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

Stability analysis of a family of optimal fourth-order methods for multiple roots

[EN] Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m >1. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2019). Stability analysis of a family of optimal fourth-order methods for multiple roots. Numerical Algorithms. 81(3):947-981. https://doi.org/10.1007/s11075-018-0577-0S94798181

Fixed point root-finding methods of fourth-order of convergence

[EN] In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub¿s conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Guasp, L.; Torregrosa Sánchez, JR. (2019). Fixed point root-finding methods of fourth-order of convergence. Symmetry (Basel). 11(6):1-15. https://doi.org/10.3390/sym11060769S115116Van Sosin, B., & Elber, G. (2017). Solving piecewise polynomial constraint systems with decomposition and a subdivision-based solver. Computer-Aided Design, 90, 37-47. doi:10.1016/j.cad.2017.05.023Kung, 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., 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-zJarratt, P. (1966). Some fourth order multipoint iterative methods for solving equations. Mathematics of Computation, 20(95), 434-434. doi:10.1090/s0025-5718-66-99924-8Sharma, J. R., & Arora, H. (2013). Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo, 51(1), 193-210. doi:10.1007/s10092-013-0097-1Hueso, J. L., Martínez, E., & Teruel, C. (2015). Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems. Journal of Computational and Applied Mathematics, 275, 412-420. doi:10.1016/j.cam.2014.06.010Ghorbanzadeh, M., & Soleymani, F. (2015). A Quartically Convergent Jarratt-Type Method for Nonlinear System of Equations. Algorithms, 8(3), 415-423. doi:10.3390/a8030415Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Blanchard, P. (1995). The dynamics of Newton’s method. Proceedings of Symposia in Applied Mathematics, 139-154. doi:10.1090/psapm/049/1315536Chicharro, 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

Modified Potra-Pták multi-step schemes with accelerated order of convergence for solving sistems of nonlinear equations

[EN] In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Ptak method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3r + 6, r = 0, 1, 2, .... The sixth order method is the special case of this multi-step scheme for r = 0. The family gives a four-step ninth order method for r = 1. As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.This research was partially supported by Ministerio de Economia y Competitividad under grants MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Arora, H.; Torregrosa Sánchez, JR.; Cordero Barbero, A. (2019). Modified Potra-Pták multi-step schemes with accelerated order of convergence for solving sistems of nonlinear equations. Mathematical and Computational Applications (Online). 24(1):1-15. https://doi.org/10.3390/mca24010003S115241Homeier, H. H. . (2004). A modified Newton method with cubic convergence: the multivariate case. Journal of Computational and Applied Mathematics, 169(1), 161-169. doi:10.1016/j.cam.2003.12.041Darvishi, M. T., & Barati, A. (2007). A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Applied Mathematics and Computation, 188(1), 257-261. doi:10.1016/j.amc.2006.09.115Cordero, 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-zCordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2011). Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation, 217(9), 4548-4556. doi:10.1016/j.amc.2010.11.006Grau-Sánchez, M., Grau, À., & Noguera, M. (2011). On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 236(6), 1259-1266. doi:10.1016/j.cam.2011.08.008Grau-Sánchez, M., Grau, À., & Noguera, M. (2011). Ostrowski type methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 218(6), 2377-2385. doi:10.1016/j.amc.2011.08.011Grau-Sánchez, M., Noguera, M., & Amat, S. (2013). On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. Journal of Computational and Applied Mathematics, 237(1), 363-372. doi:10.1016/j.cam.2012.06.005Sharma, J. R., & Arora, H. (2013). On efficient weighted-Newton methods for solving systems of nonlinear equations. Applied Mathematics and Computation, 222, 497-506. doi:10.1016/j.amc.2013.07.066Cordero, 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.2007.01.062Chicharro, 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

Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems

[EN] For solving nonlinear systems of big size, such as those obtained by applying finite differences for approximating the solution of diffusion problem and heat conduction equations, three-step iterative methods with eighth-order local convergence are presented. The computational efficiency of the new methods is compared with those of some known ones, obtaining good conclusions, due to the particular structure of the iterative expression of the proposed methods. Numerical comparisons are made with the same existing methods, on standard nonlinear systems and a nonlinear one-dimensional heat conduction equation by transforming it in a nonlinear system by using finite differences. From these numerical examples, we confirm the theoretical results and show the performance of the presented schemes.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and by Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Gómez, E.; Torregrosa Sánchez, JR. (2017). Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems. Complexity. 1-11. https://doi.org/10.1155/2017/6457532S11

Chaos and convergence of a family generalizing Homeier's method with damping parameters

[EN] In this paper, a family of parametric iterative methods for solving nonlinear equations, including Homeier's scheme, is presented. Its local convergence is obtained and the dynamical behavior on quadratic polynomials of the resulting family is studied in order to choose those values of the parameter that ensure stable behavior. To get this aim, the analysis of fixed and critical points and the associated parameter plane show the dynamical richness of the family and allow us to find members of this class with good numerical properties and also other ones with pathological conduct. To check the stable behavior of the good selected ones, the discretized planar 1D-Bratu problem is solved. Some of those chosen members of the family achieve good results when Homeier's scheme fails.This research was supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P.Cordero Barbero, A.; Franques, A.; Torregrosa Sánchez, JR. (2016). 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Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations

[EN] The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.This research was partially supported by Ministerio de Ciencia, Innovación y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089Cordero Barbero, A.; Jaiswal, J.; Torregrosa Sánchez, JR. (2019). Stability analysis of fourth-order iterative method for finding multiple roots of nonlinear equations. Applied Mathematics and Nonlinear Sciences. 4(1):43-56. https://doi.org/10.2478/AMNS.2019.1.00005S435641Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.32186

Memorizing Schroder's Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

[EN] In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schroder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots.This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Cordero Barbero, A.; Neta, B.; Torregrosa Sánchez, JR. (2021). Memorizing Schroder's Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity. Mathematics. 9(20):1-13. https://doi.org/10.3390/math9202570S11392

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

Study of iterative methods though the Cayley quadratic test

[EN] Many iterative methods for solving nonlinear equations have been developed recently. The main advantage claimed by their authors is the improvement of the order of convergence. In this work, we compare their dynamical behavior on quadratic polynomials with the one of Newton's scheme. This comparison is defined in what we call Cayley Quadratic Test (CQT) which can be used as a first test to check the efficiency of such methods. Moreover we make a brief insight in cubic polynomials. (C) 2014 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Babajee, D.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2016). Study of iterative methods though the Cayley quadratic test. Journal of Computational and Applied Mathematics. 291:358-369. https://doi.org/10.1016/j.cam.2014.09.020S35836929