On high-order iterative schemes for the matrix pth root avoiding the use of inverses

This paper is devoted to the approximation of matrix pth roots. We present and analyze a family of algorithms free of inverses. The method is a combination of two families of iterative methods. The first one gives an approximation of the matrix inverse. The second family computes, using the first method, an approximation of the matrix pth root. We analyze the computational cost and the convergence of this family of methods. Finally, we introduce several numerical examples in order to check the performance of this combination of schemes. We conclude that the method without inverse emerges as a good alternative since a similar numerical behavior with smaller computational cost is obtained.The research of the authors S.A. and S.B. was funded in part by Programa de Apoyo a la investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by PID2019-108336GB-100 (MINECO/FEDER). The research of the author M.Á.H.-V. was supported in part by Spanish MCINN PGC2018-095896-B-C21. The research of the author Á.A.M. was funded in part by Programa de Apoyo a la investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18 and by Spanish MCINN PGC2018-095896-B-C21

Collaborative learning: implementation of JigSaw technique in Google

[EN] The main purpose of this research is checking the effectiveness of some online collaborative learning techniques through the development and implementation of a Google-based environment which will let us develop the collaborative technique known as Jigsaw. The research has been carried out on maths students of a school in the Principality of Asturias.28 students from different school years and classes took part in the research and several teachers at the school also participated in it. We have used a quasiexperimental design with pre-test and post-test measures as well as an equivalent control group. We have also carried out a detailed study of the requirements needed to extract information about the problems of the implementation of online collaborative activities in the classroom and about the creation of such activities focused on the use of classroom blogs and interactive animations. The results of the analysis show that a methodology such as this exerts a positive influence not only on the students’ motivation but also on their academic achievements’ the aims of the work, the main results obtained, and the conclusions drawn.Orcos, L.; Arias, R.; Aris, N.; Magreñán, ÁA. (2016). Collaborative learning: implementation of JigSaw technique in Google. En 2nd. International conference on higher education advances (HEAD'16). Editorial Universitat Politècnica de València. 373-380. https://doi.org/10.4995/HEAD16.2015.2772OCS37338

On the convergence of a damped-secant method with modified right-hand side vector

[EN] We present a convergence analysis for a Damped Secant method with modified right-hand side vector in order to approximate a locally unique solution of a nonlinear equation in a Banach spaces setting. In the special case when the method is defined on Ri , our method provides computable error estimates based on the initial data. Such estimates were not given in relevant studies such as (Herceg et al., 1996; Krejic´, 2002). Numerical examples further validating the theoretical results are also presented in this study.The authors thank to the anonymous referee for his/her valuable comments and for the suggestions to improve the final version of the paper. This work is partially supported by UNIR Research Support Strategy 2013-2015, under the CYBERSE-CURITICS.es Research Group [http://research.unir.net].Argyros, IK.; Cordero Barbero, A.; Magreñán Ruiz, ÁA.; Torregrosa Sánchez, JR. (2015). On the convergence of a damped-secant method with modified right-hand side vector. Applied Mathematics and Computation. 252:315-323. doi:10.1016/j.amc.2014.12.029S31532325

On the convergence of a higher order family of methods and its dynamics

[EN] In this paper, we present the study of the local convergence of a higher-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 by means of studying the associated rational function. Parameter spaces are shown and the study of the stability of all the fixed points is presented. (C) 2016 Elsevier B.V. All rights reserved.This research was supported by Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación 3 [2015–2017]. Research group: Modelación matemática aplicada a la ingeniería(MOMAIN), by the grant SENECA 19374/PI/14 and by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-{01,02}-P.Argyros, IK.; Cordero Barbero, A.; Alberto Magreñán, A.; Torregrosa Sánchez, JR. (2017). On the convergence of a higher order family of methods and its dynamics. Journal of Computational and Applied Mathematics. 309:542-562. https://doi.org/10.1016/j.cam.2016.04.022S54256230

