A class of four parametric with- and without memory root finding methods

[EN] In this paper, we have constructed a derivative¿free weighted eighth¿order iterative method with and without memory for solving nonlinear equations. This method is an optimal method as it satisfies the Kung¿Traub conjecture. We have used four accelerating parameters, a univariate and a multivariate weight function at the second and third step of the method, respectively. This method is converted into with¿memory method 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 comparison of our methods is done with the recent methods of respective domain.This research was partially supported by Ministerio de Economía 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 class of four parametric with- and without memory root finding methods. Computational and Mathematical Methods. 1-14. https://doi.org/10.1002/cmm4.1024S114Cordero, A., Junjua, M.-D., Torregrosa, J. R., Yasmin, N., & Zafar, F. (2018). Efficient Four-Parametric with-and-without-Memory Iterative Methods Possessing High Efficiency Indices. Mathematical Problems in Engineering, 2018, 1-12. doi:10.1155/2018/8093673Zafar, F., Akram, S., Yasmin, N., & Junjua, M.-D. (2016). On the construction of three step derivative free four-parametric methods with accelerated order of convergence. Journal of Nonlinear Sciences and Applications, 09(06), 4542-4553. doi:10.22436/jnsa.009.06.92King, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Herzberger, J. (1974). Über Matrixdarstellungen für Iterationsverfahren bei nichtlinearen Gleichungen. Computing, 12(3), 215-222. doi:10.1007/bf02293107Jay, L. O. (2001). Bit Numerical Mathematics, 41(2), 422-429. doi:10.1023/a:1021902825707Chun, C., & Neta, B. (2015). On the new family of optimal eighth order methods developed by Lotfi et al. Numerical Algorithms, 72(2), 363-376. doi:10.1007/s11075-015-0048-9Gdawiec, K. (2017). Fractal patterns from the dynamics of combined polynomial root finding methods. Nonlinear Dynamics, 90(4), 2457-2479. doi:10.1007/s11071-017-3813-

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. Comput. 227, 567–592 (2014)Lotfi, T., Magreñán, Á.A., Mahdiani, K., Rainer, J.J.: A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: dynamic study and approach. Appl. Math. Comput. 252, 347–353 (2015)Chicharro, F.I., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, 1–11 (2013)Cordero, A., Lotfi, T., Torregrosa, J.R., Assari, P., Mahdiani, K.: Some new bi-accelerator two-point methods for solving nonlinear equations. Comput. Appl. Math. 35(1), 251–267 (2016)Cordero, A., Lotfi, T., Bakhtiari, P., Torregrosa, J.R.: An efficient two-parametric family with memory for nonlinear equations. Numer. Algorithms 68(2), 323–335 (2015)Lotfi, T., Mahdiani, K., Bakhtiari, P., Soleymani, F.: Constructing two-step iterative methods with and without memory. Comput. Math. Math. 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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

An optimal sixteenth order family of methods for solving nonlinear equations and their basins of attraction

We propose a new family of iterative methods for finding the simple roots of nonlinear equation. The proposed method is four-point method with convergence order 16, which consists of four steps: the Newton step, an optional fourth order iteration scheme, an optional eighth order iteration scheme and the step constructed using the divided difference. By reason of the new iteration scheme requiring four function evaluations and one first derivative evaluation per iteration, the method satisfies the optimality criterion in the sense of Kung-Traub\u27s conjecture and achieves a high efficiency index $16^{1/5} approx 1.7411$. Computational results support theoretical analysis and confirm the efficiency. The basins of attraction of the new presented algorithms are also compared to the existing methods with encouraging results

On improved three-step schemes with high efficiency index and their dynamics

This paper presents an improvement of the sixth-order method of Chun and Neta as a class of three-step iterations with optimal efficiency index, in the sense of Kung-Traub conjecture. Each member of the presented class reaches the highest possible order using four functional evaluations. Error analysis will be studied and numerical examples are also made to support the theoretical results. We then present results which describe the dynamics of the presented optimal methods for complex polynomials. The basins of attraction of the existing optimal methods and our methods are presented and compared to illustrate their performances.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT Republica Dominicana.Babajee, DKR.; Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR. (2014). On improved three-step schemes with high efficiency index and their dynamics. Numerical Algorithms. 65(1):153-169. https://doi.org/10.1007/s11075-013-9699-6S153169651Pang, J.S., Chan, D.: Iterative methods for variational and complementary problems. Math. Program. 24(1), 284–313 (1982)Sun, D.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91(1), 123–140 (1996)Chun, C., Neta, B.: A new sixth-order scheme for nonlinear equations. Appl. Math. Lett. 25, 185–189 (2012)Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)Neta, B.: A new family of high-order methods for solving equations. Int. J. Comput. Math. 14, 191–195 (1983)Neta, B.: On Popovski’s method for nonlinear equations. Appl. Math. Comput. 201, 710–715 (2008)Chun, C., Neta, B.: Some modifications of Newton’s method by the method of undeterminate coefficients. Comput. Math. Appl. 56, 2528–2538 (2008)Chun, C., Lee, M.Y., Neta, B., Dzunic, J.: On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: A family of modified Ostrowski’s methods with optimal eighth order of convergence. Appl. Math. Lett. 24, 2082–2086 (2011)Heydari, M., Hosseini, S.M., Loghmani, G.B.: On two new families of iterative methods for solving nonlinear equations with optimal order. Appl. Anal. Dis. Math. 5, 93–109 (2011)Neta, B., Petkovic, M.S.: Construction of optimal order nonlinear solvers using inverse interpolation. Appl. Math. Comput. 217, 2448–2445 (2010)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., Karimi Vanani, S., Khan, M., Sharifi, M.: Some modifications of King’s family with optimal eighth order of convergence. Math. Comput. Model. 55, 1373–1380 (2012)Soleymani, F., Karimi Vanani, S., Jamali Paghaleh, M.: A class of three-step derivative-free root solvers with optimal convergence order. J. Appl. Math. 2012, Article ID 568740, 15 pp. (2012). doi: 10.1155/2012/568740Soleymani, F., Sharifi, M., Mousavi, B.S.: An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. J. Optim. Theory Appl. 153, 225–236 (2012)Stewart, B.D.: Attractor basins of various root-finding methods. M.S. Thesis, Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA (2001)Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Scientia 10, 3–35 (2004)Amat, S., Busquier, S., Plaza, S.: Dynamics of the King and Jarratt iterations. Aequ. Math. 69, 212–223 (2005)Amat, S., Busquier, S., Plaza, S.: Chaotic dynamics of a third-order Newton type method. J. Math. Anal. Appl. 366, 24–32 (2010)Neta, B., Chun, C., Scott, M.: A note on the modified super-Halley method. Appl. Math. Comput. 218, 9575–9577 (2012)Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)Ardelean, G.: A comparison between iterative methods by using the basins of attraction. Appl. Math. Comput. 218, 88–95 (2011)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Babajee, D.K.R.: Analysis of higher order variants of Newton’s method and their applications to differential and integral equations and in ocean acidification. Ph.D. Thesis, University of Mauritius (2010