# Ángel Alberto MagreñánUniversidad de La Rioja (Spain) | UNIRIOJA · Mathematics and Computation

Ángel Alberto Magreñán

Ph. D.

Mathematics education with educational technology

## About

215

Publications

21,349

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1,839

Citations

Citations since 2016

Introduction

2020 Mathematics Young Investigator Award: https://www.mdpi.com/journal/mathematics/announcements/1972
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Young researcher award: https://www.unirioja.es/apnoticias/servlet/Noticias?codnot=6579&accion=detnot

Additional affiliations

September 2018 - January 2022

**Universidad de La Rioja**

Position

- Professor (Associate)

September 2018 - present

March 2016 - September 2018

Education

September 2011 - June 2012

September 2009 - September 2013

September 2003 - June 2011

## Publications

Publications (215)

In recent years, the way in which mathematics is taught has changed radically and there has been an increase in the percentage of online classes. One of the biggest challenges that teachers face in this type of online scenario is the need to motivate students so that they do not lose the thread and continue with their usual rhythm and seek methodol...

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 t...

One of the most studied problems in numerical mathematics is the solution of nonlinear equations H(y)=0
(1)
,
where H : D ⊂ ℝm → ℝm is a nonlinear operator, H ≡ (H1, H2, …, Hm) with Hi : D ⊆ ℝm → ℝ, 1 ≤ i ≤ m, and D is a non-empty open convex domain. We suppose that H has a simple root y∗ ∈ D. Iterative methods are a powerful tool for solving these...

The work of Probability from the Compulsory Secondary Education stage and during the Baccalaureate is not generally contextualised and is, mainly, based on the rote learning of formulas. Therefore, when students arrive at university, a significant lack of knowledge related to the calculation of probabilities is evident. Teachers have to advocate fo...

As accessibility of Newton’s method is very restrictive under a center-Lipschitz condition for the first derivative of the operator involved, we construct a hybrid iterative method to increase the accessibility of the method, where the modified Newton method is used to predict a good starting approximation for Newton’s method. A boundary value prob...

In this article, we introduce a new Steffensen-type method with the advantage that its behavior is very similar to Newton’s method; therefore, it is a very remarkable way of avoiding the drawback that Newton’s method presents for nondifferentiable operators. In our study, we perform an exhaustive comparative study between the semilocal convergence...

INTRODUCTION. One of the main objectives of Mathematics education is to motivate students since their interest in Mathematics is very low in many cases and, in others, even null. The use of different technologies has grown a lot in the last decades and several authors in the area have demonstrated their effectiveness in the classroom. In addition,...

Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problem...

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although...

An efficient modification of the Chebyshev method is constructed from approximating the second derivative of the operator involved by combinations of the operator in different points and it is used to locate, separate and approximate the solutions of a Chandrasekhar integral equation from analysing its global convergence.

An this article, we propose a new research related to the convergence of the frozen Potra and Schmidth-Schwetlick schemes when we apply to equations. The purpose of this study is to introduce a comparison between two solutions to equations under the same conditions. In particular, we show the convergence radius and the uniqueness ball coincidence,...

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 ma...

In this manuscript, we introduce the higher-order optimal derivative-free family of Chebyshev–Halley’s iterative technique to solve the nonlinear equation having the multiple roots. The designed scheme makes use of the weight function and one parameter α to achieve the fourth-order of convergence. Initially, the convergence analysis is performed fo...

In this work we are going to use the Kurchatov-Schmidt-Schwetlick-like solver (KSSLS) and the Kurchatov-like solver (KLS) to locate a zero, denoted by x∗ of operator F. We define F as F:D⊆B1⟶B2 where B1 and B2 stand for Banach spaces, D⊆B1 be a convex set and F be a differentiable mapping according to Fréchet. Under these conditions, for all n=0,1,...

We establish a global convergence result for an efficient third-order iterative process which is constructed from Chebyshev’s method by approximating the second derivative of the operator involved by combinations of the operator. In particular, from the use of auxiliary points, we provide domains of restricted global convergence that allow obtainin...

