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

Optimal eighth-order iterative methods for approximating multiple zeros of nonlinear functions

[EN] It is well known that the optimal iterative methods are of more significance than the non-optimal ones in view of their efficiency and convergence speed. There are only a few number of optimal iterative methods for finding multiple zeros with eighth order of convergence. In this paper, we propose a new family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity. We present an extensive convergence analysis which confirms theoretically eighth-order convergence of the presented scheme. Moreover, we consider several real life problems that contain simple as well as multiple zeros in order to compare our proposed methods with the existing eighth-order iterative schemes. Some dynamical aspects of the presented methods are also discussed. Finally, we conclude on the basis of obtained numerical results that the proposed family of iterative methods perform better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Junjua, M.; Torregrosa Sánchez, JR. (2020). Optimal eighth-order iterative methods for approximating multiple zeros of nonlinear functions. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 114(2):1-17. https://doi.org/10.1007/s13398-020-00794-7S1171142Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R.: On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput. 265(15), 520–532 (2015)Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R.: An eighth-order family of optimal multiple root finders and its dynamics. Numer. Algorithm (2017). https://doi.org/10.1007/s11075-017-0361-6Behl, 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. Algorithm 71(4), 775–796 (2016)Chicharro, F.I., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. (2013) (Article ID 780153, 9 pages)Geum, Y.H., Kim, Y.I., Neta, B.: 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)Geum, Y.H., Kim, Y.I., Neta, B.: 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)Geum, Y.H., Kim, Y.I., Neta, B.: Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with dynamics be-hind their purely imaginary extraneous fixed points. J. Comput. Appl. Math. 333, 131–156 (2018)Hueso, J.L., Martínez, E., Teruel, C.: Determination of multiple roots of nonlinear equations and applications. J. Math. Chem. 53, 880–892 (2015)Jay, L.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. 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)Liu, B., Zhou, X.: A new family of fourth-order methods for multiple roots of nonlinear equations. Nonlinear Anal. Model. Control 18(2), 143–152 (2013)Neta, B.: Extension of Murakami’s high-order non-linear solver to multiple roots. Int. J. Comput. Math. 87(5), 1023–1031 (2010)Ostrowski, A.M.: Solution of Equations and Systems of Equations. Academic Press, New York (1960)Petković, M.S., Neta, B., Petković, L.D., Dz̆unić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, New York (2013)Shacham, M.: Numerical solution of constrained nonlinear algebraic equations. Int. J. Numer. Method Eng. 23, 1455–1481 (1986)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)Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)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 for finding multiple zeros and its dynamic. Egypt. Math. Soc. 21, 346–353 (2013)Thukral, R.: Introduction to higher-order iterative methods for finding multiple roots of nonlinear equations. J. Math. (2013). https://doi.org/10.1155/2013/404635. (Article ID 404635, 3 pages)Thukral, R.: A new family of fourth-order iterative methods for solving nonlinear equations with multiple roots. Numer. Math. Stoch. 6(1), 37–44 (2014)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)Zafar, F., Cordero, A., Quratulain, R., Torregrosa, J.R.: Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. J. Math. Chem. (2017). https://doi.org/10.1007/s10910-017-0813-1Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the muliplte roots of nonlinear equations. Comput. Math. Appl. 235, 4199–4206 (2011)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

Second derivative free sixth order continuation method for solving nonlinear equations with applications

In this paper, we deal with the study of convergence analysis of modified parameter based family of second derivative free continuation method for solving nonlinear equations. We obtain the order of convergence is at least five and especially, for parameter α=2 sixth order convergence. Some application such as Max Planck’s conservative law, multi-factor effect are discussed and demonstrate the efficiency and performance of the new method (for α=2 ). We compare the absolutely value of function at each iteration |f(xn)| and |xn−ξ| with our method and Potra and Pták method [1], Kou et al. method [2]. We observed that our method is more efficient than existing methods. Also, the Dynamics of the method are studied for a special case of the parameter in convergence

