Diseño, análisis y estabilidad de métodos iterativos con memoria para la resolución de ecuaciones y sistemas no lineales

[ES] El diseño de métodos iterativos para resolver ecuaciones y sistemas de ecuaciones no lineales es una tarea importante y desafiante en el campo del Análisis Numérico. La no linealidad es una característica de muchos de los fenómenos físicos. Mecánica de fluidos y plasma, dinámica de gases, reacciones químicas, combustión, ecología, biomecánica, problemas de modelado económico, teoría del transporte y muchos otros fenómenos están todos gobernados inherentemente por ecuaciones no lineales. Por esta razón, una proporción cada vez mayor de la investigación matemática moderna se dedica al análisis de sistemas y procesos no lineales. En una era caracterizada por la disponibilidad de grandes cantidades de datos, el procesado y análisis de esta información se traduce de forma directa en la resolución de problemas cuya dimensión es cada vez mayor. Aunque en las últimas décadas se ha producido un desarrollo exponencial en la computación, sigue siendo esencial el diseño de algoritmos iterativos que garanticen la convergencia a la solución de un problema de forma rápida y eficiente. Siguiendo estas premisas, el objetivo fundamental que se persigue con el diseño de nuevos métodos iterativos, siendo también uno de los principales objetivos de la presente Tesis Doctoral, es la aproximación de soluciones de problemas no lineales garantizando un cierto equilibrio entre la velocidad con que se obtiene dicha aproximación, la fiabilidad de la misma y el coste computacional requerido en el conjunto de todo el proceso iterativo. Poder medir o cuantificar este equilibrio es, por tanto, una de las piezas esenciales de todo este proceso. Por medio del orden de convergencia, somos capaces de comparar la velocidad con que los esquemas iterativos se aproximan a la solución buscada. El coste computacional requerido a cada iteración del proceso es directamente proporcional al número de iteraciones que se necesitan para aproximar esta solución. Por tanto, acelerar la convergencia de un método se convierte en una necesidad en el diseño de esquemas eficientes. Por otro lado, todo algoritmo iterativo requiere de al menos un punto inicial para comenzar el proceso de cálculo de las iteraciones sucesivas. Por este motivo, el estudio de la influencia de las estimaciones iniciales en la convergencia de un método es también de una gran relevancia, ya que permite determinar la estabilidad de éste en función de los iterados iniciales. Este estudio se realiza utilizando herramientas de dinámica discreta, tanto real como compleja, para determinar, además de otras caracterizaciones, los puntos iniciales más adecuados y los métodos más estables de una familia de esquemas iterativos. El análisis numérico y dinámico realizado en esta memoria hace posible la propuesta de métodos iterativos eficientes que aproximen soluciones de problemas multidimensionales no lineales y ecuaciones en derivadas parciales de forma eficaz. A partir del estudio completo desarrollado utilizando las herramientas anteriormente descritas, presentamos esta Tesis Doctoral para la obtención del título de Doctora en Matemáticas.[EN] The design of iterative methods for solving nonlinear equations and nonlinear systems is an important and challenging task in the Numerical Analysis. Nonlinearity is a characteristic of many of the physical phenomena. Fluid and plasma mechanics, gas dynamics, chemical reactions, combustion, ecology, biomechanics, economic modeling problems, transport theory and many other phenomena are all inherently governed by nonlinear equations. For this reason, an increasing proportion of mathematical modern research is devoted to the analysis of systems and nonlinear processes. In a period characterized by the availability of large amounts of data, the processing and analysis of this information translates directly into the resolution of problems whose dimension is increasing. Although there has been an exponential development in computing in the last decades, it is still essential to design iterative algorithms that guarantee the convergence to the solution of a problem in a fast and efficient way. Following these assumptions, the main goal followed in the design of new iterative methods, being also one of the main objectives of this Doctoral Thesis, is the approximation of the solutions of nonlinear problems ensuring a certain balancing between the speed at which this approximation is obtained, the reliability of the approximation and the computational cost required in the whole iterative process. Being able to measure or quantify this balance is therefore one of the essential parts of this whole process. By means of the order of convergence, we are able to compare the speed with which the iterative schemes approximate the requested solution. The required computational cost for each iteration of the process is directly proportional to the number of iterations needed to approximate this solution. Therefore, accelerating the convergence of a method becomes a need in the design of efficient schemes. On the other hand, any iterative algorithm requires at least a starting point to begin the calculation of the successive iterations. For this reason, the study of the influence of initial estimates on the convergence of a method is also of high relevance, since it allows to determine the stability of the method depending on the initial iterations. This study is carried out using tools of discrete dynamics, both real and complex, to determine, in addition to other characterizations, the most suitable starting points and the most stable methods in a family of iterative schemes. The numerical and dynamical analysis carried out in this work makes it possible to propose efficient iterative methods that approximate solutions to nonlinear multidimensional problems and partial differential equations in an effective way. Based on the complete study developed using the tools described above, we present this Doctoral Thesis for gaining the title of Doctor in Mathematics.[CA] El diseny de mètodes iteratius per resoldre equacions i sistemes d'equacions no lineals es una tasca important i desafiant al domini de l'Anàlisi Numèric. La no linealitat és una característica de molts dels fenòmens físics. Mecànica de fluids i plasma, dinàmica de gasos, reaccions químiques, combustió, ecologia, biomecànica, problemes de models econòmics, teoria del transport i molts altres fenòmens estan tots governats inherentment per equacions no lineals. Per aquest motiu, una proporció cada vegada major de la investigació matemàtica moderna es dedica a l'anàlisi de sistemes i processos no lineals. En una era caracteritzada per la disponibilitat de grans quantitats de dades, el processat i anàlisi d'aquesta informació es tradueix de forma directa en la resolució de problemes la dimensió dels quals es cada vegada major. Malgrat que a les últimes dècades s'ha produït un desenvolupament exponencial a la computació, segueix sent essencial el diseny d'algorismes iteratius que garanteixen la convergència a la solució d'un problema de forma ràpida i eficient. Seguint aquestes premisses, l'objectiu fonamental que es persegueix amb el disseny de nous mètodes iteratius, sent també un dels principals objectius de la present Tesi Doctoral, és l'aproximació de solucions de problemes no lineals garantint un cert equilibri entre la velocitat amb què obtenen aquesta aproximació, la fiabilitat de la mateixa i el cost computacional requerit al conjunt de tot el procés iteratiu. Poder medir o quantificar aquest equilibri és, per tant, una de les peces essencials de tot aquest procés. Mitjançant l'ordre de convergència, tenim la capacitat de comparar la velocitat amb la qual els esquemes iteratius s'aproximen a la solució buscada. El cost computacional requerit a cada iteració del procés és directament proporcional al nombre d'iteracions que es necessiten per a aproximar aquesta solució. Per tant, accelerar la convergència d'un mètode es converteix en una necessitat al disseny d'esquemes eficients. D'una altra banda, tot algorisme iteratiu requereix d'almenys un punt inicial per començar el procés de càlcul de les iteracions successives. Per aquest motiu, l'estudi de la influència de les estimacions inicials a la convergència d'un mètode és també molt rellevant, ja que permet determinar l'estabilitat d'aquest en funció dels iterats inicials. Aquest estudi es realitza utilitzant eines de dinàmica discreta, tant real com complexa, per determinar, a més d'altres caracteritzacions, els punts inicials més adients i els mètodes més estables d'una familia d'esquemes iteratius. L'anàlisi numèric i dinàmic realitzat en aquesta memòria fa possible la proposta de mètodes iteratius eficients que aproximen solucions de problemes multidimensionals no lineals i equacions en derivades parcials de forma eficaç. A partir de l'estudi complet desenvolupat utilitzant les eines descrites anteriorment, presentem aquesta Tesi Doctoral per a l'obtenció del títol de Doctora en Matemàtiques.Garrido Saez, N. (2020). Diseño, análisis y estabilidad de métodos iterativos con memoria para la resolución de ecuaciones y sistemas no lineales [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/149573TESI

