Ángel Alberto Magreñán

Different anomalies in a Jarratt family of iterative root-finding methods☆

Abstract In this paper, the behavior of a Jarratt family of iterative methods applied to quadratic polynomials is studied. Some anomalies are found in this family be means of studying the dynamical behavior of this fourth-order family of methods. Parameter spaces are shown and the study of the stability of all the fixed points is presented. Dynamical planes for members with good and bad dynamical behavior are also provided.

A new tool to study real dynamics

In this paper, the author presents a graphical tool that allows to study the real dynamics of iterative methods whose iterations depends on one parameter in an easy and compact way. This tool gives the information as previous tools such as Feigenbaum diagrams and Lyapunov exponents for every initial point. The convergence plane can be used, inter alia, to find the elements of a family that have good convergence properties, to see how the basins of attraction changes along the elements of the family, to study two-point methods such as Secant method or even to study two-parameter families of iterative methods. To show the applicability of the tool an example of the dynamics of the Damped Newtons method applied to a cubic polynomial is presented in this paper.

On the convergence of an optimal fourth-order family of methods and its dynamics

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.

On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions

In this work we study Newtons method for solving nonlinear equations with operators defined between two Banach spaces. Together with the classical Kantorovich theory, we consider a center-Lipschitz condition for the Frechet derivative of the involved operator. This fact allow us to obtain a majorizing sequence for the sequence defined in Banach spaces and to give conditions for the convergence. In this way, we obtain a generalization of Kantorovichs theorem that improves the values of the universal constant that appears in it as well as the radius where the solution is located and where it is unique. Finally we illustrate the main theoretical result by means of some examples.

Reducing Chaos and Bifurcations in Newton-Type Methods

We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newtons method with the same derivative. We introduce a damping factor in order to reduce the bad zones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.

On a two-step relaxed Newton-type method

We propose a new two-step relaxed Newton-type method for the approximation of nonlinear equations in Banach spaces. The method is free of any bilinear operator. Moreover, in each iteration, we only approximate an associated linear system. We analyze its semilocal convergence under @w-conditioned divided differences. Finally, we include several practical advantages of the method.

A study on the local convergence and the dynamics of Chebyshev---Halley---type methods free from second derivative

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), Bruns and Bailey (Chem. Eng. Sci 32, 257–264, 1977), Candela and Marquina (Computing 44, 169–184, 1990), (Computing 45(4):355–367, 1990), Chicharro et al. (2013), Chun (Appl. Math. Comput, 190(2):1432–1437, 1990), Cordero et al. (Appl.Math. Lett. 26, 842–848, 2013), Cordero et al. (Appl. Math. Comput. 219, 8568–8583, 2013), Cordero and Torregrosa (Appl. Math. Comput. 190, 686–698, 2007), Ezquerro and Hernández (Appl. Math. Optim. 41(2):227–236, 2000), (BIT Numer. Math. 49, 325–342, 2009), (J. Math. Anal. Appl. 303, 591–601, 2005), Gutiérrez and Hernández (Comput. Math. Applic. 36(7):1–8, 1998), Ganesh and Joshi (IMA J. Numer. Anal. 11, 21–31, 1991), Hernández (Comput. Math. Applic. 41(3–4):433–455, 2001), Hernández and Salanova (Southwest J. Pure Appl. Math. 1, 29–40, 1999), Jarratt (Math. Comput. 20(95):434–437, 1996), Kou and Li (Appl. Math. Comput. 189, 1816–1821, 2007), Li (Appl. Math. Comput. 235, 221–225, 2014), Ren et al. (Numer. Algorithm. 52(4):585–603, 2009), Wang et al. (Numer. Algorithm. 57, 441–456, 2011), Kou et al. (Numer. Algorithm. 60, 369–390, 2012) 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.

Original Article: Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane

The real dynamics of a family of fourth-order iterative methods is studied when it is applied on quadratic polynomials. A Scaling Theorem is obtained and the conjugacy classes are analyzed. The convergence plane is used to obtain the same kind of information as from the parameter space, and even more, in complex dynamics.

The “Gauss-Seidelization” of iterative methods for solving nonlinear equations in the complex plane

Abstract In this paper we introduce a process we have called “Gauss-Seidelization” for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss–Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphasize how the Gauss-Seidelization process influences on the dynamical behavior of an iterative method for solving nonlinear equations.

Stability study of eighth-order iterative methods for solving nonlinear equations

In this paper, we study the stability of the rational function associated to a known family of eighth-order iterative schemes on quadratic polynomials. The asymptotic behavior of the fixed points corresponding to the rational function is analyzed and the parameter space is shown, in which we find choices of the parameter for which there exists convergence to cycles or even chaotical behavior showing the complexity of the family. Moreover, some elements of the family with good stability properties are obtained.