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...

Actas de las Jornadas Internacionales de Investigación y Práctica Docente en Alta Capacidad Matemática, celebradas virtualmente en la Universidad de la Rioja, en Noviembre de 2020

The main aim of this study is to approximate the square root of matrices using an efficient iterative method. We analyse the convergence of the iterative method and the order of convergence of this kind of methods and we will apply our theoretical results to specific examples in order to prove that results.

Based on the modification of family of iterative processes of Chebyshev-Halley presented by M. Kansal et al. in [2], whose convergence is cubic, we will present in this talk a comparison between the behaviour of some members of the family in which some different means are used. We will also study the influence of that mean in the convergence of the...

La llamada Robótica Educativa (RE) y los drones están aumentando su presencia en contextos educativos, formales y no formales, cuando el objetivo es desarrollar procesos de enseñanza-aprendizaje más atractivos, motivadores y que facilitan el rendimiento. Un marco propicio para ello es el del modelo integrado STEM. El objetivo de esta investigación...

La llamada Robótica Educativa (RE) y los drones están aumentando su presencia en contextos educativos, formales y no formales, cuando el objetivo es desarrollar procesos de enseñanza-aprendizaje más atractivos, motivadores y que facilitan el rendimiento. Un marco propicio para ello es el del modelo integrado STEM. El objetivo de esta investigación...

One of the main objectives in mathematics education is to motivate students due to the fact that their interest in this area is often very low. The use of different technologies, as well as gamification in the classroom, can help us to meet this goal. In this case, it is presented the use of two techniques, which are a digital escape room, using Ge...

This study suggests a new general scheme of high convergence order for approximating the solutions of nonlinear systems. The proposed scheme is the extension of an earlier study of Parhi and Gupta. This method requires two vector-function, two Jacobian matrices, two inverse matrices, and one frozen inverse matrix per iteration. Convergence error, c...

In this paper, we propose a center Lipschitz condition for the second derivative together with the use of restricted domains in order to improve the starting points for Newton's method when compared with previous results. Moreover, we present some numerical examples validating the theoretical results.

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 presente...

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,...

In this paper, we present the local results of the convergence of the two-step King-like method to approximate the solution of nonlinear equations. In this study, we only apply conditions to the first derivative, because we only need this condition to guarantee convergence. As a result, the applicability of the method is expanded. We also use diffe...

The convergence domain for both the local and semilocal case of Newton’s method for Banach space valued operators is small in general. There is a plethora of articles that have extended the convergence criterion due to Kantorovich under variations of the convergence conditions. In this article, we use a different approach than before to increase th...

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence...

This paper deal with the study of local convergence of fourth and fifth order iterative method for solving nonlinear equations in Banach spaces. Only the premise that the first order Fréchet derivative fulfills the Lipschitz continuity condition is needed.
Under these conditions, a convergence theorem is established to study the existence and uniqu...

In this paper we study the problem of analyzing the convergence both local and semilocal of inexact Newton-like methods for approximating the solution of an equation in which there exists nondifferentiability. We will impose conditions, to ensure that the method converges, are weaker than in the ones imposed in previous results. The theoretical res...

There is a need to extend the convergence domain of iterative methods for computing a locally unique solution of Banach space valued operator equations. This is because the domain is small in general, limiting the applicability of the methods. The new idea involves the construction of a tighter set than the ones used before also containing the iter...

Stirling's method is considered as an alternative to Newton's method when the latter 1 fails to converge to a solution of a nonlinear equation. Both methods converge quadratically under 2 similar convergence criteria and require the same computational effort. But Stirling's method has 3 shortcomings too. In particular, contractive conditions are as...

El estudio dinámico de los métodos iterativos ha aumentado en las últimas décadas debido al desarrollo de los computadores, aspecto por el cual se ha visto la necesidad de incluir la enseñanza de estos métodos en los planes de estudio. En la actualidad hay varios tipos de software cuya aplicación didáctica en las aulas es de gran utilidad, pero no...

