Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters

[EN] In this paper, we propose a family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity with the introduction of two free parameters and three univariate weight functions. Also numerical experiments have applied to a number of academical test functions and chemical problems for different special schemes from this family that satisfies the conditions given in convergence result.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Zafar, F.; Cordero Barbero, A.; Quratulain, R.; Torregrosa Sánchez, JR. (2018). Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. Journal of Mathematical Chemistry. 56(7):1884-1901. https://doi.org/10.1007/s10910-017-0813-1S18841901567R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput. 265(15), 520–532 (2015)R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, V. Kanwar, An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algor. 71(4), 775–796 (2016)R. Behl, A. Cordero, S.S. Motsa, J.R. Torregrosa, An eighth-order family of optimal multiple root finders and its dynamics. Numer. Algor. (2017). doi: 10.1007/s11075-017-0361-6F.I. Chicharro, A. Cordero, J. R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. ID 780153 (2013)A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications (Prentice Hall PTR, New Jersey, 1999)J.M. Douglas, Process Dynamics and Control, vol. 2 (Prentice Hall, Englewood Cliffs, 1972)Y.H. Geum, Y.I. Kim, B. Neta, A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Appl. Math. Comput. 270, 387–400 (2015)Y.H. Geum, Y.I. Kim, B. Neta, A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 283, 120–140 (2016)J.L. Hueso, E. Martınez, C. Teruel, Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)L.O. Jay, A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)S. Li, X. Liao, L. Cheng, A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)S.G. Li, L.Z. Cheng, B. Neta, Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)B. Liu, X. Zhou, A new family of fourth-order methods for multiple roots of nonlinear equations. Nonlinear Anal. Model. Control 18(2), 143–152 (2013)M. Shacham, Numerical solution of constrained nonlinear algebraic equations. Int. J. Numer. Method Eng. 23, 1455–1481 (1986)M. Sharifi, D.K.R. Babajee, F. Soleymani, Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)J.R. Sharma, R. Sharma, Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)F. Soleymani, D.K.R. Babajee, T. Lofti, On a numerical technique forfinding multiple zeros and its dynamic. J. Egypt. Math. Soc. 21, 346–353 (2013)F. Soleymani, D.K.R. Babajee, Computing multiple zeros using a class of quartically convergent methods. Alex. Eng. J. 52, 531–541 (2013)R. Thukral, A new family of fourth-order iterative methods for solving nonlinear equations with multiple roots. J. Numer. Math. Stoch. 6(1), 37–44 (2014)R. Thukral, Introduction to higher-order iterative methods for finding multiple roots of nonlinear equations. J. Math. Article ID 404635 (2013)X. Zhou, X. Chen, Y. Song, Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. J. Comput. Math. Appl. 235, 4199–4206 (2011)X. Zhou, X. Chen, Y. Song, Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013

A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

[EN] The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev¿Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for ¿ = 2,which corresponds to an optimal method in the sense of Kung and Traub¿s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.This research was partially supported by Ministerio de Economia y Competitividad under Grant MTM2014-52016-C2-1-2-P and by the project of Generalitat Valenciana Prometeo/2016/089Behl, R.; Martínez Molada, E.; Cevallos-Alarcon, FA.; Alarcon-Correa, D. (2019). A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots. Mathematics. 7(4):1-12. https://doi.org/10.3390/math7040339S11274Gutiérrez, J. M., & Hernández, M. A. (1997). A family of Chebyshev-Halley type methods in Banach spaces. Bulletin of the Australian Mathematical Society, 55(1), 113-130. doi:10.1017/s0004972700030586Kanwar, V., Singh, S., & Bakshi, S. (2008). Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations. Numerical Algorithms, 47(1), 95-107. doi:10.1007/s11075-007-9149-4Argyros, I. K., Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., & Hilout, S. (2011). On the semilocal convergence of efficient Chebyshev–Secant-type methods. Journal of Computational and Applied Mathematics, 235(10), 3195-3206. doi:10.1016/j.cam.2011.01.005Xiaojian, Z. (2008). Modified Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 203(2), 824-827. doi:10.1016/j.amc.2008.05.092Amat, S., Hernández, M. A., & Romero, N. (2008). A modified Chebyshev’s iterative method with at least sixth order of convergence. Applied Mathematics and Computation, 206(1), 164-174. doi:10.1016/j.amc.2008.08.050Kou, J., & Li, Y. (2007). Modified Chebyshev–Halley methods with sixth-order convergence. Applied Mathematics and Computation, 188(1), 681-685. doi:10.1016/j.amc.2006.10.018Li, D., Liu, P., & Kou, J. (2014). An improvement of Chebyshev–Halley methods free from second derivative. Applied Mathematics and Computation, 235, 221-225. doi:10.1016/j.amc.2014.02.083Sharma, J. R. (2015). Improved Chebyshev–Halley methods with sixth and eighth order convergence. Applied Mathematics and Computation, 256, 119-124. doi:10.1016/j.amc.2015.01.002Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Hueso, 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/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2015). On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathematics and Computation, 265, 520-532. doi:10.1016/j.amc.2015.05.004Behl, R., Cordero, A., Motsa, S. S., Torregrosa, J. R., & Kanwar, V. (2015). An optimal fourth-order family of methods for multiple roots and its dynamics. Numerical Algorithms, 71(4), 775-796. doi:10.1007/s11075-015-0023-5Geum, Y. H., Kim, Y. I., & Neta, B. (2015). A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Applied Mathematics and Computation, 270, 387-400. doi:10.1016/j.amc.2015.08.039Geum, Y. H., Kim, Y. I., & Neta, B. (2016). A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Applied Mathematics and Computation, 283, 120-140. doi:10.1016/j.amc.2016.02.029Behl, R., Alshomrani, A. S., & Motsa, S. S. (2018). An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence. Journal of Mathematical Chemistry, 56(7), 2069-2084. doi:10.1007/s10910-018-0857-xMcNamee, 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:10.1145/290590.290592Cordero, 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