Invirtiendo la clase de matemáticas en los últimos cursos de secundaria mediante el uso de videos enriquecidos

Uno de los principales retos del profesorado en los tiempos que corren es el de motivar al estudiantado y buscar metodologías que faciliten el proceso de enseñanza-aprendizaje. El crecimiento de la tecnología y su uso en el aula ha permitido evolucionar las metodologías. En este estudio se presenta el uso de EdPuzzle como herramienta para trabajar matemáticas siguiendo el modelo de clase invertida. Para poder medir los resultados se ha realizado un pretest y se ha tomado la nota del examen de evaluación del primer trimestre, como postest. Para hacer esto se han considerado dos grupos en tercero, y otros dos en cuarto, de Educación Secundaria Obligatoria (ESO), definidos a partir del aprovechamiento que han tenido de los vídeos. A pesar de que, en un inicio, los dos grupos de tercero eran homogéneos, en términos de conocimiento previo, y los de cuarto también en los mismos términos, la comparación de las medianas de las calificaciones obtenidas por ambos grupos muestra que estas son, en líneas generales, superiores en los grupos que han aprovechado los vídeos que en aquellos que no los han aprovechado, lo que contribuye a considerar el uso de EdPuzzle como una muy buena herramienta para invertir la clase. Por último, la satisfacción del estudiantado con respecto a su uso ha mostrado ser muy elevada, y este aspecto también contribuye a que se considere como una herramienta a considerar en el aula de matemáticas.One of the main challenges for teachers is to motivate students and seek methodologies that facilitate the teaching-learning process. The growth of technology and its use in the classroom has allowed the evolution of methodologies. This study presents the use of EdPuzzle as a tool to work mathematics following the flipped classroom model. In order to measure the results, a pre-test has been carried out and the mark obtained in the first quarter evaluation exam has been taken, as a post-test. To carry out this measurement, we have considered two groups in the third year, and another two in the fourth, of Compulsory Secondary Education (ESO), defined from the use they have had of the videos. Despite the fact that, initially, the two third-year groups were homogeneous, in terms of previous knowledge, and the fourth-year groups were also homogeneous in the same terms, the comparison of the medians of the grades obtained by both groups shows that these are, in general terms, higher in the groups that have taken advantage of the videos than in those that have not taken advantage of them, which contributes to consider the use of EdPuzzle as a very good tool to reverse the class. Lastly, student satisfaction with respect to its use has shown to be very high, and this aspect also contributes to consider EdPuzzle as a tool to be taken into account in the mathematics classroom

On new means with interesting practical applications: generalized power means

Means of positive numbers appear in many applications and have been a traditional matter of study. In this work, we focus on defining a new mean of two positive values with some properties which are essential in applications, ranging from subdivision and multiresolution schemes to the numerical solution of conservation laws. In particular, three main properties are crucial—in essence, the ideas of these properties are roughly the following: to stay close to the minimum of the two values when the two arguments are far away from each other, to be quite similar to the arithmetic mean of the two values when they are similar and to satisfy a Lipchitz condition. We present new means with these properties and improve upon the results obtained with other means, in the sense that they give sharper theoretical constants that are closer to the results obtained in practical examples. This has an immediate correspondence in several applications, as can be observed in the section devoted to a particular example

Study of a biparametric family of iterative methods

[EN] The dynamics of a biparametric family for solving nonlinear equations is studied on quadratic polynomials. This biparametric family includes the c -iterative methods and the well-known Chebyshev-Halley family. We find the analytical expressions for the fixed and critical points by solving 6-degree polynomials. We use the free critical points to get the parameter planes and, by observing them, we specify some values of (alfa, c) with clear stable and unstable behaviors.This work was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-{01,02}, P11B2011-30 of the Universitat Jaume I, and PAID-SP20120474 of the Universitat Politècnica de València.Campos, B.; Cordero Barbero, A.; Magreñán Ruiz, ÁA.; Torregrosa Sánchez, JR.; Vindel Cañas, P. (2014). Study of a biparametric family of iterative methods. Abstract and Applied Analysis. 2014. https://doi.org/10.1155/2014/141643S201

