Basins of attraction for various Steffensen-Type methods

The dynamical behavior of different Steffensen-type methods is analyzed. We check the chaotic behaviors alongside the convergence radii (understood as the wideness of the basin of attraction) needed by Steffensen-type methods, that is, derivative-free iteration functions, to converge to a root and compare the results using different numerical tests. We will conclude that the convergence radii (and the stability) of Steffensen-type methods are improved by increasing the convergence order. The computer programming package MATHEMATICA provides a powerful but easy environment for all aspects of numerics. This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.The authors are indebted to the referees for some interesting comments and suggestions. This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02.Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR.; Shateyi, S. (2014). Basins of attraction for various Steffensen-Type methods. Journal of Applied Mathematics. 2014. https://doi.org/10.1155/2014/539707S2014Soleymani, F. (2011). Optimal fourth-order iterative method free from derivative. Miskolc Mathematical Notes, 12(2), 255. doi:10.18514/mmn.2011.303Zheng, Q., Zhao, P., Zhang, L., & Ma, W. (2010). Variants of Steffensen-secant method and applications. Applied Mathematics and Computation, 216(12), 3486-3496. doi:10.1016/j.amc.2010.04.058Neta, B., Scott, M., & Chun, C. (2012). Basins of attraction for several methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 218(21), 10548-10556. doi:10.1016/j.amc.2012.04.017Neta, B., & Scott, M. (2013). On a family of Halley-like methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 219(15), 7940-7944. doi:10.1016/j.amc.2013.02.035Neta, B., & Chun, C. (2013). On a family of Laguerre methods to find multiple roots of nonlinear equations. Applied Mathematics and Computation, 219(23), 10987-11004. doi:10.1016/j.amc.2013.05.002Neta, B., Chun, C., & Scott, M. (2014). Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Applied Mathematics and Computation, 227, 567-592. doi:10.1016/j.amc.2013.11.017Amat, S., Busquier, S., & Plaza, S. (2005). Dynamics of the King and Jarratt iterations. Aequationes mathematicae, 69(3), 212-223. doi:10.1007/s00010-004-2733-yChicharro, F., Cordero, A., Gutiérrez, J. M., & Torregrosa, J. R. (2013). Complex dynamics of derivative-free methods for nonlinear equations. Applied Mathematics and Computation, 219(12), 7023-7035. doi:10.1016/j.amc.2012.12.075Cordero, A., García-Maimó, J., Torregrosa, J. R., Vassileva, M. P., & Vindel, P. (2013). Chaos in King’s iterative family. Applied Mathematics Letters, 26(8), 842-848. doi:10.1016/j.aml.2013.03.012Chun, C., Lee, M. Y., Neta, B., & Džunić, J. (2012). On optimal fourth-order iterative methods free from second derivative and their dynamics. Applied Mathematics and Computation, 218(11), 6427-6438. doi:10.1016/j.amc.2011.12.013Cordero, A., Torregrosa, J. R., & Vindel, P. (2013). Dynamics of a family of Chebyshev–Halley type methods. Applied Mathematics and Computation, 219(16), 8568-8583. doi:10.1016/j.amc.2013.02.042Soleimani, F., Soleymani, F., & Shateyi, S. (2013). Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations. Discrete Dynamics in Nature and Society, 2013, 1-10. doi:10.1155/2013/301718Susanto, H., & Karjanto, N. (2009). Newton’s method’s basins of attraction revisited. Applied Mathematics and Computation, 215(3), 1084-1090. doi:10.1016/j.amc.2009.06.041Vrscay, E. R., & Gilbert, W. J. (1987). Extraneous fixed points, basin boundaries and chaotic dynamics for Schr�der and K�nig rational iteration functions. Numerische Mathematik, 52(1), 1-16. doi:10.1007/bf01401018Blanchard, P. (1984). Complex analytic dynamics on the Riemann sphere. Bulletin of the American Mathematical Society, 11(1), 85-142. doi:10.1090/s0273-0979-1984-15240-6Varona, J. L. (2002). Graphic and numerical comparison between iterative methods. The Mathematical Intelligencer, 24(1), 37-46. doi:10.1007/bf03025310Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860McMullen, C. (1987). Families of Rational Maps and Iterative Root-Finding Algorithms. The Annals of Mathematics, 125(3), 467. doi:10.2307/1971408Smale, S. (1985). On the efficiency of algorithms of analysis. Bulletin of the American Mathematical Society, 13(2), 87-122. doi:10.1090/s0273-0979-1985-15391-1Liu, Z., Zheng, Q., & Zhao, P. (2010). A variant of Steffensen’s method of fourth-order convergence and its applications. Applied Mathematics and Computation, 216(7), 1978-1983. doi:10.1016/j.amc.2010.03.028Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2012). A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations. Abstract and Applied Analysis, 2012, 1-15. doi:10.1155/2012/836901Zheng, Q., Li, J., & Huang, F. (2011). An optimal Steffensen-type family for solving nonlinear equations. Applied Mathematics and Computation, 217(23), 9592-9597. doi:10.1016/j.amc.2011.04.035Soleymani, F., Karimi Vanani, S., & Jamali Paghaleh, M. (2012). A Class of Three-Step Derivative-Free Root Solvers with Optimal Convergence Order. Journal of Applied Mathematics, 2012, 1-15. doi:10.1155/2012/56874

