The Iterative Methods To Solve Systems Of Nonlinear Equations

As is known to all, the nonlinear problem is one of the important objects in mathematical analysis in Banach space.The iterative algorithm is an efficient method in solving nonlinear problems. Nonlinear problems are considered as the most important part by scholars and engineers in studying all kinds of social phenomena and solving practical problems. With the development of society, science and technology, many mathematicians pay more and more attention to nonlinear problems. The performance of an iterative algorithm depends on the order of convergence, convergence speed and efficiency index, and even the initial values, and so on. Solving nonlinear equations or systems of nonlinear equations is perhaps the most difficult problem in all of numerical computations. Therefore, it is of significance in both theory and applications to study the high-order iterative algorithm for solving nonlinear equation, system of nonlinear equations and even for modern mathematics. The thesis consists of five parts:In Chapter 1, we introduce the research background, concept and some definitions about iterative method.In Chapter 2, we introduce some classical iterative algorithms, such as the classical Newton iterative method, the deformation of Newton iteration method, the third-order Chebyshev iteration method, Halley iteration method, super-Halley iteration method, and the fourth-order Jarratt type iterative method and so on.In Chapter 3, a new approach for solving nonlinear equation is presented, which is based on Thiele’s continued fractions. The Thiele’s continued fraction iteration algorithms with three-order and four-order convergence are constructed. Numerical examples are given to show that the new formula has higher convergence order than other iterative formulae.In Chapter 4, an iterative formula based on [1/n] Pade approximants is presented for solving nonlinear equations. The convergence of the iterative algorithm is then analyzed. Numerical examples show that the new formula has higher convergence order than other iterative formulae.In Chapter 5, we conclude this paper by giving suggestions about what can be done in the future.