A preconditioned iterative method for solving systems of nonlinear equations having unknown multiplicity

A modification to an existing iterative method for computing zeros with unknown multiplicities of nonlinear equations or a system of nonlinear equations is presented. We introduce preconditioners to nonlinear equations or a system of nonlinear equations and their corresponding Jacobians. The inclusion of preconditioners provides numerical stability and accuracy. The different selection of preconditioner offers a family of iterative methods. We modified an existing method in a way that we do not alter its inherited quadratic convergence. Numerical simulations confirm the quadratic convergence of the preconditioned iterative method. The influence of preconditioners is clearly reflected in the numerically achieved accuracy of computed solutions.Peer ReviewedPostprint (published version

Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications

A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases). To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method. To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively. Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices. Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective

Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations

[EN] There is a few number of optimal fourth-order iterative methods for obtaining the multiple roots of nonlinear equations. But, in most of the earlier studies, scholars gave the flexibility in their proposed schemes only at the second step (not at the first step) in order to explore new schemes. Unlike what happens in existing methods, the main aim of this manuscript is to construct a new fourth-order optimal scheme which will give the flexibility to the researchers at both steps as well as faster convergence, smaller residual errors and asymptotic error constants. The construction of the proposed scheme is based on the mid-point formula and weight function approach. From the computational point of view, the stability of the resulting class of iterative methods is studied by means of the conjugacy maps and the analysis of strange fixed points. Their basins of attractions and parameter planes are also given to show their dynamical behavior around the multiple roots. Finally, we consider a real-life problem and a concrete variety of standard test functions for numerical experiments and relevant results are extensively treated to confirm the theoretical development.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Behl, R.; Cordero Barbero, A.; Motsa, SS.; Torregrosa Sánchez, JR. (2018). Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations. Nonlinear Dynamics. 91(1):81-112. https://doi.org/10.1007/s11071-017-3858-6S81112911Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R., Kanwar, V.: An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algorithms 71(4), 775–796 (2016)Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. Am. Math. Soc. 11(1), 85–141 (1984)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013(2013), 1–11 (2013)Devaney, R.L.: The Mandelbrot Set, the Farey Tree and the Fibonacci sequence. Am. Math. Mon. 106(4), 289–302 (1999)Dong, C.: A family of multipoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)Hueso, J.L., Martínez, E., Teruel, C.: Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)Li, S.G., Cheng, L.Z., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)Li, S., Liao, X., Cheng, L.: A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)Petković, M.S., Neta, B., Petković, L.D., Dz̆unić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, New York (2013)Sbibih, D., Serghini, A., Tijini, A., Zidna, A.: A general family of third order method for finding multiple roots. AMC 233, 338–350 (2014)Schröder, E.: Über unendlichviele Algorithm zur Auffosung der Gleichungen. Math. Ann. 2, 317–365 (1870)Sharifi, M., Babajee, D.K.R., Soleymani, F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)Soleymani, F., Babajee, D.K.R.: Computing multiple zeros using a class of quartically convergent methods. Alex. Eng. J. 52, 531–541 (2013)Soleymani, F., Babajee, D.K.R., Lofti, T.: On a numerical technique forfinding multiple zeros and its dynamic. J. Egypt. Math. Soc. 21, 346–353 (2013)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)Zhou, X., Chen, X., Song, Y.: Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013)Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011

A note on "Convergence radius of Osada s method under Hölder continuous condition"

[EN] In this paper we revise the proofs of the results obtained in "Convergence radius of Osada's method under Holder continuous condition"[4], because the remainder of the Taylor's expansion used for the obtainment of the local convergence radius is not correct. So we perform the complete study in order to modify the equation for getting the local convergence radius, the uniqueness radius and the error bounds. Moreover a dynamical study for the third order Osada's method is also developed. (C) 2017 Elsevier Inc. All rights reserved.Hueso, J.; Martínez Molada, E.; Gupta, D.; Cevallos-Alarcon, FA. (2018). A note on "Convergence radius of Osada s method under Hölder continuous condition". Applied Mathematics and Computation. 321:689-699. https://doi.org/10.1016/j.amc.2017.11.003S68969932

