Two New Predictor-Corrector Iterative Methods with Third- and Ninth-Order Convergence for Solving Nonlinear Equations

In this paper, we suggest and analyze two new predictor-corrector iterative methods with third and ninth-order convergence for solving nonlinear equations. The first method is a development of [M. A. Noor, K. I. Noor and K. Aftab, Some New Iterative Methods for Solving Nonlinear Equations, World Applied Science Journal, 20(6),(2012):870-874.] based on the trapezoidal integration rule and the centroid mean. The second method is an improvement of the first new proposed method by using the technique of updating the solution. The order of convergence and corresponding error equations of new proposed methods are proved. Several numerical examples are given to illustrate the efficiency and performance of these new methods and compared them with the Newton's method and other relevant iterative methods. Keywords: Nonlinear equations, Predictor–corrector methods, Trapezoidal integral rule, Centroid mean, Technique of updating the solution; Order of convergence

Dynamics of a multipoint variant of Chebyshev-Halley's family

In this paper, a complex dynamical study of a parametric Chebyshev-Halley type family of iterative methods on quadratic polynomial is presented. The stability of the fixed points is analyzed in terms of the parameter of the family. We also calculate the critical points building their corresponding parameter planes which allow us to analyze the qualitative behavior of this family. Moreover, we locate some dynamical planes showing different pathological aspects of this family. (C) 2016 Elsevier Inc. All rights reserved.The authors thank to the anonymous referees for their suggestions to improve the readability of the paper. This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel, P. (2016). Dynamics of a multipoint variant of Chebyshev-Halley's family. Applied Mathematics and Computation. 284:195-208. doi:10.1016/j.amc.2016.03.009S19520828

Métodos iterativos para la resolución de problemas aplicados transformados a sistemas no lineales

[ES] La resolución de ecuaciones y sistemas no lineales es un tema de gran interés teórico-práctico, pues muchos modelos matemáticos de la ciencia o de la industria se expresan mediante sistemas no lineales o ecuaciones diferenciales o integrales que, mediante técnicas de discretización, dan lugar a dichos sistemas. Dado que generalmente es difícil, si no imposible, resolver analíticamente las ecuaciones no lineales, la herramienta más extendida son los métodos iterativos, que tratan de obtener aproximaciones cada vez más precisas de las soluciones partiendo de determinadas estimaciones iniciales. Existe una variada literatura sobre los métodos iterativos para resolver ecuaciones y sistemas, que abarca conceptos como, eficiencia, optimalidad, estabilidad, entre otros importantes temas. En este estudio obtenemos nuevos métodos iterativos que mejoran algunos conocidos en términos de orden o eficiencia, es decir que obtienen mejores aproximaciones con menor coste computacional. La convergencia de los métodos iterativos suele estudiarse desde el punto de vista local. Esto significa que se obtienen resultados de convergencia imponiendo condiciones a la ecuación en un entorno de la solución. Obviamente, estos resultados no son aplicables si no la conocemos. Otro punto de vista, que abordamos en este trabajo, es el estudio semilocal que, imponiendo condiciones en un entorno de la estimación inicial, proporciona un entorno de dicho punto que contiene la solución y garantiza la convergencia del método iterativo a la misma. Finalmente, desde un punto de vista global, estudiamos el comportamiento de los métodos iterativos en función de la estimación inicial, mediante el estudio de la dinámica de las funciones racionales asociadas a estos métodos. La presente memoria recoge los resultados de varios artículos de nuestra autoría, en los que se tratan distintos aspectos de la materia, como son, las peculiaridades de la convergencia en el caso de raíces múltiples, la posibilidad de aumentar el orden de un método óptimo de orden cuatro a orden ocho, manteniendo la optimalidad en el caso de raíces múltiples, el estudio de la convergencia semilocal en un método de alto orden, así como el comportamiento dinámico de algunos métodos iterativos.[CA] La resolució d'equacions i sistemes no lineals és un tema de gran interés teoricopràctic, perquè molts models matemàtics de la ciència o de la indústria s'expressen mitjançant sistemes no lineals o equacions diferencials o integrals que, mitjançant tècniques de discretizació, donen lloc a aquests sistemes. Atés que generalment és difícil, si no impossible, resoldre analíticament les equacions no lineals, l'eina més estesa són els mètodes iteratius, que tracten d'obtindre aproximacions cada vegada més precises de les solucions partint de determinades estimacions inicials. Existeix una variada literatura sobre els mètodes iteratius per a resoldre equacions i sistemes, que abasta conceptes com ordre d'aproximació, eficiència, optimalitat, estabilitat, entre altres importants temes. En aquest estudi obtenim nous mètodes iteratius que milloren alguns coneguts en termes d'ordre o eficiència, és a dir que obtenen millors aproximacions amb menor cost computacional. La convergència dels mètodes iteratius sol estudiar-se des del punt de vista local. Això significa que s'obtenen resultats de convergència imposant condicions a l'equació en un entorn de la solució. Òbviament, aquests resultats no són aplicables si no la coneixem. Un altre punt de vista, que abordem en aquest treball, és l'estudi semilocal que, imposant condicions en un entorn de l'estimació inicial, proporciona un entorn d'aquest punt que conté la solució i garanteix la convergència del mètode iteratiu a aquesta. Finalment, des d'un punt de vista global, estudiem el comportament dels mètodes iteratius en funció de l'estimació inicial, mitjançant l'estudi de la dinàmica de les funcions racionals associades a aquests mètodes. La present memòria recull els resultats de diversos articles de la nostra autoria, en els quals es tracten diferents aspectes de la matèria, com són, les peculiaritats de la convergència en el cas d'arrels múltiples, la possibilitat d'augmentar l'ordre d'un mètode òptim d'ordre quatre a ordre huit, mantenint l'optimalitat en el cas d'arrels múltiples, l'estudi de la convergència semilocal en un mètode d'alt ordre, així com el comportament dinàmic d'alguns mètodes iteratius.[EN] The resolution of nonlinear equations and systems is a subject of great theoretical and practical interest, since many mathematical models in science or industry are expressed through nonlinear systems or differential or integral equations that, by means of discretization techniques, give rise to such systems. Since it is generally difficult, if not impossible, to solve nonlinear equations analytically, the most widely used tool is iterative methods, which try to obtain increasingly precise approximations of the solutions based on certain initial estimates. There is a varied literature on iterative methods for solving equations and systems, which covers concepts of order of approximation, efficiency, optimality, stability, among other important topics. In this study we obtain new iterative methods that improve some known ones in terms of order or efficiency, that is, they obtain better approximations with lower computational cost. The convergence of iterative methods is usually studied locally. This means that convergence results are obtained by imposing conditions on the equation in a neighbourhood of the solution. Obviously, these results are not applicable if we do not know it. Another point of view, which we address in this work, is the semilocal study that, by imposing conditions in a neighbourhood of the initial estimation, provides an environment of this point that contains the solution and guarantees the convergence of the iterative method to it. Finally, from a global point of view, we study the behaviour of iterative methods as a function of the initial estimation, by studying the dynamics of the rational functions associated with these methods. This report collects the results of several articles of our authorship, in which different aspects of the matter are dealt with, such as the peculiarities of convergence in the case of multiple roots, the possibility of increasing the order of an optimal method from order four to order eight, maintaining optimality in the case of multiple roots, the study of semilocal convergence in a high-order method, as well as the dynamic behaviour of some iterative methods.Cevallos Alarcón, FA. (2023). Métodos iterativos para la resolución de problemas aplicados transformados a sistemas no lineales [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/19349