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. Phys. 55(2), 183–193 (2015)Cordero, A., Maimó, J.G., Torregrosa, J.R., Vassileva, M.P.: Solving nonlinear problems by Ostrowski–Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)Abad, M., Cordero, A., Torregrosa, J.R.: A family of seventh-order schemes for solving nonlinear systems. Bull. Math. Soc. Sci. Math. Roum. Tome 57(105), 133–145 (2014)Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)White, F.: Fluid Mechanics. McGraw-Hill, Boston (2003)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Soleymani, F., Babajee, D.K.R., Shateyi, S., Motsa, S.S.: Construction of optimal derivative-free techniques without memory. J. Appl. Math. (2012). doi: 10.1155/2012/49702

A family of Kurchatov-type methods and its stability

[EN] We present a parametric family of iterative methods with memory for solving of nonlinear problems including Kurchatov¿s scheme, preserving its second-order of convergence. By using the tools of multidimensional real dynamics, the stability of members of this family is analyzed on low-degree polynomials, showing some elements of this class more stable behavior than the original Kurchatov¿s method. The iteration is extended for multi-dimensional case. Computational efficiencies of proposed technique is discussed and compared with the existing methods. A couple of numerical examples are considered to test the performance of the new family of iterations.The authors thank to the anonymous referees for their valuable comments and for the suggestions that have improved the final version of the paper. This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Soleymani, F.; Torregrosa Sánchez, JR.; Haghani, FK. (2017). A family of Kurchatov-type methods and its stability. Applied Mathematics and Computation. 294:264-279. https://doi.org/10.1016/j.amc.2016.09.021S26427929

A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems

[EN] We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous iterate in the very recent literature, mostly with low orders of convergence. The proposed scheme showed its efficiency and robustness in several numerical tests, where it was also compared with the existing procedures with high orders of convergence. These numerical tests included large nonlinear systems. In addition, we show that the proposed scheme has very stable qualitative behavior, by means of the analysis of an associated multidimensional, real rational function and also by means of a comparison of its basin of attraction with those of comparison methods.This research was supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE).Behl, R.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Bhalla, S. (2021). A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems. Mathematics. 9(17):1-16. https://doi.org/10.3390/math9172122S11691

