Impact on stability by the use of memory in Traub-type schemes

[EN] In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub's method, they have been designed using linear approximations or the Newton's interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub's scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process.This research was partially supported by Ministerio de Ciencia, Innovacion y Universidades under grants PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE).Chicharro, FI.; Cordero Barbero, A.; Garrido, N.; Torregrosa Sánchez, JR. (2020). Impact on stability by the use of memory in Traub-type schemes. Mathematics. 8(2):1-16. https://doi.org/10.3390/math8020274S11682Shacham, M. (1989). An improved memory method for the solution of a nonlinear equation. Chemical Engineering Science, 44(7), 1495-1501. doi:10.1016/0009-2509(89)80026-0Balaji, G. V., & Seader, J. D. (1995). Application of interval Newton’s method to chemical engineering problems. Reliable Computing, 1(3), 215-223. doi:10.1007/bf02385253Shacham, M. (1986). Numerical solution of constrained non-linear algebraic equations. International Journal for Numerical Methods in Engineering, 23(8), 1455-1481. doi:10.1002/nme.1620230805Shacham, M., & Kehat, E. (1973). Converging interval methods for the iterative solution of a non-linear equation. Chemical Engineering Science, 28(12), 2187-2193. doi:10.1016/0009-2509(73)85008-0Amat, S., Busquier, S., & Plaza, S. (2010). Chaotic dynamics of a third-order Newton-type method. Journal of Mathematical Analysis and Applications, 366(1), 24-32. doi:10.1016/j.jmaa.2010.01.047Argyros, I. K., Cordero, A., Magreñán, Á. A., & Torregrosa, J. R. (2017). Third-degree anomalies of Traub’s method. Journal of Computational and Applied Mathematics, 309, 511-521. doi:10.1016/j.cam.2016.01.060Chicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Chicharro, F., Cordero, A., & Torregrosa, J. (2015). Dynamics and Fractal Dimension of Steffensen-Type Methods. Algorithms, 8(2), 271-279. doi:10.3390/a8020271Scott, M., Neta, B., & Chun, C. (2011). Basin attractors for various methods. Applied Mathematics and Computation, 218(6), 2584-2599. doi:10.1016/j.amc.2011.07.076Steffensen, J. F. (1933). Remarks on iteration. Scandinavian Actuarial Journal, 1933(1), 64-72. doi:10.1080/03461238.1933.10419209Wang, X., & Zhang, T. (2012). A new family of Newton-type iterative methods with and without memory for solving nonlinear equations. Calcolo, 51(1), 1-15. doi:10.1007/s10092-012-0072-2Džunić, J., & Petković, M. S. (2014). On generalized biparametric multipoint root finding methods with memory. Journal of Computational and Applied Mathematics, 255, 362-375. doi:10.1016/j.cam.2013.05.013Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2014). Multipoint methods for solving nonlinear equations: A survey. Applied Mathematics and Computation, 226, 635-660. doi:10.1016/j.amc.2013.10.072Campos, B., Cordero, A., Torregrosa, J. R., & Vindel, P. (2015). A multidimensional dynamical approach to iterative methods with memory. Applied Mathematics and Computation, 271, 701-715. doi:10.1016/j.amc.2015.09.056Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2019). Dynamics of iterative families with memory based on weight functions procedure. Journal of Computational and Applied Mathematics, 354, 286-298. doi:10.1016/j.cam.2018.01.019Chicharro, F. I., Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2017). King-Type Derivative-Free Iterative Families: Real and Memory Dynamics. Complexity, 2017, 1-15. doi:10.1155/2017/2713145Magreñán, Á. A., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2014). Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane. Mathematics and Computers in Simulation, 105, 49-61. doi:10.1016/j.matcom.2014.04.006Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-141. doi:10.1090/s0273-0979-1984-15240-6Magreñán, Á. A. (2014). A new tool to study real dynamics: The convergence plane. Applied Mathematics and Computation, 248, 215-224. doi:10.1016/j.amc.2014.09.061Chicharro, F. I., Cordero, A., & Torregrosa, J. R. (2013). Drawing Dynamical and Parameters Planes of Iterative Families and Methods. The Scientific World Journal, 2013, 1-11. doi:10.1155/2013/78015

