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In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence 4 for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the Molecular Interaction Problem.

Iterative methods for solving nonlinear equations are said to have memory when the calculation of the next iterate requires the use of more than one previous iteration. Methods with memory usually have a very stable behavior in the sense of the wideness of the set of convergent initial estimations. With the right choice of parameters, iterative methods without memory can increase their order of convergence significantly, becoming schemes with memory. In this work, starting from a simple method without memory, we increase its order of convergence without adding new functional evaluations by approximating the accelerating parameter with Newton interpolation polynomials of degree one and two. Using this technique in the multidimensional case, we extend the proposed method to systems of nonlinear equations. Numerical tests are presented to verify the theoretical results and a study of the dynamics of the method is applied to different problems to show its stability.

In this paper, we present a multidimensional real dynamical study of the Ostrowsky-Chun family of iterative methods to solve systems of nonlinear equations. This family was defined initially for solving scalar equations but, in general, scalar methods can be transferred to make them suitable to solve nonlinear systems. The complex dynamical behavior of the rational operator associated to a scalar method applied to low-degree polynomials has shown to be an efficient tool for analyzing the stability and reliability of the methods. However, a good scalar dynamical behavior does not guarantee a good one in multidimensional case. We found different real intervals where both parameters can be defined assuring a completely stable performance and also other regions where it is dangerous to select any of the parameters, as undesirable behavior as attracting elements that are not solution of the problem to be solved appear. This performance is checked on a problem of chemical wave propagation, Fisher's equation, where the difference in numerical results provided by those elements of the class with good stability properties and those showed to be unstable, is clear.

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Las microrredes son sistemas de potencia de baja tensión, que pueden operar en modo aislado o con conexión a la red eléctrica. Estos sistemas cuentan con fuentes de generación distribuida, sistemas de almacenamiento, cargas distribuidas, sistemas de comunicación y sistemas de monitoreo y control. Los sistemas de comunicación implementados en las microrredes son un área de estudio de gran importancia, debido a que para asegurar la continuidad y calidad del servicio eléctrico es necesario que la información de los sistemas se transmita de manera confiable y rápida. Por esta razón se utilizan diferentes topologías de red y protocolos de comunicación, los cuales tienen distintas ventajas dependiendo su aplicación. Este artículo presenta una revisión de literatura sobre las topologías de red y los protocolos de comunicación utilizados en las microrredes, ventajas y desventajas para cada aplicación, así como los protocolos utilizados por las entidades del sector eléctrico de la República Dominicana.

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.