Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE SEMI-LOCAL CONVERGENCE OF CONTRAHARMONIC-MEAN NEWTON'S METHOD (CHMN)

  • Argyros, Ioannis K. (Department of Mathematical Sciences Cameron University) ;
  • Singh, Manoj Kumar (Department of Mathematics Institute of Science Banaras Hindu University)
  • Received : 2021.09.14
  • Accepted : 2021.12.08
  • Published : 2022.10.01
Download PDF
⟨ Previous Next ⟩

Abstract

The main objective of this work is to investigate the study of the local and semi-local convergence of the contraharmonic-mean Newton's method (CHMN) for solving nonlinear equations in a Banach space. We have performed the semi-local convergence analysis by using generalized conditions. We examine the theoretical results by comparing the CHN method with the Newton's method and other third order methods by Weerakoon et al. using some test functions. The theoretical and numerical results are also supported by the basins of attraction for a selected test function.

Keywords

References

  1. S. Amat, S. Busquier, and S. Plaza, Review of some iterative root-finding methods from a dynamical point of view, Sci. Ser. A Math. Sci. (N.S.) 10 (2004), 3-35.
  2. A. Amiri, A. Cordero, M. T. Darvishi, and J. R. Torregrosa, Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems, Appl. Math. Comput. 323 (2018), 43-57. https://doi.org/10.1016/j.amc.2017.11.040
  3. A. Amiri, A. Cordero, M. T. Darvishi, and J. R. Torregrosa, Stability analysis of Jacobian-free Newton's iterative method, Algorithms (Basel) 12 (2019), no. 11, Paper No. 236, 26 pp. https://doi.org/10.3390/a12110236
  4. A. Amiri, A. Cordero, M. T. Darvishi, and J. R. Torregrosa, Stability analysis of Jacobian-free iterative methods for solving nonlinear systems by using families of mth power divided differences, J. Math. Chem. 57 (2019), no. 5, 1344-1373. https://doi.org/10.1007/s10910-018-0971-9
  5. I. K. Argyros, Computational theory of iterative methods, Studies in Computational Mathematics, 15, Elsevier B. V., Amsterdam, 2007.
  6. I. K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer, New York, 2008.
  7. I. K. Argyros and A. Magrenan, Iterative Methods and Their Dynamics with Applications, CRC Press, Boca Raton, FL, 2017.
  8. J. Chen, I. K. Argyros, and R. P. Agarwal, Majorizing functions and two-point Newtontype methods, J. Comput. Appl. Math. 234 (2010), no. 5, 1473-1484. https://doi.org/10.1016/j.cam.2010.02.024
  9. M. T. Darvishi, A two-step high order Newton-like method for solving systems of nonlinear equations, Int. J. Pure Appl. Math. 57 (2009), no. 4, 543-555.
  10. M. T. Darvishi, Some three-step iterative methods free from second order derivative for finding solutions of systems of nonlinear equations, Int. J. Pure Appl. Math. 57 (2009), no. 4, 557-573.
  11. M. A. Hernandez-Veron and E. Martinez, On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions, Numer. Algorithms 70 (2015), no. 2, 377-392. https://doi.org/10.1007/s11075-014-9952-7
  12. P. Jarratt, Some fourth order multipoint iterative methods for solving equations, Math. Comp. 20 (1966), 434-437. https://doi.org/10.1090/S0025-5718-66-99924-8
  13. L. V. Kantorovich and G. P. Akilov, Funtional Analysis, Pergamon Press, Oxford, 1982.
  14. K. Madhu, Semilocal convergence of sixth order method by using recurrence relations in Banach spaces, Appl. Math. E-Notes 18 (2018), 197-208.
  15. A. M. Ostrowski, Solution of equations and systems of equations, second edition, Pure and Applied Mathematics, Vol. 9, Academic Press, New York, 1966.
  16. P. K. Parida and D. K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math. 206 (2007), no. 2, 873-887. https://doi.org/10.1016/j.cam.2006.08.027
  17. L. B. Rall, Computational solution of nonlinear operator equations, corrected reprint of the 1969 original, Robert E. Krieger Publishing Co., Inc., Huntington, NY, 1979.
  18. M. K. Singh, A six-order variant of Newton's method for solving non linear equations, Comput. Methods Sci. Tech. 15 (2009), no. 2, 185-193. https://doi.org/10.12921/cmst.2009.15.02.185-193
  19. M. K. Singh and A. K. Singh, Variant of Newton's method using Simpson's 3/8th rule, Int. J. Appl. Comput. Math. 6 (2020), no. 1, Paper No. 20, 13 pp. https://doi.org/10.1007/s40819-020-0770-4
  20. E. R. Vrscay and W. J. Gilbert, Extraneous fixed points, basin boundaries and chaotic dynamics for Schroder and Konig rational iteration functions, Numer. Math. 52 (1988), no. 1, 1-16. https://doi.org/10.1007/BF01401018
  21. X. Wang, J. Kou, and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011), no. 4, 441-456. https://doi.org/10.1007/s11075-010-9438-1
  22. S. Weerakoon and T. G. I. Fernando, A variant of Newton's method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000), no. 8, 87-93. https://doi.org/10.1016/S0893-9659(00)00100-2
  23. Q. Wu and Y. Zhao, Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space, Appl. Math. Comput. 175 (2006), no. 2, 1515-1524. https://doi.org/10.1016/j.amc.2005.08.043