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

On a novel fourth-order algorithm for solving systems of nonlinear equations

This paper focuses on solving systems of nonlinear equations numerically. We propose an efficient iterative scheme including two steps and fourth order of convergence. The proposed method does not require the evaluation of second or higher order Frechet derivatives per iteration to proceed and reach fourth order of convergence. Finally, numerical results illustrate the efficiency of the method.The authors would like to thank the referees for the valuable comments and for the suggestions to improve the readability of the paper. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Babajee, DKR.; Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR. (2012). On a novel fourth-order algorithm for solving systems of nonlinear equations. Journal of Applied Mathematics. 2012. https://doi.org/10.1155/2012/1654522012Babajee, D. K. R., Dauhoo, M. Z., Darvishi, M. T., Karami, A., & Barati, A. (2010). Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 233(8), 2002-2012. doi:10.1016/j.cam.2009.09.035Darvishi, M. T., & Barati, A. (2007). A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Applied Mathematics and Computation, 188(1), 257-261. doi:10.1016/j.amc.2006.09.115Soleymani, F., Khattri, S. K., & Karimi Vanani, S. (2012). Two new classes of optimal Jarratt-type fourth-order methods. Applied Mathematics Letters, 25(5), 847-853. doi:10.1016/j.aml.2011.10.030Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2010). Accelerated methods of order 2p for systems of nonlinear equations. Journal of Computational and Applied Mathematics, 233(10), 2696-2702. doi:10.1016/j.cam.2009.11.018Dayton, B. H., Li, T.-Y., & Zeng, Z. (2011). Multiple zeros of nonlinear systems. Mathematics of Computation, 80(276), 2143-2143. doi:10.1090/s0025-5718-2011-02462-2Haijun, W. (2008). New third-order method for solving systems of nonlinear equations. Numerical Algorithms, 50(3), 271-282. doi:10.1007/s11075-008-9227-2Noor, M. A., Waseem, M., Noor, K. I., & Al-Said, E. (2012). Variational iteration technique for solving a system of nonlinear equations. Optimization Letters, 7(5), 991-1007. doi:10.1007/s11590-012-0479-3Frontini, M., & Sormani, E. (2004). Third-order methods from quadrature formulae for solving systems of nonlinear equations. Applied Mathematics and Computation, 149(3), 771-782. doi:10.1016/s0096-3003(03)00178-4Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2009). A modified Newton-Jarratt’s composition. Numerical Algorithms, 55(1), 87-99. doi:10.1007/s11075-009-9359-zCordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06