Estudio de la dinámica del método de Newton amortiguado

La Tesis Doctoral defendida se sitúa en la frontera de dos líneas de investigación de gran relevancia matemática, como son los sistemas dinámicos y la resolución numérica de ecuaciones no lineales mediante procesos iterativos. En concreto, hemos realizado un estudio del conocido como método de Newton amortiguado, que es una modificación del método de Newton clásico. Dicho método consiste en la generación de una sucesión dependiente de un parámetro amortiguador que, en condiciones adecuadas, converge a la solución buscada. La tesis pone en evidencia la importancia del parámetro amortiguador, no solo en la convergencia del método sino también en sus propiedades dinámicas. La tesis presenta tres enfoques diferenciados. El primero de ellos tiene como objetivo profundizar en el análisis de la dinámica real del método, utilizando entre otras técnicas diagramas de Feigenbaum o exponentes de Lyapunov, que nos permiten encontrar comportamientos extraños (convergencia hacia ciclos, comportamiento caótico, etc.) para diferentes valores reales del parámetro amortiguador. En el segundo enfoque, dedicado a la dinámica compleja del método, se enfatiza en el conocimiento de las cuencas de atracción de las soluciones, en muchos casos, con una intrincada estructura fractal. Nos apoyaremos para ello en el carácter de los puntos fijos, la aparición de ciclos atractores y en los planos de parámetros asociados a los puntos críticos libres. Finalmente, se hace un estudio del método para operadores definidos entre espacios de Banach, obteniendo resultados sobre convergencia local y semilocal. Esta generalización permite abordar problemas tales como sistemas de ecuaciones no lineales, ecuaciones diferenciales o integrales, problemas de optimización, etc.The Doctoral Thesis defended lies on the border of two lines of mathematical research of great relevance, such as dynamical systems and the numerical solution of nonlinear equations by iterative processes. Specifically, we studied the iterative method known as damped Newton method, which is a modification of the classical Newton method. This method generates a sequence depending on a damping parameter, which in suitable conditions, converges to the desired solution. The thesis shows the importance of the damping parameter, not only in the convergence of the method but also in their dynamic properties. The thesis presents three different approaches. The first one is focused in the analysis of the real dynamics of the method, using among other techniques Feigenbaum diagrams and Lyapunov exponents which allow us to find strange behaviors (convergence to cycles, chaotical behaviour, etc.) for different values of the damping parameter. In the second approach, dedicated to the complex dynamics of the method, the main aim consist on distinguishing the basins of attraction associated to the solutions, which have in many cases an intricate fractal structure. We rely on the character of this fixed point, the attracting cycles and in the parameters planes associated to the iteration of the free critical points. Finally, we study the method applied to operators defined between Banach spaces, obtaining results on the local and semilocal convergence. This generalization allows us to find the solution of different problems such as systems of nonlinear equations, differential or integral equations, optimization problems, etc

Real dynamics for damped Newton's method applied to cubic polynomials

In this paper we study the real dynamics of the damped Newton's methods applied to cubic polynomials, but instead of taking a value of the damping factor (0,1), we consider all values of R. The method for unusual values of presents different behaviors such as convergence to n-cycles or even chaos

A unified convergence analysis for secant-type methods

We present a unified local and semilocal convergence analysis for secant-type methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis includes the computation of the bounds on the limit points of the majorizing sequences involved. Under the same computational cost our semilocal convergence criteria can be weaker; the error bounds more precise and in the local case the convergence balls can be larger and the error bounds tighter than in earlier studies such as [13,714,16,20,21] at least for the cases of Newtons method and the secant method. Numerical examples are also presented to illustrate the theoretical results obtained in this study