In this paper, we present the study of the semilocal and local convergence of an optimal fourth-order family of methods. Moreover, the dynamical behavior of this family of iterative methods applied ...to quadratic polynomials is studied. Some anomalies are found in this family be means of studying the dynamical behavior. Parameter spaces are shown and the study of the stability of all the fixed points is presented.
Didactics of mathematics has been recently considered, for less than a century, as scientific discipline as itself. The study of this discipline has significantly grown in the last decades since many ...authors have focused their efforts in the study of the relations of the knowledge and the processes of teaching-learning of mathematics. This book presents eight original contributions of authors from ten different universities, and even from different countries, related to (1) Learning and metacognition; (2) A methodology to teach mathematics; (3) A study related to mathematics in China; (4) Collaborative learning in Mathematics in Secondary Education; (5) Intervention to teach notable products in Secondary Education; (6) The use of holography in geometry teaching in Secondary Education; (7) Problem Based Learning in University for advanced mathematics teaching; (8) Flip teaching in University. This monograph is required reading for all researchers in mathematics education and contains different useful material for mathematics educators and teacher trainers interested in the theory and practice of mathematics education. As such this monograph is suitable to teachers of mathematics in different educational levels. Researchers, graduate students and seminars will find this book really helpful for their daily work. This book is also recommended to researchers in different disciplines, such as general education, didactics or general mathematics.
We present a local convergence analysis for general multi-point-Chebyshev–Halley-type methods (MMCHTM) of high convergence order in order to approximate a solution of an equation in a Banach space ...setting. MMCHTM includes earlier methods given by others as special cases. The convergence ball for a class of MMCHTM methods is obtained under weaker hypotheses than before. Numerical examples are also presented in this study.
In this paper, we are interested to justified two typical hypotheses that appear in the convergence analysis,
|
λ
|
≤
2
and
z
0
sufficient close to
z
∗
. In order to proof these ideas, the dynamics ...of a damped two-step Newton-type method for solving nonlinear equations and systems is presented. We present the parameter space for values of the damping factor in the complex plane, focusing our attention in such values for which the fixed points related to the roots are attracting. Moreover, we study the stability of the strange fixed points, showing that there exists attracting cycles and chaotical behavior for some choices of the damping factor.
We study the local convergence of Chebyshev–Halley methods with six and eight order of convergence to approximate a locally unique solution of a nonlinear equation. In Sharma (2015) (see Theorem 1, ...p. 121) the convergence of the method was shown under hypotheses reaching up to the third derivative. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamics of these methods are also studied. Finally, numerical examples examining dynamical planes are also provided in this study to solve equations in cases where earlier studies cannot apply.
In this work, a uniparametric generalization of the iterative method due to Kurchatov is presented. The iterative model presented is derivative‐free and approximates the solution of nonlinear ...equations when the operator is non‐differenciable. As the accessibility of the Kurchatov method is usually a problem in the application of the method, since the set of initial guesses that guarantee the convergence of the method is small, the main objective of this work is to improve the Kurchatov iterative method in its accessibility while maintaining and even increasing its speed of convergence. For this purpose, we introduce a variable parameter in the iterative function of the Kurchatov method that allows us to get a better approximation of the derivative by using a symmetric uniparameteric first‐order divided difference operator. We perform a complex dynamic study that corroborate the improvements in the accessibility region. Moreover, a complete analysis of the local and semilocal convergence is established for the new uniparametric iterative method. Finally, we apply the theoretical results to solve a nonlinear integral equation showing the usefulness of the work.
We study the local convergence of Chebyshev-Halley-type methods of convergence order at least five to approximate a locally unique solution of a nonlinear equation. Earlier studies such as Behl (
...2013
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) show convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamical analyses of these methods are also studied. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.
A generic family of sixth-order modified Newton-like multiple-zero finders have been proposed in Geum et al. (2016). Among them we select a specific family of iterative methods with uniparametric ...bivariate polynomial weight functions and study their dynamics by investigating the relevant parameter spaces and dynamical planes, via Möbius conjugacy map applied to a prototype polynomial of the form (z−a)m(z−b)m. The resulting dynamics is best illustrated through a variety of parameter spaces as well as dynamical planes.
A class of two-point quartic-order simple-zero finders and their dynamics are investigated in this paper by extending King’s fourth-order family of methods. With the introduction of an error ...corrector having a weight function dependent on a function-to-function ratio, higher-order convergence is obtained. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for a prototype quadratic polynomial as well as parameter spaces and dynamical planes.
This paper is devoted to the semilocal analysis of a high‐order Steffensen‐type method with frozen divided differences. The methods are free of bilinear operators and derivatives, which constitutes ...the main limitation of the classical high‐order iterative schemes. Although the methods are more demanding, a semilocal convergence analysis is presented using weaker conditions than the classical Steffensen method.
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