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

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

On the optimality of some multi-point methods for finding multiple roots of nonlinear equation

This paper deals with the problem of determining the multiple roots of nonlinear equations, where the multiplicity of the roots is known. The paper contains some remarks on the optimality of the recently published methods [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] and [X. Zhou, X. Chen, Y. Song, Families of third- and fourth-order methods for multiple roots of nonlinear equations, Appl. Math. Comput., 219(11):6030–6038, 2013]. Separate analysis of odd and even multiplicity, has shown the cases where those methods lose their optimal convergence properties. Numerical experiments are made and they support theoretical analysis

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

On Some Optimal Multiple Root-Finding Methods and their Dynamics

Finding multiple zeros of nonlinear functions pose many difficulties for many of the iterative methods. In this paper, we present an improved optimal class of higher-order methods for multiple roots having quartic convergence. The present approach of deriving an optimal class is based on weight function approach. In terms of computational cost, all the proposed methods require three functional evaluations per full iteration, so that their efficiency indices are 1.587 and, are optimal in the sense of Kung-Traub conjecture. It is found by way of illustrations that they are useful in high precision computing enviroments. Moreover, basins of attraction of some of the higher-order methods in the complex plane are also given

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

Third-Order and Fourth-Order Iterative Methods Free from Second Derivative for Finding Multiple Roots of Nonlinear Equations

Abstract. In this paper, we present two new families of thirdorder and fourth-order methods for finding multiple roots of nonlinear equations. Each of them requires one evaluation of the function and two of its first derivative per iteration. Several numerical examples are given to illustrate the performance of the presented methods

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

Higher-order families of multiple root finding methods suitable for non-convergent cases and their dynamics

In this paper, we present many new one-parameter families of classical Rall’s method (modified Newton’s method), Schröder’s method, Halley’s method and super-Halley method for the first time which will converge even though the guess is far away from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods

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. 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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. 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