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

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

Generating root-finder iterative methods of second order: convergence and stability

[EN] In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are members of this family for particular choices of the weight function. The dynamical behavior of one of these choices is presented, analyzing the stability of the fixed points and the critical points of the rational function obtained when the iterative expression is applied on low degree polynomials. Several numerical tests are given to compare different elements of the proposed family on non-polynomial problems.This research was partially supported by the Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.Chicharro López, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2019). Generating root-finder iterative methods of second order: convergence and stability. Axioms. 8(2):1-14. https://doi.org/10.3390/axioms8020055S1148

Symmetry in the Multidimensional Dynamical Analysis of Iterative Methods with Memory

[EN] In this paper, new tools for the dynamical analysis of iterative schemes with memory for solving nonlinear systems of equations are proposed. These tools are in concordance with those of the scalar case and provide interesting results about the symmetry and wideness of the basins of attraction on different iterative procedures with memory existing in the literature.This research was partially supported by Grant PGC2018-095896-B-C22, funded by MCIN/ AEI/10.13039/5011000113033 by ERDF A way of making Europe, European Union; by the internal research project ADMIREN of Universidad Internacional de La Rioja (UNIR); and partially supported by Universitat Politècnica de València Contrato Predoctoral PAID-01-20-17 (UPV).Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR.; Triguero-Navarro, P. (2022). Symmetry in the Multidimensional Dynamical Analysis of Iterative Methods with Memory. Symmetry (Basel). 14(3):1-18. https://doi.org/10.3390/sym1403044211814