Equations and systems of nonlinear equations: from high order numerical methods to fast Eigensolvers for structured matrices and applications

A parametrized multi-step Newton method is constructed for widening the region of convergence of classical multi-step Newton method. The second improvement is proposed in the context of multistep Newton methods, by introducing preconditioners to enhance their accuracy, without disturbing their original order of convergence and the related computational cost (in most of the cases). To find roots with unknown multiplicities preconditioners are also effective when they are applied to the Newton method for roots with unknown multiplicities. Frozen Jacobian higher order multistep iterative method for the solution of systems of nonlinear equations are developed and the related results better than those obtained when employing the classical frozen Jacobian multi-step Newton method. To get benefit from the past information that is produced by the iterative method, we constructed iterative methods with memory for solving systems of nonlinear equations. Iterative methods with memory have a greater rate of convergence, if compared with the iterative method without memory. In terms of computational cost, iterative methods with memory are marginally superior comparatively. Numerical methods are also introduced for approximating all the eigenvalues of banded symmetric Toeplitz and preconditioned Toeplitz matrices. Our proposed numerical methods work very efficiently, when the generating symbols of the considered Toeplitz matrices are bijective

An alternative analysis for the local convergence of iterative methods for multiple roots including when the multiplicity is unknown

[EN] In this paper we propose an alternative for the study of local convergence radius and the uniqueness radius for some third-order methods for multiple roots whose multiplicity is known. The main goal is to provide an alternative that avoids the use of sophisticated properties of divided differences that are used in already published papers about local convergence for multiple roots. We defined the local study by using a technique taking into consideration a bounding condition for the derivative of the function with i=1,2. In the case that the method uses first and second derivative in its iterative expression and i=1 in case the method only uses first derivative. Furthermore we implement a numerical analysis in the following sense. Since the radius of local convergence for high-order methods decreases with the order, we must take into account the analysis of ITS behaviour when we introduce a new iterative method. Finally, we have used these iterative methods for multiple roots for the case where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behaviour of the corresponding estimations and how this fact affect to the original method.This work was supported by Secretaria de Educacion Superior, Ciencia, Tecnologia e Innovacion (Convocatoria Abierta 2015 fase II).Alarcon, D.; Hueso, JL.; Martínez Molada, E. (2020). An alternative analysis for the local convergence of iterative methods for multiple roots including when the multiplicity is unknown. International Journal of Computer Mathematics. 97(1-2):312-329. https://doi.org/10.1080/00207160.2019.1589460S312329971-2Argyros, I. (2003). On The Convergence And Application Of Newton’s Method Under Weak HÖlder Continuity Assumptions. International Journal of Computer Mathematics, 80(6), 767-780. doi:10.1080/0020716021000059160Hueso, J. L., Martínez, E., & Teruel, C. (2014). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry, 53(3), 880-892. doi:10.1007/s10910-014-0460-8McNamee, J. M. (1998). A comparison of methods for accelerating convergence of Newton’s method for multiple polynomial roots. ACM SIGNUM Newsletter, 33(2), 17-22. doi:10.1145/290590.290592Ortega, J. M. (1974). Solution of Equations in Euclidean and Banach Spaces (A. M. Ostrowski). SIAM Review, 16(4), 564-564. doi:10.1137/1016102Osada, N. (1994). An optimal multiple root-finding method of order three. Journal of Computational and Applied Mathematics, 51(1), 131-133. doi:10.1016/0377-0427(94)00044-1Schr�der, E. (1870). Ueber unendlich viele Algorithmen zur Aufl�sung der Gleichungen. Mathematische Annalen, 2(2), 317-365. doi:10.1007/bf01444024Vander Stracten, M., & Van de Vel, H. (1992). Multiple root-finding methods. Journal of Computational and Applied Mathematics, 40(1), 105-114. doi:10.1016/0377-0427(92)90045-yZhou, X., Chen, X., & Song, Y. (2013). On the convergence radius of the modified Newton method for multiple roots under the center–Hölder condition. Numerical Algorithms, 65(2), 221-232. doi:10.1007/s11075-013-9702-