Iterative Methods For Multiple Roots Of Nonlinear Equation

Finding the roots of nonlinear equations is very important in numerical analysis and has many applications in engineering and other applied sciences. Among the methods for finding roots, iterative method is the most used one, which generates a sequence from the iterative scheme and converges to a root of the nonlinear equations. It’s known that the information of higher order derivatives of the target function f(x) is favorable for constructing higher-order iteration. Nevertheless, sometimes, it is expensive or impossible to be calculated, hence the practical applications of this type methods are restricted rigorously.There are a large number of iterative methods derived for simple roots. The techniques used to construct higher-order method for simple roots are more abundant and mature. However, these techniques become more complicated or invalid when they are used to handle the case of multiple roots. On the other hand, both theoretical analysis and experimental tests show that, when the higher-order iterative method for simple roots is applied to find the multiple roots without any modifications, the convergence order of it reduces to one only. A simple but persuasive example is the classical Newton’s method, which is quadratically convergent for simple roots while linearly convergent for multiple ones. So, how to construct a higher-order, especially optimal order, iteration for solving multiple roots is worth studying. Until now, there are not so many effective methods proposed yet. Most of the known iterative methods of optimal order for finding multiple roots are based on the knowledge of multiplicity, i.e., the order of the root is assumed to be known. On the other hand, in the case of simple root, many results on the estimate of the radius of the convergence ball have been given. But in the case of multiple roots, the situation is reversed. Only a very few results have been given.In this thesis, we shall only discuss the iterative method for finding multiple roots of nonlinear equation.First, we propose a family of iterative methods, which is of optimal fourth-order and contains all optimal order iterative methods known already. Upon careful study, we find that the first step of those optimal order iterations for simple roots is usually the classical Newton’s method or some others of order two. However, this rule seems invalid in constructing the high-order iterative method for multiple roots. Then, we try to present some methods of this type. We also give two iterations without the information of multiplicity, derivative and the initial guesses. Although the orders of these two methods are not optimal, the advantages of them are self-explanatory.Convergence radius is crucial important to an iterative method. However, to the iterative method for multiple roots, there is only few work on this field. Until very re-cently, based on high-order divided difference and multiple integral, Ren and Argyros give the convergence radius of the modified Newton’s method. Using this technique, we estimate the convergence radius of Osada’s method for multiple roots. Further-more, we propose a new technique based on the Taylor expansion with integral form remainder and enlarge the radius of Osada’s method. Compared with that proposed by Ren and Argyros, our treatment is simpler and more efficient. Then, we re-consider the convergence radius of the modified Newton’s method and have better results suc-cessfully.The last part of this thesis discusses the dynamical behaviors of some iterative methods on complex plan. Compared with some other results published already, the presented methods in this thesis perform very well and almost no chaos behaviors are found. Although it’s shown Halley’s method is the best, it requires the evalua-tion of the second order derivative per iteration, which makes Halley’s method more unpractical in real applications.