Dynamical analysis of a family of Traub-type iterative methods for solving nonlinear problems

This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and by the internal research project MICoCo of Universidad Internacional de La Rioja (UNIR).Chicharro, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2022). Dynamical analysis of a family of Traub-type iterative methods for solving nonlinear problems. Universitat Politècnica de València. 107-112. http://hdl.handle.net/10251/19242010711

Modifying Kurchatov's method to find multiple roots

Spanish Ministerio de Ciencia, Innovación y Universidades PGC2018-095896-B-C22, by the project ADMIREN of Universidad Internacional de La Rioja and was partially supported by Universitat Politècnica de València PAID-01-20-17 (UPV).Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR.; Triguero-Navarro, P. (2022). Modifying Kurchatov's method to find multiple roots. Universitat Politècnica de València. 257-262. http://hdl.handle.net/10251/19242225726

Iterative schemes for finding all roots simultaneously of nonlinear equations

[EN] In this paper, we propose a procedure that can be added to any iterative scheme in order to turn it into an iterative method for approximating all roots simultaneously of any nonlinear equations. By applying this procedure to any iterative method of order p, we obtain a new scheme of order of convergence 2p. Some numerical tests allow us to confirm the theoretical results and to compare the proposed schemes with other known methods for simultaneous roots of polynomial and non-polynomial functions.This research was partially supported by Grant PGC2018-095896-B-C22, funded by MCIN/AEI/10.13039/5011000113033 by " ERDF A way of making Europe", European Union, also by the internal research project Grant ADMIREN, Universidad Internacional de La Rioja (UNIR), Spain and by Contrato Predoctoral, Universitat Politecnica de Valencia, Spain PAID-01-20-17 (UPV) and Ayuda Movilidad Estancia Doctorado, Universitat Politecnica de Valencia, Spain 01/10/2021.Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR.; Triguero-Navarro, P. (2022). Iterative schemes for finding all roots simultaneously of nonlinear equations. Applied Mathematics Letters. 134:1-7. https://doi.org/10.1016/j.aml.2022.1083251713

Wide stability in a new family of optimal fourth-order iterative methods

"This is the peer reviewed version of the following article: Chicharro, 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, which has been published in final form at https://doi.org/10.1002/cmm4.1023. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] A new family of two¿steps fourth¿order iterative methods for solving nonlinear equations is introduced based on the weight functions procedure. This family is optimal in the sense of Kung¿Traub conjecture and it is extended to design a class of iterative schemes with four step and seventh order of convergence. We are interested in analyzing the dynamical behavior of different elements of the fourth¿order class. This analysis gives us important information about the stability of these members of the family. The methods are also tested with nonlinear functions and compared with other known schemes. The results show the good features of the introduced class.This research was partially supported by the Ministerio de Ciencia, Innovación y Universidades PGC-2018-095896-B-C22 and by the Generalitat Valenciana PROMETEO/2016/089. The authors also want to thank the anonymous referees for their suggestions and comments that have improved the final version of this paper.Chicharro López, FI.; Cordero Barbero, A.; Garrido-Saez, N.; Torregrosa Sánchez, JR. (2019). Wide stability in a new family of optimal fourth-order iterative methods. Computational and Mathematical Methods. 1(2):1-14. https://doi.org/10.1002/cmm4.1023S11412Amat, S., & Busquier, S. (Eds.). (2016). Advances in Iterative Methods for Nonlinear Equations. SEMA SIMAI Springer Series. doi:10.1007/978-3-319-39228-8Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Soleymani, F. (2011). Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations. Advances in Numerical Analysis, 2011, 1-10. doi:10.1155/2011/270903ChicharroFI CorderoA TorregrosaJR.Memory and dynamics for a family of King‐type iterative methods. In: Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE);2017;Rota Spain.Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Chicharro, F., Cordero, A., & Torregrosa, J. (2015). Dynamics and Fractal Dimension of Steffensen-Type Methods. Algorithms, 8(2), 271-279. doi:10.3390/a8020271Chicharro, 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:10.1155/2013/780153Varona, J. L. (2002). Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer, 24(1), 37-46. doi:10.1007/bf03025310Amat, S., Busquier, S., & Magreñán, Á. A. (2013). Reducing Chaos and Bifurcations in Newton-Type Methods. Abstract and Applied Analysis, 2013, 1-10. doi:10.1155/2013/72670