The changes in our society in recent years and the consequent idiosyncrasy of young people demand new teaching methodologies. The methodology known as flip teaching, in which pupils study the subject before the class experience, using the material given by the teacher, makes it possible to turn the classroom into a place to solve different problems...

Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems....

This paper is devoted to the construction and analysis of some new nonlinear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four point int...

We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is...

The main purpose of this work is the development of an application that allows compulsory secondary education teachers the assessment of the students’ maths competence. An equation of prediction is obtained in a pilot study through the multivariate technique of multiple regression using for its analysis, by means of SPSS, the results gathered throu...

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with th...

We first present a local convergence analysis for some families of fourth and six order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies have used hypotheses on the fourth Fréchet-derivative of the operator involved. We use hypotheses only on the first Fréchet-derivative in...

We present local convergence results for inexact iterative procedures of high convergence order in a normed space in order to approximate a locally unique solution. The hypotheses involve only Lipschitz conditions on the first Fréchet-derivative of the operator involved. Earlier results involve Lipschitz-type hypotheses on higher than the first Fré...

The aim of this article is to present the correct version of the main theorem 3.2 given in Guo and Duff (2011), concerning the semi-local convergence analysis of the Newton-HSS (NHSS) method for solving systems of nonlinear equations. Our analysis also includes the corrected upper bound on the initial point.

The aim of this paper is to present a new semi-local convergence analysis for Newton’s method in a Banach space setting. The novelty of this paper is that by using more precise Lipschitz constants than in earlier studies and our new idea of restricted convergence domains, we extend the applicability of Newton’s method as follows: The convergence do...

This study aims to implement and evaluate a methodological proposal using the hologram as a teaching medium for the learning of concepts related to areas and volumes of geometrical bodies. The study has been carried out with a sample of 78 students in the third year of secondary education from a privately-owned but state-funded school in Madrid. Th...

The principal objective of this study is to propose two derivative free iteration functions. Both are applicable to each earlier optimal multi-point derivative free scheme of order four and eight whose first sub step should be Steffensen’s type method to develop more advanced optimal iteration techniques of order eight and sixteen, respectively. Bo...

The study of the dynamics and the analysis of local convergence of an iterative method, when approximating a locally unique solution of a nonlinear equation, is presented in this article. We obtain convergence using a center-Lipschitz condition where the ball radii are greater than previous studies. We investigate the dynamics of the method. To val...

The the study of the dynamics and the analysis of local convergence of an iterative method when approximating a locally unique solution of a non-linear equation is presented in this article. We are going to obtain the convergence using a center-Lipschitz condition where the ball radii are greater than previous studies. We also do the study of the d...

The present work gathers an educational experience based on the application of the personalized Kumon Mathematics Method, carried out in the school year 2015-2016, in which 30,849 students and 230 teachers from several educational centers throughout Spain have participated. We start with a theoretical foundation of the Kumon Method and continue wit...

In this study, a new highly efficient sixth-order family of iterative methods for solving nonlinear equations are presented along with their convergence properties. Further, we extend this family to the multidimensional case preserving the same order of convergence. For the implementation of the proposed techniques for system of nonlinear equations...

MSC: 65H05 65H99 41A25 65B99 Keywords: Parameter space Möbius map Dynamical plane Multiple-zero Sixth-order Conjugacy a b s t r a c t 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...

In this paper we establish the construction of a family of free derivative of point to point iterative processes, with quadratic convergence, from two known data in each previous iteration. Besides, we study the accessibility of this family by means of the basins of attraction and the convergence balls. We provide a local convergence analysis for t...

We present a local as well a semilocal convergence analysis of secant-like methods under g eneral conditions in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The new conditions are more flexible than in earlier studies. This way we expand the applicability of these methods, since the new convergen...

We present a new semilocal convergence analysis for Secant 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 on the parameters involved our converg...

We study the local convergence of a Newton-like method of convergence order six to approximate a locally unique solution of a nonlinear equation. Earlier studies show convergence under hypotheses on the seventh derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative although only the first derivati...