Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior

In this paper, we present a family of optimal, in the sense of Kung-Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methodsThis research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by the Center of Excellence for Mathematics, University of Shahrekord, Iran.Cordero Barbero, A.; Fardi, M.; Ghasemi, M.; Torregrosa Sánchez, JR. (2014). Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior. Calcolo. 51(1):17-30. https://doi.org/10.1007/s10092-012-0073-11730511Bi, W., Ren, H., Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 255, 105–112 (2009)Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. Am. Math. Soc. 11(1), 85–141 (1984)Chun, C.: Some variants of Kings fourth-order family of methods for nonlinear equations. Appl. Math. Comput. 190, 57–62 (2007)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: New modifications of Potra-Pták’s method with optimal fourth and eighth order of convergence. J. Comput. Appl. Math. 234, 2969–2976 (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., Torregrosa, J.R., Vassileva, M.P.: A family of modified Ostrowski’s method with optimal eighth order of convergence. Appl. Math. Lett. 24(12), 2082–2086 (2011)Douady, A., Hubbard, J.H.: On the dynamics of polynomials-like mappings. Ann. Sci. Ec. Norm. Sup. (Paris) 18, 287–343 (1985)Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)Liu, L., Wang, X.: Eighth-order methods with high efficiency index for solving nonlinear equations. Appl. Math. Comput. 215, 3449–3454 (2010)Ostrowski, A.M.: Solutions of equations and systems of equations. Academic Press, New York (1966)Sharma, J.R., Sharma, R.: A family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algoritms 54, 445–458 (2010)Soleymani, F., Karimi Banani, S., Khan, M., Sharifi, M.: Some modifications of King’s family with optimal eighth order of convergence. Math. Comput. Model. 55, 1373–1380 (2012)Thukral, R., Petkovic, M.S.: A family of three-point methods of optimal order for solving nonlinear equations. J. Comput. Appl. Math. 233, 2278–2284 (2010

Three New Optimal Fourth-Order Iterative Methods to Solve Nonlinear Equations

We present new modifications to Newton's method for solving nonlinear equations. The analysis of convergence shows that these methods have fourth-order convergence. Each of the three methods uses three functional evaluations. Thus, according to Kung-Traub's conjecture, these are optimal methods. With the previous ideas, we extend the analysis to functions with multiple roots. Several numerical examples are given to illustrate that the presented methods have better performance compared with Newton's classical method and other methods of fourth-order convergence recently published

Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior

In this paper, we present a family of optimal, in the sense of Kung-Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methodsThis research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by the Center of Excellence for Mathematics, University of Shahrekord, Iran.Cordero Barbero, A.; Fardi, M.; Ghasemi, M.; Torregrosa Sánchez, JR. (2014). Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior. Calcolo. 51(1):17-30. doi:10.1007/s10092-012-0073-1Senia173051

Choosing the most stable members of Kou's family of iterative methods

[EN] In this manuscript, we analyze the dynamical anomalies of a parametric family of iterative schemes designed by Kou et al. It is known that its order of convergence is three for any arbitrary value of the parameter, but it has order four (and it is optimal in the sense of Kung-Traub's conjecture) when a specific value is selected. Among all the elements of this family, one can choose this fourth-order element or any of the infinite members of third order of convergence, if only the speed of convergence is considered. However, the stability of the methods plays an important role in their reliability when they are applied on different problems. This is the reason why we analyze in this paper the dynamical behavior on quadratic polynomials of the mentioned family. The study of fixed points and their stability, joint with the critical points and their associated parameter planes, show the richness of the class and allow us to find members of it with excellent numerical properties, as well as other ones with very unstable behavior. Some test functions are analyzed for confirming the theoretical results. (C) 2017 Elsevier B.V. All rights reserved.This research was supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Guasp, L.; Torregrosa Sánchez, JR. (2018). Choosing the most stable members of Kou's family of iterative methods. Journal of Computational and Applied Mathematics. 330:759-769. https://doi.org/10.1016/j.cam.2017.02.012S75976933

A general class of four parametric with and without memory iterative methods for nonlinear equations