Suitable Approximations for the Self-Accelerating Parameters in Iterative Methods With Memory

This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Chicharro, FI.; Garrido, N.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2020). Suitable Approximations for the Self-Accelerating Parameters in Iterative Methods With Memory. 42-47. http://hdl.handle.net/10251/179836424

Generalized high-order classes for solving nonlinear systems and their applications

[EN] A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher's problem, showing the good performance of the new methods.This research was partially supported by both Ministerio de Ciencia, Innovacion y Universidades and Generalitat Valenciana, under grants PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE) and PROMETEO/2016/089, respectively.Chicharro, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2019). Generalized high-order classes for solving nonlinear systems and their applications. Mathematics. 7(12):1-14. https://doi.org/10.3390/math7121194S114712Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2014). Multipoint methods for solving nonlinear equations: A survey. Applied Mathematics and Computation, 226, 635-660. doi:10.1016/j.amc.2013.10.072Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Cordero, A., Gómez, E., & Torregrosa, J. R. (2017). Efficient High-Order Iterative Methods for Solving Nonlinear Systems and Their Application on Heat Conduction Problems. Complexity, 2017, 1-11. doi:10.1155/2017/6457532Sharma, J. R., & Arora, H. (2016). Improved Newton-like methods for solving systems of nonlinear equations. SeMA Journal, 74(2), 147-163. doi:10.1007/s40324-016-0085-xAmiri, A., Cordero, A., Taghi Darvishi, M., & Torregrosa, J. R. (2018). Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems. Applied Mathematics and Computation, 323, 43-57. doi:10.1016/j.amc.2017.11.040Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). A modified Newton-Jarratt’s composition. Numerical Algorithms, 55(1), 87-99. doi:10.1007/s11075-009-9359-zChicharro, F. I., Cordero, A., Garrido, N., & Torregrosa, J. R. (2019). Wide stability in a new family of optimal fourth‐order iterative methods. Computational and Mathematical Methods, 1(2), e1023. doi:10.1002/cmm4.1023FISHER, R. A. (1937). THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES. Annals of Eugenics, 7(4), 355-369. doi:10.1111/j.1469-1809.1937.tb02153.xSharma, J. R., Guha, R. K., & Sharma, R. (2012). An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numerical Algorithms, 62(2), 307-323. doi:10.1007/s11075-012-9585-7Soleymani, F., Lotfi, T., & Bakhtiari, P. (2013). A multi-step class of iterative methods for nonlinear systems. Optimization Letters, 8(3), 1001-1015. doi:10.1007/s11590-013-0617-6Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06

Generalizing Traub's method to a parametric iterative class for solving multidimensional nonlinear problems

[EN] In this work, we modify the iterative structure of Traub's method to include a real parameter alpha$$\alpha$$. A parametric family of iterative methods is obtained as a generalization of Traub, which is also a member of it. The cubic order of convergence is proved for any value of alpha$$\alpha$$. Then, a dynamical analysis is performed after applying the family for solving a system cubic polynomials by means of multidimensional real dynamics. This analysis allows to select the best members of the family in terms of stability as a preliminary study to be generalized to any nonlinear function. Finally, some iterative schemes of the family are used to check numerically the previous developments when they are used to approximate the solutions of academic nonlinear problems and a chemical diffusion reaction problem.ERDF A way of making Europe, Grant/Award Number: PGC2018-095896-B-C22; MICoCo of Universidad Internacional de La Rioja (UNIR), Grant/Award Number: PGC2018-095896-B-C22Chicharro, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2023). Generalizing Traub's method to a parametric iterative class for solving multidimensional nonlinear problems. Mathematical Methods in the Applied Sciences. 1-14. https://doi.org/10.1002/mma.937111

Paired SSB optical OFDM channels for high spectral efficient signal transmission over DWDM networks

