Widening basins of attraction of optimal iterative methods

[EN] In this work, we analyze the dynamical behavior on quadratic polynomials of a class of derivative-free optimal parametric iterative methods, designed by Khattri and Steihaug. By using their parameter as an accelerator, we develop different methods with memory of orders three, six and twelve, without adding new functional evaluations. Then a dynamical approach is made, comparing each of the proposed methods with the original ones without memory, with the following empiric conclusion: Basins of attraction of iterative schemes with memory are wider and the behavior is more stable. This has been numerically checked by estimating the solution of a practical problem, as the friction factor of a pipe and also of other nonlinear academic problems.This research was supported by Islamic Azad University, Hamedan Branch, Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and Generalitat Valenciana PROMETEO/2016/089.Bakhtiari, P.; Cordero Barbero, A.; Lotfi, T.; Mahdiani, K.; Torregrosa Sánchez, JR. (2017). Widening basins of attraction of optimal iterative methods. Nonlinear Dynamics. 87(2):913-938. https://doi.org/10.1007/s11071-016-3089-2S913938872Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Lett. 25, 2209–2217 (2012)Amat, S., Busquier, S., Bermúdez, C., Magreñán, Á.A.: On the election of the damped parameter of a two-step relaxed Newton-type method. Nonlinear Dyn. 84(1), 9–18 (2016)Chun, C., Neta, B.: An analysis of a family of Maheshwari-based optimal eighth order methods. Appl. Math. Comput. 253, 294–307 (2015)Babajee, D.K.R., Cordero, A., Soleymani, F., Torregrosa, J.R.: On improved three-step schemes with high efficiency index and their dynamics. Numer. Algorithms 65(1), 153–169 (2014)Argyros, I.K., Magreñán, Á.A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)Petković, M., Neta, B., Petković, L., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, London (2013)Ostrowski, A.M.: Solution of Equations and System of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)Khattri, S.K., Steihaug, T.: Algorithm for forming derivative-free optimal methods. Numer. Algorithms 65(4), 809–824 (2014)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Cordero, A., Soleymani, F., Torregrosa, J.R., Shateyi, S.: Basins of Attraction for Various Steffensen-Type Methods. J. Appl. Math. 2014, 1–17 (2014)Devaney, R.L.: The Mandelbrot Set, the Farey Tree and the Fibonacci sequence. Am. Math. Mon. 106(4), 289–302 (1999)McMullen, C.: Families of rational maps and iterative root-finding algorithms. Ann. Math. 125(3), 467–493 (1987)Chicharro, F., Cordero, A., Gutiérrez, J.M., Torregrosa, J.R.: Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219, 70237035 (2013)Magreñán, Á.A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)Neta, B., Chun, C., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Appl. Math. 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Some new efficient multipoint iterative methods for solving nonlinear systems of equations

It is attempted to put forward a new multipoint iterative method of sixth-order convergence for approximating solutions of nonlinear systems of equations. It requires the evaluation of two vector-function and two Jacobian matrices per iteration. Furthermore, we use it as a predictor to derive a general multipoint method. Convergence error analysis, estimating computational complexity, numerical implementation and comparisons are given to verify applicability and validity for the proposed methods.This research was supported by Islamic Azad University - Hamedan Branch, Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and Universitat Politecnica de Valencia SP20120474.Lotfi, T.; Bakhtiari, P.; Cordero Barbero, A.; Mahdiani, K.; Torregrosa Sánchez, JR. (2015). Some new efficient multipoint iterative methods for solving nonlinear systems of equations. International Journal of Computer Mathematics. 92(9):1921-1934. https://doi.org/10.1080/00207160.2014.946412S1921193492

Two optimal general classes of iterative methods with eighth-order

Two new three-step classes of optimal iterative methods to approximate simple roots of nonlinear equations, satisfying the Kung-Traub's conjecture, are designed. The development of the methods and their convergence analysis are provided joint with a generalization of both processes. In order to check the goodness of the theoretical results, some concrete methods are extracted and numerical and dynamically compared with some known methods.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Cordero Barbero, A.; Lotfi, T.; Mahdiani, K.; Torregrosa Sánchez, JR. (2014). Two optimal general classes of iterative methods with eighth-order. Acta Applicandae Mathematicae. 134(1):61-74. https://doi.org/10.1007/s10440-014-9869-0S61741341Higham, N.J.: Funstions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)Chun, C., Kim, Y.: Several new third-order iterative methods for solving nonlinear equations. Acta Appl. Math. 109(3), 1053–1063 (2010)Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A family of iterative methods with sixth and seventh order convergence for nonlinear equations. Math. Comput. Model. 52, 1490–1496 (2010)Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)Wang, H., Liu, H.: Note on a cubically convergent Newton-type method under weak conditions. Acta Appl. Math. 110(2), 725–735 (2010)Ostrowski, A.M.: Solution of Equations and Systems of Equations. Prentice-Hall, Englewood Cliffs (1964)Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)Petković, M.S., Neta, B., Petković, L.D., Dz̆nić, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam (2013)Petković, M.S., Petković, L.D.: Families of optimal multipoint methods for solving polynomial equations. Appl. Anal. Discrete Math. 4, 1–22 (2010)Soleymani, F.: Two novel classes of two-step optimal methods for all the zeros in an interval. Afr. Math. (2012). doi: 10.1007/s13370-012-0112-8Džunić, J., Petković, M.S., Petković, L.D.: A family of optimal three-point methods for solving nonlinear equations using two parametric functions. Appl. Math. Comput. 217(19), 7612–7619 (2011)Thukral, R., Petković, M.S.: A family of three-point methods of optimal order for solving nonlinear equation. J. Comput. Appl. Math. 233(9), 2278–2284 (2010)Obrechkoff, N.: Sur la solution numeriue des equations. God. Sofij. Univ. 56(1), 73–83 (1963)Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)Petković, M.S.: Multipoint methods for solving nonlinear equations: a survey. Appl. Math. Comput. 226, 635–660 (2014)Džunić, J., Petković, M.S.: A family of three-point methods of Ostrowski’s type for solving nonlinear equations. J. Appl. Math. 2012, 425867 (2012)Soleymani, F., Vanani, S.K., Afghani, A.: A general three-step class of optimal iterations for nonlinear equations. Math. Probl. Eng. 2011, 469512 (2011). 10 pp.Geum, Y.H., Kim, Y.I.: A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots. Appl. Math. Lett. 24, 929–935 (2011)Geum, Y.H., Kim, Y.I.: A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-variable linear fraction and a two-variable generic function. Comput. Math. Appl. 61, 708–714 (2011)Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, 780153 (2013). 11 pp

A stable family with high order of convergence for solving nonlinear equations

[EN] Recently, Li et al. (2014) have published a new family of iterative methods, without memory, with order of convergence five or six, which are not optimal in the sense of Kung and Traub’s conjecture. Therefore, we attempt to modify this suggested family in such a way that it becomes optimal. To this end, we consider the same two first steps of the mentioned family, and furthermore, we introduce a better approximation for f 0 ðzÞ in the third step based on interpolation idea as opposed to the Taylor’s series used in the work of Li et al. Theoretical, dynamical and numerical aspects of the new family are described and investigated in details.This research was supported by Islamic Azad University, Hamedan Branch and Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02.Cordero Barbero, A.; Lotfi, T.; Mahdiani, K.; Torregrosa Sánchez, JR. (2015). A stable family with high order of convergence for solving nonlinear equations. Applied Mathematics and Computation. 254:240-251. doi:10.1016/j.amc.2014.12.141S24025125

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