[EN] In this paper, we have constructed a derivative-free weighted eighth-order iterative class of methods with and without-memory for solving nonlinear equations. These methods are optimal as they satisfy Kung-Traub's conjecture. We have used four accelerating parameters, univariate and multivariate weight functions at the second and third step of the method respectively. This family of schemes is converted into with-memory one by approximating the parameters using Newton's interpolating polynomials of appropriate degree to increase the order of convergence to 15.51560 and the efficiency index is nearly two. Numerical and dynamical comparison of our methods is done with some recent methods of the same order applying them on some applied problems from chemical engineering, such as fractional conversion in a chemical reactor. The stability of the proposed schemes and their comparison with existing ones is made by using dynamical planes of the different methods, showing the wideness of the sets of converging initial estimations for all the test functions. The proposed schemes show to have good stability properties, as in their eighth-order version as well as in the case of methods with memory.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.; Rafi, A. (2019). A general class of four parametric with and without memory iterative methods for nonlinear equations. Journal of Mathematical Chemistry. 57(5):1448-1471. https://doi.org/10.1007/s10910-018-00996-wS14481471575F. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World 2013, 11 (2013)A. Cordero, M. Junjua, J.R. Torregrosa, N. Yasmin, F. Zafar, Efficient four parametric with and without-memory iterative methods possessing high efficiency indices. Math. Probl. Eng. 2018, 12 (2018)J.M. Douglas, Process Dynamics and Control, vol. 2 (Prentice Hall, Englewood Cliffs, NJ, 1972)J. Herzberger, Über Matrixdarstellungen für Iterationverfahren bei nichtlinearen Gleichungen. Computing 12(3), 215–222 (1974)L.O. Jay, A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)R.F. King, A family of fourth order methods for non-linear equations. SIAM J. Numer. Anal. 10(5), 876–879 (1973)T. Lotfi, P. Assari, New three- and four-parametric iterative with-memory methods with efficiency index near 2. Appl. Math. 270, 1004–1010 (2015)M. Shacham, Numerical solution of constrained nonlinear algebraic equations. Int. J. Numer. Method Eng. 23, 1455–1481 (1986)I.F. Steffensen, Remarks on iteration. Skand. Aktuarietidskr. 16, 64–72 (1933)J.F. Traub, Iterative Methods for the Solution of Equations (Prentice Hall, New York, 1964)F. Zafar, S. Akram, N. Yasmin, M. Junjua, On the construction of three step derivative free four-parametric methods with accelerated order of convergence. J. Nonlinear Sci. Appl. 9, 4542–4553 (2016

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

CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems

[EN] The third-order iterative method designed by Weerakoon and Fernando includes the arithmetic mean of two functional evaluations in its expression. Replacing this arithmetic mean with different means, other iterative methods have been proposed in the literature. The evolution of these methods in terms of order of convergence implies the inclusion of a weight function for each case, showing an optimal fourth-order convergence, in the sense of Kung-Traub's conjecture. The analysis of these new schemes is performed by means of complex dynamics. These methods are applied on the solution of the nonlinear Colebrook-White equation and the nonlinear system of the equilibrium conversion, both frequently used in Chemistry.This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE) and Generalitat Valenciana PROMETEO/2016/089.Chicharro, FI.; Cordero Barbero, A.; Martínez, TH.; Torregrosa Sánchez, JR. (2020). CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems. Journal of Mathematical Chemistry. 58(3):555-572. https://doi.org/10.1007/s10910-019-01085-2S555572583O. Ababneh, New Newton’s method with third order convergence for solving nonlinear equations. World Acad. Sci. Eng. Technol. 61, 1071–1073 (2012)S. Amat, S. Busquier, Advances in iterative methods for nonlinear equations, chapter 5. SEMA SIMAI Springer Series. (Springer, Berlin, 2016), vol. 10, pp. 79–111R. Behl, Í. Sarría, R. González, Á.A. Magreñán, Highly efficient family of iterative methods for solving nonlinear models. J. Comput. Appl. Math. 346, 110–132 (2019)B. Campos, J. Canela, P. Vindel, Convergence regions for the Chebyshev-Halley family. Commun. Nonlinear Sci. Numer. Simul. 56, 508–525 (2018)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 780513, 1–11 (2013)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Dynamics of iterative families with memory based on weight functions procedure. J. Comput. Appl. Math. 354, 286–298 (2019)C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. 161, 367–381 (1937)A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications (Prentice-Hall, Englewood Cliffs, 1999)A. Cordero, J. Franceschi, J.R. Torregrosa, A.C. Zagati, A convex combination approach for mean-based variants of Newton’s method. Symmetry 11, 1062 (2019)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)T. Lukić, N. Ralević, Geometric mean Newton’s method for simple and multiple roots. Appl. Math. Lett. 21, 30–36 (2008)A. Özban, Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004)M. Petković, B. Neta, L. Petković, J. Dz̆unić, Multipoint Methods for Solving Nonlinear Equations (Academic Press, Cambridge, 2013)E. Shashi, Transmission Pipeline Calculations and Simulations Manual, Fluid Flow in Pipes (Elsevier, London, 2015), pp. 149–234M.K. Singh, A.K. Singh, A new-mean type variant of Newton’s method for simple and multiple roots. Int. J. Math. Trends Technol. 49, 174–177 (2017)K. Verma, On the centroidal mean Newton’s method for simple and multiple roots of nonlinear equations. Int. J. Comput. Sci. Math. 7, 126–143 (2016)S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)Z. Xiaojian, A class of Newton’s methods with third-order convergence. Appl. Math. Lett. 20, 1026–1030 (2007