Some new bi-accelerator two-point methods for solving nonlinear equations

In this work, we extract some new and efficient two-point methods with memory from their corresponding optimal methods without memory, to estimate simple roots of a given nonlinear equation. Applying two accelerator parameters in each iteration, we try to increase the convergence order from four to seven without any new functional evaluation. To this end, firstly we modify three optimal methods without memory in such a way that we could generate methods with memory as efficient as possible. Then, convergence analysis is put forward. Finally, the applicability of the developed methods on some numerical examples is examined and illustrated by means of dynamical tools, both in smooth and in nonsmooth functions.The authors thank to the anonymous referees for their suggestions to improve the final version of the paper. The second author would like to thank Hamedan Brach of Islamic Azad University for partial financial support in this research.Cordero Barbero, A.; Lotfi, T.; Torregrosa Sánchez, JR.; Assari, P.; Mahdiani, K. (2016). Some new bi-accelerator two-point methods for solving nonlinear equations. Computational and Applied Mathematics. 35(1):251-267. doi:10.1007/s40314-014-0192-1S251267351Babajee DKR (2012) Several improvements of the 2-point third order midpoint iterative method using weight functions. Appl Math Comput 218:7958–7966Chicharro FI, Cordero A, Torregrosa JR (2013) Drawing dynamical and parameters planes of iterative families and methods. Sci World J. Article ID 780153, 11 ppChun C, Lee MY (2013) A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Appl Math Comput 223:506–519Cordero A, Hueso JL, Martínez E, Torregrosa JR (2010) New modifications of Potra–Ptàk’s method with optimal fourth and eighth orders of convergence. J Comput Appl Math 234:2969–2976Cordero A, Lotfi T, Bakhtiari P, Torregrosa JR (2014) An efficient two-parametric family with memory for nonlinear equations. Numer Algor. doi: 10.1007/s11075-014-9846-8Geum YH, Kim YI (2011) A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots. Appl Math Lett 24:929–935Heydari M, Hosseini SH, Loghmani GB (2011) On two new families of iterative methods for solving nonlinear equations with optimal order. Appl Anal Discret Math 5:93–109Jay IO (2001) A note on Q-order of convergence. BIT Numer Math 41:422–429Khattri SK, Steihaug T (2013) Algorithm for forming derivative-free optimal methods. Numer Algor. doi: 10.1007/s11075-013-9715-xKou J, Wang X, Li Y (2010) Some eighth-order root-finding three-step methods. Commun Nonlinear Sci Numer Simul 15:536–544Kung HT, Traub JF (1974) Optimal order of one-point and multipoint iteration. J Assoc Comput Math 21:634–651Liu X, Wang X (2012) A convergence improvement factor and higher-order methods for solving nonlinear equations. Appl Math Comput 218:7871–7875Lotfi T, Tavakoli E (2014) On a new efficient Steffensen-like iterative class by applying a suitable self-accelerator parameter. Sci World J. Article ID 769758, 9 pp. doi: 10.1155/2014/769758Lotfi T, Soleymani F, Shateyi S, Assari P, Khaksar Haghani F (2014a) New mono- and biaccelerator iterative methods with memory for nonlinear equations. Abstr Appl Anal. Article ID 705674, 8 pp. doi: 10.1155/2014/705674Lotfi T, Soleymani F, Noori Z, Kiliman A, Khaksar Haghani F (2014b) Efficient iterative methods with and without memory possessing high efficiency indices. Discret Dyn Nat Soc. Article ID 912796, 9 pp. doi: 10.1155/2014/912796Magreñan AA (2014) A new tool to study real dynamics: the convergence plane. arXiv:1310.3986 [math.NA]Ortega JM, Rheimbolt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New YorkOstrowski AM (1966) Solutions of equations and systems of equations. Academic Press, New York-LondonPetković MS, Ilić S, Džunić J (2010) Derivative free two-point methods with and without memory for solving nonlinear equations. Appl Math Comput 217(5):1887–1895Petković MS, Neta B, Petković LD, Džunić J (2014) Multipoint methods for solving nonlinear equations: a survey. Appl Math Comput 226(2):635–660Ren H, Wu Q, Bi W (2009) A class of two-step Steffensen type methods with fourth-order convergence. Appl Math Comput 209:206–210Soleymani F, Sharifi M, Mousavi S (2012) An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. J Optim Theory Appl 153:225–236Soleimani F, Soleymani F, Shateyi S (2013) Some iterative methods free from derivatives and their basins of attraction for nonlinear equations. Discret Dyn Nat Soc. Article ID 301718, 10 ppThukral R (2011) Eighth-order iterative methods without derivatives for solving nonlinear equation. ISRN Appl Math. Article ID 693787, 12 ppTraub JF (1964) Iterative methods for the solution of equations. Prentice Hall, New YorkZheng Q, Li J, Huang F (2011) An optimal Steffensen-type family for solving nonlinear equations. Appl Math Comput 217:9592–959

New fourth and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis

[EN] In this paper, a two-step class of fourth-order iterative methods for solving systems of nonlinear equations is presented. We further extend the two-step class to establish a new sixth-order family which requires only one additional functional evaluation. The convergence analysis of the proposed classes is provided under several mild conditions. A complete dynamical analysis is made, by using real multidimensional discrete dynamics, in order to select the most stable elements of both families of fourth- and sixth-order of convergence. To get this aim, a novel tool based on the existence of critical points has been used, the parameter line. The analytical discussion of the work is upheld by performing numerical experiments on some application-oriented problems. We provide an implementation of the proposed scheme on nonlinear optimization problem and zero-residual nonlinear least-squares problems taken from the constrained and unconstrained testing environment test set. Finally, based on numerical results, it has been concluded that our methods are comparable with the existing ones of similar nature in terms of order, efficiency, and computational time and also that the stability results provide the most efficient member of each class of iterative schemes.This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089Kansal, M.; Cordero Barbero, A.; Bhalla, S.; Torregrosa Sánchez, JR. (2021). New fourth and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis. Numerical Algorithms. 87(3):1017-1060. https://doi.org/10.1007/s11075-020-00997-4S1017106087