An efficient two-parametric family with memory for nonlinear equations

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• 作者：Alicia Cordero (1)
  Taher Lotfi (2)
  Parisa Bakhtiari (3)
  Juan R. Torregrosa (1)
  
  1. Instituto de Matem谩tica Multidisciplinar ; Universitat Polit猫cnica de Val猫ncia ; Camino de Vera ; s/n ; 46022 ; Valencia ; Spain
  2. Department of Applied Mathematics ; Hamedan Branch ; Islamic Azad University ; Hamedan ; Iran
  3. Young Researchers and Elite Club Young Researchers and Elite Club ; Hamedan Branch ; Islamic Azad University ; Hamedan ; Iran
  
• 关键词：Multipoint iterative method ; Nonlinear equation ; Optimal order ; Method with memory ; Kung ; Traub鈥檚 conjecture
• 刊名：Numerical Algorithms
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：68
• 期：2
• 页码：323-335
• 全文大小：1,502 KB
• 参考文献：1. Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643鈥?51 (1974) CrossRef
  2. Cordero, A., Hueso, J.L., Mart铆nez, E., Torregrosa, J.R.: A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equation. J. Comput. Appl. Math. 252, 95鈥?02 (2013) CrossRef
  3. 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鈥?1508 (2012) CrossRef
  4. D啪uni膰, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. Algorithms. 63(3), 549鈥?69 (2013) CrossRef
  5. D啪uni膰, J., Petkovi膰, M.S.: On generalized multipoint root-solvers with memory. J. Comput. Appl. Math. 236, 2909鈥?920 (2012) CrossRef
  6. Petkovi膰, M.S., Neta, B., Petkovi膰, L.D., D啪uni膰, J. (ed.).: Multipoint methods for solving nonlinear equations. Elsevier (2013)
  7. Sharma, J.R., Sharma, R.: A new family of modified Ostrowski鈥檚 methods with accelerated eighth order convergence. Numer. Algorithms 54, 445鈥?58 (2010) CrossRef
  8. Soleymani, F., Shateyi, S.: Two optimal eighth-order derivative-free classes of iterative methods. Abstr. Appl. Anal. 2012(318165), 14 (2012). doi:10.1155/2012/318165
  9. Soleymani, F., Sharma, R., Li, X., Tohidi, E.: An optimized derivative-free form of the Potra-Pt谩k methods. Math. Comput. Model. 56, 97鈥?04 (2012) CrossRef
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  11. Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, New York (1964)
  12. Wang, X., D啪uni膰, J., Zhang, T.: On an efficient family of derivative free three-point methods for solving nonlinear equations. Appl. Math. Comput. 219, 1749鈥?760 (2012) CrossRef
  13. Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592鈥?597 (2011) CrossRef
  14. Ortega, J.M., Rheinboldt, W.G. (ed.).: Iterative Solutions of Nonlinear Equations in Several Variables, Ed. Academic Press, New York (1970)
  15. Jay, I.O.: A note on / Q-order of convergence. BIT Numer. Math. 41, 422鈥?29 (2001) CrossRef
  16. Blanchard, P.: Complex Analytic Dynamics on the Riemann Sphere. Bull. AMS 11(1), 85鈥?41 (1984) CrossRef
  17. Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. arXiv:1307.6705 [math.NA]
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

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 71/3 = 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.