A multistep Steffensen-type method for solving nonlinear systems of equations

[EN] 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.This work was supported supported in part by by Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19374/PI/14, by the project of Generalitat Valenciana Prometeo/2016/089 and the projects MTM2015-64382-P (MINECO/FEDER), MTM2014-52016-C2-1-P and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and InnovationAmat, S.; Argyros, IK.; Busquier, S.; Hernández-Verón, MA.; Magreñán, AA.; Martínez Molada, E. (2020). A multistep Steffensen-type method for solving nonlinear systems of equations. Mathematical Methods in the Applied Sciences. 43(13):7518-7536. https://doi.org/10.1002/mma.5599S75187536431

A class of efficient high-order iterative methods with memory for nonlinear equations and their dynamics

[EN] In this paper we obtain some theoretical results about iterative methods with memory for nonlinear equations. The class of algorithms we consider focus on incorporating memory without increasing the computational cost of the algorithm. This class uses for the predictor step of each iteration a quantity that has already been calculated in the previous iteration, typically the quantity governing the slope from the previous corrector step. In this way we do not introduce any extra computation, and more importantly, we avoid new function evaluations, allowing us to obtain high-order iterative methods in a simple way. A specific class of methods of this type is introduced, and we prove the convergence order is 2(n) + 2(n-2) with n + 1 function evaluations. An exhaustive efficiency study is performed to show the competitiveness of these methods. Finally, we test some specific examples and explore the effect that this predictor may have on the convergence set by setting a dynamical study.Ministerio de Economia y Competitividad de Espana, Grant/Award Number: MTM2014-52016-C2-2-P; Generalitat Valenciana Prometeo, Grant/Award Number: /2016/089Howk, CL.; Hueso, J.; Martínez Molada, E.; Teruel-Ferragud, C. (2018). A class of efficient high-order iterative methods with memory for nonlinear equations and their dynamics. Mathematical Methods in the Applied Sciences. 41(17):7263-7282. https://doi.org/10.1002/mma.4821S72637282411

CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior

[EN] A family of fourth-order iterative methods without memory, for solving nonlinear systems, and its seventh-order extension, are analyzed. By using complex dynamics tools, their stability and reliability are studied by means of the properties of the rational function obtained when they are applied on quadratic polynomials. The stability of their fixed points, in terms of the value of the parameter, its critical points and their associated parameter planes, etc. give us important information about which members of the family have good properties of stability and whether in any of them appear chaos in the iterative process. The conclusions obtained in this dynamical analysis are used in the numerical section, where an academical problem and also the chemical problem of predicting the diffusion and reaction in a porous catalyst pellet are solved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Guasp, L.; Torregrosa Sánchez, JR. (2018). CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior. Journal of Mathematical Chemistry. 56(7):1902-1923. https://doi.org/10.1007/s10910-017-0814-0S19021923567S. Amat, S. Busquier, Advances in Iterative Methods for Nonlinear Equations (Springer, Berlin, 2016)S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)S. Amat, S. Busquier, S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods. Comput. Appl. Math. 189, 22–33 (2006)I.K. Argyros, Á.A. Magreñn, On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)D.K.R. Babajee, A. Cordero, J.R. Torregrosa, Study of multipoint iterative methods through the Cayley quadratic test. Comput. Appl. Math. 291, 358–369 (2016). doi: 10.1016/J.CAM.2014.09.020P. Blanchard, The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, Article ID 780153 (2013)C. Chun, M.Y. Lee, B. Neta, J. Džunić, On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)A. Cordero, E. Gómez, J.R. Torregrosa, Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems. Complexity 2017, Article ID 6457532 (2017)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, Reading, 1989)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first application to atomic problems. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. Math. Chem. 49, 1384–1415 (2011)Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. Math. Chem. 51(9), 2361–2385 (2013)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)B. Neta, C. Chun, M. Scott, Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)M.S. Petković, B. Neta, L.D. Petković, J. Džunić, Multipoint Methods for Solving Nonlinear Equations (Elsevier, Amsterdam, 2013)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. Math. Chem. 52(1), 255–267 (2014)R. Singh, G. Nelakanti, J. Kumar, A new effcient technique for solving two-point boundary value problems for integro-differential equations. Math. Chem. 52, 2030–2051 (2014