An efficient two-parametric family with memory for nonlinear equations

A new two-parametric family of derivative-free iterative methods for solving nonlinear equations is presented. First, a new biparametric family without memory of optimal order four is proposed. The improvement of the convergence rate of this family is obtained by using two self-accelerating parameters. These varying parameters are calculated in each iterative step employing only information from the current and the previous iteration. The corresponding R-order is 7 and the efficiency index 7(1/3) = 1.913. Numerical examples and comparison with some existing derivative-free optimal eighth-order schemes are included to confirm the theoretical results. In addition, the dynamical behavior of the designed method is analyzed and shows the stability of the scheme.The second author wishes to thank the Islamic Azad University, Hamedan Branch, where the paper was written as a part of the research plan, for financial support.Cordero Barbero, A.; Lotfi, T.; Bakhtiari, P.; Torregrosa Sánchez, JR. (2015). An efficient two-parametric family with memory for nonlinear equations. Numerical Algorithms. 68(2):323-335. doi:10.1007/s11075-014-9846-8S323335682Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equation. J. Comput. Appl. Math. 252, 95–102 (2013)Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Pseudocomposition: a technique to design predictor-corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218, 11496–11508 (2012)Džunić, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. Algorithms. 63(3), 549–569 (2013)Džunić, J., Petković, M.S.: On generalized multipoint root-solvers with memory. J. Comput. Appl. Math. 236, 2909–2920 (2012)Petković, M.S., Neta, B., Petković, L.D., Džunić, J. (ed.).: Multipoint methods for solving nonlinear equations. Elsevier (2013)Sharma, J.R., Sharma, R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algorithms 54, 445–458 (2010)Soleymani, F., Shateyi, S.: Two optimal eighth-order derivative-free classes of iterative methods. Abstr. Appl. Anal. 2012(318165), 14 (2012). doi: 10.1155/2012/318165Soleymani, F., Sharma, R., Li, X., Tohidi, E.: An optimized derivative-free form of the Potra-Pták methods. Math. Comput. Model. 56, 97–104 (2012)Thukral, R.: Eighth-order iterative methods without derivatives for solving nonlinear equations. ISRN Appl. Math. 2011(693787), 12 (2011). doi: 10.5402/2011/693787Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)Wang, X., Džunić, J., Zhang, T.: On an efficient family of derivative free three-point methods for solving nonlinear equations. Appl. Math. Comput. 219, 1749–1760 (2012)Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)Ortega, J.M., Rheinboldt, W.G. (ed.).: Iterative Solutions of Nonlinear Equations in Several Variables, Ed. Academic Press, New York (1970)Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)Blanchard, P.: Complex Analytic Dynamics on the Riemann Sphere. Bull. AMS 11(1), 85–141 (1984)Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. arXiv: 1307.6705 [math.NA

Basin attractors for derivative-free methods to find simple roots of nonlinear equations

 Many methods exist for solving nonlinear equations. Several of these methods are derivative-free. One of the oldest is the secant method where the derivative is replaced by a divided difference. Clearly such method will need an additional starting value. Here we consider several derivative-free methods and compare them using the idea of basin of attraction

A Class of Steffensen-Type Iterative Methods for Nonlinear Systems

A class of iterative methods without restriction on the computation of Fréchet derivatives including multisteps for solving systems of nonlinear equations is presented. By considering a frozen Jacobian, we provide a class of m-step methods with order of convergence m+1. A new method named as Steffensen-Schulz scheme is also contributed. Numerical tests and comparisons with the existing methods are included

A multidimensional dynamical approach to iterative methods with memory

[EN] A dynamical approach on the dynamics of iterative methods with memory for solving nonlinear equations is made. We have designed new methods with memory from Steffensen’ or Traub’s schemes, as well as from a parametric family of iterative procedures of third- and fourth-order of convergence. We study the local order of convergence of the new iterative methods with memory. We define each iterative method with memory as a discrete dynamical system and we analyze the stability of the fixed points of its rational operator associated on quadratic polynomials. As far as we know, there is no dynamical study on iterative methods with memory and the techniques of complex dynamics used in schemes without memory are not useful in this context. So, we adapt real multidimensional dynamical tools to afford this task. The dynamical behavior of Secant method and the versions of Steffensen’ and Traub’s schemes with memory, applied on quadratic polynomials, are analyzed. Different kinds of behavior occur, being in general very stable but pathologic cases as attracting strange fixed points are also found. Finally, a modified parametric family of order four, applied on quadratic polynomials, is also studied, showing the bifurcations diagrams and the appearance of chaos.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P.Campos, B.; Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vindel, P. (2015). A multidimensional dynamical approach to iterative methods with memory. Applied Mathematics and Computation. 271:701-715. https://doi.org/10.1016/j.amc.2015.09.05670171527