A class of efficient high-order iterative methods with memory for nonlinear equations and their dynamics

[EN] In this paper we obtain some theoretical results about iterative methods with memory for nonlinear equations. The class of algorithms we consider focus on incorporating memory without increasing the computational cost of the algorithm. This class uses for the predictor step of each iteration a quantity that has already been calculated in the previous iteration, typically the quantity governing the slope from the previous corrector step. In this way we do not introduce any extra computation, and more importantly, we avoid new function evaluations, allowing us to obtain high-order iterative methods in a simple way. A specific class of methods of this type is introduced, and we prove the convergence order is 2(n) + 2(n-2) with n + 1 function evaluations. An exhaustive efficiency study is performed to show the competitiveness of these methods. Finally, we test some specific examples and explore the effect that this predictor may have on the convergence set by setting a dynamical study.Ministerio de Economia y Competitividad de Espana, Grant/Award Number: MTM2014-52016-C2-2-P; Generalitat Valenciana Prometeo, Grant/Award Number: /2016/089Howk, CL.; Hueso, J.; Martínez Molada, E.; Teruel-Ferragud, C. (2018). A class of efficient high-order iterative methods with memory for nonlinear equations and their dynamics. Mathematical Methods in the Applied Sciences. 41(17):7263-7282. https://doi.org/10.1002/mma.4821S72637282411

CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior

[EN] A family of fourth-order iterative methods without memory, for solving nonlinear systems, and its seventh-order extension, are analyzed. By using complex dynamics tools, their stability and reliability are studied by means of the properties of the rational function obtained when they are applied on quadratic polynomials. The stability of their fixed points, in terms of the value of the parameter, its critical points and their associated parameter planes, etc. give us important information about which members of the family have good properties of stability and whether in any of them appear chaos in the iterative process. The conclusions obtained in this dynamical analysis are used in the numerical section, where an academical problem and also the chemical problem of predicting the diffusion and reaction in a porous catalyst pellet are solved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Guasp, L.; Torregrosa Sánchez, JR. (2018). CMMSE2017: On two classes of fourth- and seventh-order vectorial methods with stable behavior. Journal of Mathematical Chemistry. 56(7):1902-1923. https://doi.org/10.1007/s10910-017-0814-0S19021923567S. Amat, S. Busquier, Advances in Iterative Methods for Nonlinear Equations (Springer, Berlin, 2016)S. Amat, S. Busquier, S. Plaza, Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)S. Amat, S. Busquier, S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods. Comput. Appl. Math. 189, 22–33 (2006)I.K. Argyros, Á.A. Magreñn, On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)D.K.R. Babajee, A. Cordero, J.R. Torregrosa, Study of multipoint iterative methods through the Cayley quadratic test. Comput. Appl. Math. 291, 358–369 (2016). doi: 10.1016/J.CAM.2014.09.020P. Blanchard, The dynamics of Newton’s method. Proc. Symp. Appl. Math. 49, 139–154 (1994)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, Article ID 780153 (2013)C. Chun, M.Y. Lee, B. Neta, J. Džunić, On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)A. Cordero, E. Gómez, J.R. Torregrosa, Efficient high-order iterative methods for solving nonlinear systems and their application on heat conduction problems. Complexity 2017, Article ID 6457532 (2017)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley Publishing Company, Reading, 1989)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: formalism and first application to atomic problems. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. Math. Chem. 49, 1384–1415 (2011)Á.A. Magreñán, Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction-diffusion model arising in mathematical chemistry. Math. Chem. 51(9), 2361–2385 (2013)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon. Proc. Comput. Sci. 3, 361–364 (2011)B. Neta, C. Chun, M. Scott, Basins of attraction for optimal eighth-order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)M.S. Petković, B. Neta, L.D. Petković, J. Džunić, Multipoint Methods for Solving Nonlinear Equations (Elsevier, Amsterdam, 2013)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. Math. Chem. 52(1), 255–267 (2014)R. Singh, G. Nelakanti, J. Kumar, A new effcient technique for solving two-point boundary value problems for integro-differential equations. Math. Chem. 52, 2030–2051 (2014

On the choice of the best members of the Kim family and the improvement of its convergence

The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results

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

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. 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