Local convergence balls for nonlinear problems with multiplicity and their extension to eight-order of convergence

[EN] The main contribution of this study is to present a new optimal eighth-order scheme for locating zeros with multiplicity m > 1. An extensive convergence analysis is presented with the main theorem in order to demonstrate the optimal eighth-order convergence of the proposed scheme. Moreover, a local convergence study for the optimal fourth-order method defined by the first two steps of the new method is presented, allowing us to obtain the radius of the local convergence ball. Finally, numerical tests on some real-life problems, such as a Van der Waals equation of state, a conversion Chemical engineering problem and two standard academic test problems are presented, which confirm the theoretical results established in this paper and the efficiency of this proposed iterative method. We observed from the numerical experiments that our proposed iterative methods have good values for convergence radii. Further, they have not only faster convergence towards the desired zero of the involved function but they also have both smaller residual error and a smaller difference between two consecutive iterations than current existing techniques.This research was partially supported by Ministerio de Economia y Competitividad under grant MTM2014-52016-C2-2-P and by the project of Generalitat Valenciana Prometeo/2016/089.Behl, R.; Martínez Molada, E.; Cevallos-Alarcon, FA.; Alshomrani, AS. (2019). Local convergence balls for nonlinear problems with multiplicity and their extension to eight-order of convergence. Mathematical Problems in Engineering. 2019:1-18. https://doi.org/10.1155/2019/1427809S1182019Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2013). Basic concepts. Multipoint Methods, 1-26. doi:10.1016/b978-0-12-397013-8.00001-7Shengguo, L., Xiangke, L., & Lizhi, C. (2009). A new fourth-order iterative method for finding multiple roots of nonlinear equations. Applied Mathematics and Computation, 215(3), 1288-1292. doi:10.1016/j.amc.2009.06.065Neta, B. (2010). Extension of Murakami’s high-order non-linear solver to multiple roots. International Journal of Computer Mathematics, 87(5), 1023-1031. doi:10.1080/00207160802272263Li, S. G., Cheng, L. Z., & Neta, B. (2010). Some fourth-order nonlinear solvers with closed formulae for multiple roots. Computers & Mathematics with Applications, 59(1), 126-135. doi:10.1016/j.camwa.2009.08.066Zhou, X., Chen, X., & Song, Y. (2011). Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics, 235(14), 4199-4206. doi:10.1016/j.cam.2011.03.014Sharifi, M., Babajee, D. K. R., & Soleymani, F. (2012). Finding the solution of nonlinear equations by a class of optimal methods. Computers & Mathematics with Applications, 63(4), 764-774. doi:10.1016/j.camwa.2011.11.040Soleymani, F., & Babajee, D. K. R. (2013). Computing multiple zeros using a class of quartically convergent methods. Alexandria Engineering Journal, 52(3), 531-541. doi:10.1016/j.aej.2013.05.001Soleymani, F., Babajee, D. K. R., & Lotfi, T. (2013). On a numerical technique for finding multiple zeros and its dynamic. Journal of the Egyptian Mathematical Society, 21(3), 346-353. doi:10.1016/j.joems.2013.03.011Zhou, X., Chen, X., & Song, Y. (2013). Families of third and fourth order methods for multiple roots of nonlinear equations. Applied Mathematics and Computation, 219(11), 6030-6038. doi:10.1016/j.amc.2012.12.041Hueso, 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/s10910-014-0460-8Behl, R., Cordero, A., Motsa, S. S., & Torregrosa, J. R. (2015). On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathematics and Computation, 265, 520-532. doi:10.1016/j.amc.2015.05.004Zafar, F., Cordero, A., Quratulain, R., & Torregrosa, J. R. (2017). Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. Journal of Mathematical Chemistry, 56(7), 1884-1901. doi:10.1007/s10910-017-0813-1Geum, Y. H., Kim, Y. I., & Neta, B. (2018). Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points. Journal of Computational and Applied Mathematics, 333, 131-156. doi:10.1016/j.cam.2017.10.033Geum, Y. H., Kim, Y. I., & Magreñán, Á. A. (2018). A study of dynamics via Möbius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions. Journal of Computational and Applied Mathematics, 344, 608-623. doi:10.1016/j.cam.2018.06.006Chun, C., & Neta, B. (2015). An analysis of a family of Maheshwari-based optimal eighth order methods. Applied Mathematics and Computation, 253, 294-307. doi:10.1016/j.amc.2014.12.064Thukral, R. (2013). Introduction to Higher-Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations. Journal of Mathematics, 2013, 1-3. doi:10.1155/2013/404635Geum, Y. H., Kim, Y. I., & Neta, B. (2016). A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Applied Mathematics and Computation, 283, 120-140. doi:10.1016/j.amc.2016.02.029Argyros, 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:10.1080/0020716021000059160Zhou, 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/s11075-013-9702-2Bi, W., Ren, H., & Wu, Q. (2011). Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative. Numerical Algorithms, 58(4), 497-512. doi:10.1007/s11075-011-9466-5Zhou, X., & Song, Y. (2014). Convergence radius of Osada’s method under center-Hölder continuous condition. Applied Mathematics and Computation, 243, 809-816. doi:10.1016/j.amc.2014.06.068Cordero, 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.2007.01.062Balaji, 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. (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-

Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics

[EN] In this paper, we propose a family of iterative methods for finding multiple roots, with known multiplicity, by means of the introduction of four univariate weight functions. With the help of these weight functions, that play an important role in the development of higher order convergent iterative techniques, we are able to construct three-point eight-order optimal multiple-root finders. Also, numerical experiments have been applied to a number of test equations for different special schemes from this family satisfying the conditions given in the convergence analysis. We have also compared the basins of attraction of some proposed and known methods in order to check the wideness of the sets of converging initial points for each problem. (C) 2018 Elsevier B.V. All rights reserved.This research was partially supported by Ministerio de Economia y Competitividad, Spain MTM2014-52016-C2-2-P, MTM2015-64013-P and Generalitat Valenciana, Spain PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Sultana, S.; Torregrosa Sánchez, JR. (2018). Optimal iterative methods for finding multiple roots of nonlinear equations using weight functions and dynamics. Journal of Computational and Applied Mathematics. 342:352-374. https://doi.org/10.1016/j.cam.2018.03.033S35237434

Derivative-Free King's Scheme for Multiple Zeros of Nonlinear Functions

[EN] There is no doubt that the fourth-order King's family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King's family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m >= 2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung-Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.This research was partially supported by the project PGC2018-095896-B-C22 of the Spanish Ministry of Economy and Competitiveness.Behl, R.; Bhalla, S.; Martínez Molada, E.; Alsulami, MA. (2021). Derivative-Free King's Scheme for Multiple Zeros of Nonlinear Functions. Mathematics. 9(11):1-14. https://doi.org/10.3390/math9111242S11491

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

An optimal sixteenth order family of methods for solving nonlinear equations and their basins of attraction

We propose a new family of iterative methods for finding the simple roots of nonlinear equation. The proposed method is four-point method with convergence order 16, which consists of four steps: the Newton step, an optional fourth order iteration scheme, an optional eighth order iteration scheme and the step constructed using the divided difference. By reason of the new iteration scheme requiring four function evaluations and one first derivative evaluation per iteration, the method satisfies the optimality criterion in the sense of Kung-Traub\u27s conjecture and achieves a high efficiency index $16^{1/5} approx 1.7411$. Computational results support theoretical analysis and confirm the efficiency. The basins of attraction of the new presented algorithms are also compared to the existing methods with encouraging 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