some new interesting methods. By analysis of conver-gence and tested on several numerical examples, we show the methods are competitive to Newton's method and other methods of the same order.In the third chapter,we present some new variants of Chebyshev-Halley methods free from second derivative. We first introduce the classical Chebyshev-Halley methods, which is known to converge cubically, but it is observed that the methods depend on the second derivatives in each process, this making its practical utility restricted rigorously. Secondly, In order to avoid the calculation of second derivative, we consider approximating the function around the point by a quadratic model, and impose the tangency conditions, we obtain some new variants of the Chebyshev-Halley meth-ods for solving the nonlinear equations.and the new methods require two function and one first derivative evaluations per iteration, which shown to be third-order convergent at least, numerical examples show that the new methods are comparable with the well known existing methods and give better numerical results in many aspects.In the fourth chapter, we present several new variants of Cauchy meth-ods free from second derivative, the methods have three parameters. In order to avoid the calculation of second derivative, we consider the same technique of the third chapter to construct new variants of Cauchy methods with three parameters. In order to avoid the calculation of the square roots, we derive some forms free from square roots by Taylor approximation, and obtain some new methods.The new methods require two function and one first derivative evaluations per iteration, and have third-order convergent at least, numerical examples show that the new methods are efficient.In the fifth chapter, we firstly use the curvature of the circle to con-struct a family of third-order convergence iterative methods, the methods avoid calculating the second derivative of the function, we put the methods to the multidimensional case, the convergence analysis and numerical ex-periments show that the family of iterative methods have some effectiveness and competitiveness. Secondly, we present a family of predictor-corrector iterative methods, these methods are not only avoid computing the second derivative, but also have at least third-order convergence, if we select the special value of the parameters,we can obtain fourth-order iterative meth-ods. Finally,we give the convergence analysis and numerical results show that the family of iterative methods have at least equal performance as compared with the other methods of the same order.
Keywords/Search Tags:Newton's method, Iterative methods, Nonlinear equations, Order of convergence
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