Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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LOCAL CONVERGENCE FOR SOME THIRD-ORDER ITERATIVE METHODS UNDER WEAK CONDITIONS

  • Argyros, Ioannis K. (Department of Mathematical Sciences Cameron University) ;
  • Cho, Yeol Je (Department of Mathematics Education and the RINS Gyeongsang National University, Department of Mathematics King Abdulaziz University) ;
  • George, Santhosh (Department of Mathematical and Computational Sciences NIT)
  • Received : 2015.04.18
  • Published : 2016.07.01
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Abstract

The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. S. Amat, M. A. Hernandez, and N. Romero, A modified Chebyshev's iterative method with at least sixth order of convergence, Appl. Math. Comput. 206 (2008), no. 1, 164-174. https://doi.org/10.1016/j.amc.2008.08.050
  2. I. K. Argyros, Convergence and Application of Newton-type Iterations, Springer, 2008.
  3. I. K. Argyros, Y. J. Cho, and S. George, On the "Terra incognita" for the Newton-Kantrovich method, J. Korean Math. Soc. 51 (2014), no. 2, 251-266. https://doi.org/10.4134/JKMS.2014.51.2.251
  4. I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical Methods for Equations and its Applications, Taylor & Francis, CRC Press, New York, 2012.
  5. I. K. Argyros, Y. J. Cho, and S. K. Khattri, On a new semilocal convergence analysis for the Jarratt method, J. Inequal. Appl. 2013 (2013), 194, 16 pp. https://doi.org/10.1186/1029-242X-2013-16
  6. I. K. Argyros, Y. J. Cho, and H. M. Ren, Convergence of Halley's method for operators with the bounded second derivative in Banach spaces, J. Inequal. Appl. 2013 (2013), 260, 12 pp. https://doi.org/10.1186/1029-242X-2013-12
  7. I. K. Argyros and S. Hilout, Computational methods in nonlinear Analysis, World Scientific Publ. House, New Jersey, USA, 2013.
  8. V. Candela and A. Marquina, Recurrence relations for rational cubic methods I: The Halley method, Computing 44 (1990), no. 2, 169-184. https://doi.org/10.1007/BF02241866
  9. J. Chen, Some new iterative methods with three-order convergence, Appl. Math. Comput. 181 (2006), no. 2, 1519-1522. https://doi.org/10.1016/j.amc.2006.02.037
  10. A. Cordero and J. Torregrosa, Variants of Newton's method using fifth order quadrature formulas, Appl. Math. Comput. 190 (2007), no. 1, 686-698. https://doi.org/10.1016/j.amc.2007.01.062
  11. J. A. Ezquerro and M. A. Hernandez, A uniparametric Halley-type iteration with free second derivative, Internat. Int. J. Pure Appl. Math. 6 (2003), no. 1, 103-114.
  12. J. A. Ezquerro and M. A. Hernandez, On the R-order of the Halley method, J. Math. Anal. Appl. 303 (2005), no. 2, 591-601. https://doi.org/10.1016/j.jmaa.2004.08.057
  13. J. A. Ezquerro and M. A. Hernandez, New iterations of R-order four with reduced computational cost, BIT 49 (2009), no. 2, 325-342. https://doi.org/10.1007/s10543-009-0226-z
  14. M. Frontini and E. Sormani, Some variants of Newton's method with third order con-vergence, Appl. Math. Comput. 140 (2003), no. 2-3, 419-426. https://doi.org/10.1016/S0096-3003(02)00238-2
  15. J. M. Gutierrez and M. A. Hernandez, Recurrence relations for the super-Halley method, Comput. Math. Appl. 36 (1998), no. 7, 1-8.
  16. M. A. Hernandez, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001), no. 3-4, 433-455. https://doi.org/10.1016/S0898-1221(00)00286-8
  17. M. A. Hernandez and M. A. Salanova, Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces, Southwest J. Pure Appl. Math. 1 (1999), no. 1, 29-40.
  18. M. V. Kanwar, V. K. Kukreja, and S. Singh, On some third-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 171 (2005), no. 1, 272-280. https://doi.org/10.1016/j.amc.2005.01.057
  19. J. Kou and Y. Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007), no. 2, 1816-1821. https://doi.org/10.1016/j.amc.2006.12.062
  20. A. Y. Ozban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), no. 6, 677-682. https://doi.org/10.1016/S0893-9659(04)90104-8
  21. S. K. Parhi and D. K. Gupta, Semilocal convergence of a Stirling-like method in Banach spaces, Int. J. Comput. Methods 7 (2010), no. 2, 215-228. https://doi.org/10.1142/S0219876210002210
  22. M. S. Petkovic, B. Neta, L. Petkovic, and J. Dzunic, Multipoint methods for solving nonlinear equations, Elsevier, 2013.
  23. F. A. Potra and V. Ptak, Nondiscrete induction and iterative processes, Research Notes in Mathematics, Vol. 103, Pitman Publ., Boston, MA, 1984.
  24. L. B. Rall, Computational solution of nonlinear operator equations, Robert E. Krieger, New York, 1979.
  25. H. Ren, Q.Wu, and W. Bi, New variants of Jarratt method with sixth-order convergence, Numer. Algorithms 52 (2009), no. 4, 585-603. https://doi.org/10.1007/s11075-009-9302-3
  26. W. C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, In: Mathematical models and numerical methods (Papers, Fifth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1975), pp. 129-142, Banach Center Publ., 3, PWN, Warsaw, 1978.
  27. J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall Englewood Cliffs, New Jersey, USA, 1964.
  28. 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
  29. X. Xiao and H. Yin, A new class of methods with higher order of convergence for solving systems of nonlinear equations, (submitted for publication).
  30. X. Wang and J. Kou, Convergence for modified Halley-like methods with less computa-tion of inversion, J. Difference Equ. Appl. 19 (2013), no. 9, 1483-1500. https://doi.org/10.1080/10236198.2012.761979

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