[EN] A new high spectral efficient SSB-OOFDM DWDM transmission system has been experimentally demonstrated. The proposed transmitter employs paired optical channels consisting of two SSB modulated OFDM signals using opposite sidebands in order to allow an efficient use of the spectrum with optical carriers separation under 10 GHz. Moreover, different paired channels are multiplexed into the 25 GHz grid DWDM fiber transmission link. Optical carrier spacing of 8.75 GHz in paired channels has been demonstrated allowing 40.8 Gb/s signal transmission rate over a 25 GHz paired channel bandwidth.The research leading to these results has received funding from the national project TEC2011-26642 (NEWTON) funded by the Ministerio de Ciencia y Tecnología and the Research Excellency Award Programme GVA PROMETEO 2013/012NEXT GENERATION MICROWAVE PHOTONIC TECHNOLOGIES.Chicharro López, FI.; Ortega Tamarit, B.; Mora Almerich, J. (2016). Paired SSB optical OFDM channels for high spectral efficient signal transmission over DWDM networks. Optics Communications. 370:239-244. https://doi.org/10.1016/j.optcom.2016.03.007S23924437

On the evaluation of an optical OFDM radio over FSO system with IM-DD for high-speed indoor communications

En aquest article es proposa un nou sistema híbrid de ràdio sobre fibra (RoF) i òptica d’espai lliure (FSO) basat en la transmissió de senyals òptics OFDM (OOFDM) per proporcionar enllaços de comunicació sense fils òptics d’alta capacitat. Es proposa un esquema de transmissió de baix cost basat en IM-DD per simplificar i reutilitzar la infraestructura existent per desplegar un enllaç FSO d’alt ample de banda. Aquest article mostra els resultats experimentals basats en la implementació de l'esquema proposat sota una transmissió digital OOFDM de 16 QAM a través de 10 km de fibra monomode (SMF) i enllaç FSO de 300 mm. Els resultats mostren la viabilitat de desplegar un sistema de comunicació de 20 Gbit / s sobre un senyal OFDM d’amplada de banda RF de 5 GHz mitjançant l’esquema IM-DD per a la transmissió òptica sense cap tècnica de multiplexació òptica.JCI-2012-14805A novel radio over fiber (RoF) and free space optics (FSO) hybrid system based on optical OFDM (OOFDM) signal transmission is proposed in this paper to provide high capacity optical wireless indoor communication links. A low cost transmission scheme based on IM-DD is proposed to simplify and reuse existing infrastructure to deploy a high bandwidth FSO link. This paper shows the experimental results based on the implementation of the proposed scheme under a 16-QAM OOFDM digital transmission over 10 km Single Mode Fiber (SMF) and 300 mm FSO link. The results show the feasibility to deploy a 20 Gbit/s communication system over a 5 GHz RF bandwidth OFDM signal by using IM-DD scheme for the optical transmission without any optical multiplexing technique

The Enhancement of Academic Performance in Online Environments

Distance education has been gaining popularity for the last years. The proficiency in online environments of both learners and teachers explains the success of this methodology. An evaluation of graduate students’ performance within numerical analysis is discussed. In order to improve the marks obtained by the students, specific actions have been performed over the years and data from different classes has been analyzed using statistical tools. The results show that the actions proposed seemed to help the students in their learning process

A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed

Stability comparison of self-accelerating parameter approximation on one-step iterative methods

The authors were supported by the internal research project ADMIREN of Universidad Internacional de La Rioja (UNIR). The first author was also partially supported by PGC2018-095896-BC22 (MCIU/AEI/FEDER, UE).Chicharro, FI.; Garrido, N.; Pérez, D.; Sarría, Í. (2021). Stability comparison of self-accelerating parameter approximation on one-step iterative methods. Universitat Politècnica de València. 90-95. http://hdl.handle.net/10251/182196S909

Different approximations of the parameter for low-order iterative methods with memory

[EN] A technique for generating iterative methods for solving nonlinear equations with memory can be constructed from a method without memory that includes a parameter, provided the parameter is present in the error equation. Generally, the parameter depends on the evaluation of the function and its derivatives in the solution. However, this information is not available. So this parameter is approximated using interpolation techniques, taking the current iterate ¿¿ and the previous iterates ¿¿¿1, ¿¿¿2, . . . In this paper we explore different interpolation techniques to obtain both the convergence order of the new methods and their stability characteristics.The authors were supported by the internal research project ADMIREN of Universidad Internacional de La Rioja (UNIR). The first author was also partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Chicharro, FI.; Garrido, N.; Sarría, Í.; Orcos, L. (2021). Different approximations of the parameter for low-order iterative methods with memory. Servicio de Publicaciones de la Universidad de Oviedo. 130-134. http://hdl.handle.net/10251/180525S13013