Concepción Bermúdez

On two families of high order Newton type methods

Abstract We study a general class of high order Newton type methods. The schemes consist of the application of several steps of Newton type methods with frozen derivatives. We are interested to improve the order of convergence in each sub-step. In particular, we should finish the computation after some stop criteria and before the full computation of the current approximation. We prove that only two sequences of parameters can be derived verifying these properties. One corresponds to a very well known family and the other is a little (but not natural) modification. Finally, we study some dynamical aspects of these families in order to find differences. Surprisingly, the less natural family seems to have a simpler dynamic.

On a third‐order Newton‐type method free of bilinear operators

This paper is devoted to the study of a third-order Newton-type method. The method is free of bilinear operators, which constitutes the main limitation of the classical third-order iterative schemes. First, a global convergence theorem in the real case is presented. Second, a semilocal convergence theorem and some examples are analyzed, including quadratic equations and integral equations. Finally, an approximation using divided differences is proposed and used for the approximation of boundary-value problems. Copyright

On the dynamics of the Euler iterative function

Abstract The dynamics of Euler’s third-order iterative method which is used to find roots of non-linear equations applied to complex polynomials of degrees three and four is studied. The conjugacy classes of this method are found explicitly.

On an efficient k -step iterative method for nonlinear equations

This paper is devoted to the construction and analysis of an efficient k -step iterative method for nonlinear equations. The main advantage of this method is that it does not need to evaluate any high order Frechet derivative. Moreover, all the k -step have the same matrix, in particular only one LU decomposition is required in each iteration. We study the convergence order, the efficiency and the dynamics in order to motivate the proposed family. We prove, using some recurrence relations, a semilocal convergence result in Banach spaces. Finally, a numerical application related to nonlinear conservative systems is presented.

Expanding the Applicability of a Third Order Newton-Type Method Free of Bilinear Operators

This paper is devoted to the semilocal convergence, using centered hypotheses, of a third order Newton-type method in a Banach space setting. The method is free of bilinear operators and then interesting for the solution of systems of equations. Without imposing any type of Frechet differentiability on the operator, a variant using divided differences is also analyzed. A variant of the method using only divided differences is also presented.

On the dynamics of some Newton's type iterative functions

The dynamics of a family of Newtons type iterative methods for second- and third-degree complex polynomials is studied. The conjugacy classes of these methods are presented. Classical properties of rational maps are used.

Wavelets for the Maxwell’s equations: An overview

Abstract In recent years wavelets decompositions have been widely used in computational Maxwell’s curl equations, to effectively resolve complex problems. In this paper, we review different types of wavelets that we can consider, the Cohen–Daubechies–Feauveau biorthogonal wavelets, the orthogonal Daubechies wavelets and the Deslauries–Dubuc interpolating wavelets. We summarize the main features of these frameworks and we propose some possible future works.