Some results on B-matrices and doubly B-matrices

A real matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row means was called in [4] a B-matrix. In [5], the class of doubly B-matrices was introduced as a generalization of the previous class. We present several characterizations and properties of these matrices and for the class of B-matrices we consider corresponding questions for subdirect sums of two matrices (a general ‘sum’ of matrices introduced in [1] by S.M. Fallat and C.R. Johnson, of which the direct sum and ordinary sum are special cases), for the Hadamard product of two matrices and for the Kronecker product and sum of two matrices.Fundação para a Ciência e Tecnologia (FCT)Ministerio de Ciencia y Tecnología (Espanha

The completion problem for N-matrices

An $n\times m$ matrix is called an $N$-matrix if all principal minors are negative. In this paper, we are interested in $N$-matrix completion problems, that is, when a partial $N$-matrix has an $N$-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial $N$-matrix does not have an $N$-matrix completion. Here, we prove that a combinatorially symmetric partial $N$-matrix has an $N$-matrix completion if the graph of its specified entries is a 1-chordal graph. We also prove that there exists an $N$-matrix completion for a partial $N$-matrix whose associated graph is an undirected cycle.Fundação para a Ciência e a Tecnologia (FCT) – Programa Operacional “Ciência, Tecnologia, Inovação” (POCTI) DGI - BFM2001-0081-C03-0

N_0 completions on partial matrices

An $n\times n$ matrix is called an $N_0$-matrix if all its principal minors are nonpositive. In this paper, we are interested in $N_0$-matrix completion problems, that is, when a partial $N_0$-matrix has an $N_0$-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial $N_0$-matrix does not have an $N_0$-matrix completion. Here, we prove that a combinatorially symmetric partial $N_0$-matrix, with no null main diagonal entries, has an $N_0$-matrix completion if the graph of its specified entries is a 1-chordal graph or a cycle. We also analyze the mentioned problem when the partial matrix has some null main diagonal entries.Fundação para a Ciência e a Tecnologia(FCT) através do programa POCTISpanish DGI grant number MTM2007-6447

Design, Analysis, and Applications of Iterative Methods for Solving Nonlinear Systems

In this chapter, we present an overview of some multipoint iterative methods for solving nonlinear systems obtained by using different techniques such as composition of known methods, weight function procedure, and pseudo-composition, etc. The dynamical study of these iterative schemes provides us valuable information about their stability and reliability. A numerical test on a specific problem coming from chemistry is performed to compare the described methods with classical ones and to confirm the theoretical results

Dynamical analysis of an iterative method with memory on a family of third-degree polynomials

Qualitative analysis of iterative methods with memory has been carried out a few years ago. Most of the papers published in this context analyze the behaviour of schemes on quadratic polynomials. In this paper, we accomplish a complete dynamical study of an iterative method with memory, the Kurchatov scheme, applied on a family of cubic polynomials. To reach this goal we transform the iterative scheme with memory into a discrete dynamical system defined on R2. We obtain a complete description of the dynamical planes for every value of parameter of the family considered. We also analyze the bifurcations that occur related with the number of fixed points. Finally, the dynamical results are summarized in a parameter line. As a conclusion, we obtain that this scheme is completely stable for cubic polynomials since the only attractors that appear for any value of the parameter, are the roots of the polynomial.This paper is supported by the MCIU grant PGC2018-095896-B-C22. The first and the last authors are also supported by University Jaume I grant UJI-B2019-18. Moreover, the authors would like to thank the anonymous reviewers for their comments and suggestions

Herramientas de la teoría de grafos para la modelización

[EN] In graph theory, networks play an important role in a lot of type of problems. In this paper we present some of them, sometimes without a clear relation with the classic definition of a network. They are enunciated in a real neighbourhood and the first step to solve them consists of translating the conditions of the problem to a graph. In order to get a good modelization, it must to take in mind the objective to reach. After the problem is modelized we apply the techniques or methods studied in graph theory. This type of problems increases the interest of the students, helps them to see the wide applicability of networks and trains them in the use of mathematical modelization.[ES] Dentro de la teoría de grafos las redes ocupan un papel destacado por la amplia variedad de problemas que resuelven. En esta ocasión presentamos varios de estos problemas, algunos sin relación aparente a priori con la definición clásica de red. Están enunciados en contextos reales y su resolución pasa, en primer lugar, por definir un grafo que represente la situación, teniendo presente, puesto que influye en su definición, el objetivo a determinar. Una vez modelizado el problema, ya dentro de la teoría de grafos, es el momento de aplicar las técnicas o métodos estudiados. Este tipo de problemas, además de incentivar el interés del alumno, le ayudan a vislumbrar la amplia aplicabilidad de las redes y le entrenan en general en el uso de la modelización.Jordan, C.; Torregrosa, JR. (2011). Herramientas de la teoría de grafos para la modelización. Modelling in Science Education and Learning. 4:275-287. doi:10.4995/msel.2011.3088SWORD2752874G. Chartrand, O.R. Oellerman, Applied and algorithmic graph theory, McGraw Hill 1993. A.Dolan, J.Aldous, Networks and algorithms, Wiley 1993.J. L. Gross, J. Yellen, Graph theory and its applications, Chapman&Hall, 2006.G. Hernández Pe-alver, Grafos. Teoría y algoritmos, UPM 2003

Two weighted-order classes of iterative root-finding methods

In this paper we design, by using the weight function technique, two families of iterative schemes with order of convergence eight. These weight functions depend on one, two and three variables and they are used in the second and third step of the iterative expression. Dynamics on polynomial and non-polynomial functions is analysed and they are applied on the problem of preliminary orbit determination by using a modified Gauss method. Finally, some standard test functions are to check the reliability of the proposed schemes and allow us to compare them with other known methods.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT 2011-1-B1-33 Republica Dominicana.Artidiello Moreno, SDJ.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Two weighted-order classes of iterative root-finding methods. International Journal of Computer Mathematics. 92(9):1790-1805. https://doi.org/10.1080/00207160.2014.887201S1790180592

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

[EN] 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. (C) 2015 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-{01,02}.Cordero Barbero, A.; Magrenán, A.; Quemada, C.; Torregrosa Sánchez, JR. (2016). Stability study of eighth-order iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics. 291:348-357. https://doi.org/10.1016/j.cam.2015.01.006S34835729

Totally nonpositive completions on partial matrices

An n £ n real matrix is said to be totally no positive if every minor is no positive. In this paper, we are interested in totally no positive completion problems, that is, does A partial totally no positive matrix have a totally no positive matrix completion? This Problem has, in general, a negative answer. Therefore, we analyze the question: for which Labelled graphs G does every partial totally no positive matrix, whose associated graph is G, have a totally no positive completion? Here we study the mentioned problem when G Is a choral graph or an undirected cycle.Spanish DGI grant number BFM2001-0081-C03-02 and Generalitat Valenciana GRUPOS03/062Fundação para a Ciência e a Tecnologia (FCT