Program Abstracts, 98th Session, Iowa Academy of Science, April 25-26, 1986

Presentation abstracts from the annual meeting of the Iowa Academy of Sciencehttps://scholarworks.uni.edu/ias_docs/1039/thumbnail.jp

Spectral/hp Finite Element Models for Fluids and Structures

We consider the application of high-order spectral/hp finite element technology to the numerical solution of boundary-value problems arising in the fields of fluid and solid mechanics. For many problems in these areas, high-order finite element procedures offer many theoretical and practical computational advantages over the low-order finite element technologies that have come to dominate much of the academic research and commercial software of the last several decades. Most notably, we may avoid various forms of locking which, without suitable stabilization, often plague low-order least-squares finite element models of incompressible viscous fluids as well as weak-form Galerkin finite element models of elastic and inelastic structures. The research documented in this dissertation includes applications of spectral/hp finite element technology to an analysis of the roles played by the linearization and minimization operators in least-squares finite element models of nonlinear boundary value problems, a novel least-squares finite element model of the incompressible Navier-Stokes equations with improved local mass conservation, weak-form Galerkin finite element models of viscoelastic beams and a high-order seven parameter continuum shell element for the numerical simulation of the fully geometrically nonlinear mechanical response of isotropic, laminated composite and functionally graded elastic shell structures. In addition, we also present a simple and efficient sparse global finite element coefficient matrix assembly operator that may be readily parallelized for use on shared memory systems. We demonstrate, through the numerical simulation of carefully chosen benchmark problems, that the finite element formulations proposed in this study are efficient, reliable and insensitive to all forms of numerical locking and element geometric distortions

Pomobabble: Postmodern Newspeak and Constitutional Meaning for the Uninitiated

A parody of postmodern writing

The Victorian Naturalist

v.114 (1997