UX of Social Network Edmodo in Undergraduate Engineering Students

The main objective of this research is to describe the use that students make of an academic SNS (social network service) and detail the relationship between socio-demographic and academic factors associated with the use of EDMODO and the perception of the contribution to the acquisition of skills for the future career. In the analysis of user experience, participants positively evaluated EDMODO and found that the level of satisfaction is positively associated with the academic results obtained, and negatively with perceived usefulness in terms of the impact on their grades

UX of Social Network Edmodo in Undergraduate Engineering Students

The main objective of this research is to describe the use that students make of an academic SNS (social network service) and detail the relationship between socio-demographic and academic factors associated with the use of EDMODO and the perception of the contribution to the acquisition of skills for the future career. In the analysis of user experience, participants positively evaluated EDMODO and found that the level of satisfaction is positively associated with the academic results obtained, and negatively with perceived usefulness in terms of the impact on their grades

Use of the hologram as a tool to work Geometry contents in Secondary Education

[EN] We present in this paper a methodology for geometric concepts learning in high school students, using the hologram as a tool. To do this, the foundations are first established to make the hologram a means in the teaching-learning process to continue with the description of the development of this methodology.[ES] Presentamos en este trabajo una propuesta metodológica para el aprendizaje de conceptos geométricos en alumnos de secundaria, empleando el holograma como herramienta. Para ello, se establecen las bases para hacer del holograma un medio en el proceso de enseñanza-aprendizaje, pasando a continuación a describir el desarrollo de dicha metodología.Orcos Palma, L.; Jordan-Lluch, C.; Magreñán, AA. (2018). Uso del holograma como herramienta para trabajar contenidos de geometría en Educación Secundaria. Pensamiento Matemático. VIII(2):91-100. http://hdl.handle.net/10251/137998S91100VIII

Purely Iterative Algorithms for Newton’s Maps and General Convergence

The aim of this paper is to study the local dynamical behaviour of a broad class of purely iterative algorithms for Newton’s maps. In particular, we describe the nature and stability of fixed points and provide a type of scaling theorem. Based on those results, we apply a rigidity theorem in order to study the parameter space of cubic polynomials, for a large class of new root finding algorithms. Finally, we study the relations between critical points and the parameter space

Starting points for Newton’s method under a center Lipschitz condition for the second derivative

We analyze the semilocal convergence of Newton's method under a center Lipschitz condition for the second derivative of the operator involved different from that used by other authors until now. In particular, we propose to center the Lipschitz condition for the second derivative in a different point from that where Newton's method starts. This allows us to obtain different starting points for Newton's method and modify the domain of starting points. (C) 2016 Elsevier B.V. All rights reserved

Solving non-differentiable Hammerstein integral equations via first-order divided differences

[EN] In this paper, the behavior of derivative-free techniques to approximate solutions of nonlinear Hammerstein-type integral equations in Banach space C([alpha,beta]) as alternatives against the well-known Newton's method is examined. In particular, from the well-known Kurchatov method, we construct a uniparametric family of derivative-free iterative schemes in order to achieve an approximation of a solution of a Hammerstein-type integral equation with a non-differentiable Nemyskii operator. We compare the uniparametric family built and the Kurchatov method, obtaining improvements in the precision and also in the accessibility through studying its dynamic behavior.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C21-C22.Hernández-Verón, MA.; Magreñán, AA.; Martínez Molada, E.; Villalba, EG. (2023). Solving non-differentiable Hammerstein integral equations via first-order divided differences. Numerical Algorithms. https://doi.org/10.1007/s11075-023-01715-