대한수학회지 (Journal of the Korean Mathematical Society)

  • 제54권1호
  • /
  • Pages.17-33
  • /
  • 2017
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

IMPROVED CONVERGENCE OF STEFFENSEN'S METHOD FOR APPROXIMATING FIXED POINTS OF OPERATORS IN BANACH SPACE

  • Argyros, Ioannis K. (Department of Mathematical Sciences Cameron University) ;
  • Ren, Hongmin (College of Information and Engineering Hangzhou Polytechnic)
  • 투고 : 2015.09.27
  • 발행 : 2017.01.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

We present a new local as well as a semilocal convergence analysis for Steffensen's method in order to locate fixed points of operators on a Banach space setting. Using more precise majorizing sequences we show under the same or less computational cost that our convergence criteria can be weaker than in earlier studies such as [1-13], [21, 22]. Numerical examples are provided to illustrate the theoretical results.

키워드

참고문헌

  1. S. Amat, S. Busquier, and V. F. Candela, A class of quasi Newton generalized Steffensen's methods on Banach spaces, J. Comput. Appl. Math. 149 (2002), no. 2, 397-406. https://doi.org/10.1016/S0377-0427(02)00484-3
  2. I. K. Argyros, An error analysis for the Steffensen method under generalized Zabrejko-Nguen-type assumptions, Rev. Anal. Numer. Theor. Approx. 25 (1996), no. 1-2, 11-22.
  3. I. K. Argyros, On the convergence of Steffensen-Galerkin methods, Atti Sem. Mat. Fis. Univ. Modena 48 (2000), no. 2, 355-370.
  4. I. K. Argyros, On the convergence of Steffensen-Galerkin methods, Ann. Univ. Sci. Budapest. Sect. Comput. 21 (2002), 3-18.
  5. I. K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton like methods in Banach space, J. Math. Anal. Appl. 298 (2004), no. 2, 374-397. https://doi.org/10.1016/j.jmaa.2004.04.008
  6. I. K. Argyros, Convergence and Application of Newton-type Iterations, Springer, 2008.
  7. I. K. Argyros, An improved local convergence analysis for Newton-Steffensen-type methods, J. Appl. Math. Computing 32 (2010), no. 1, 111-118. https://doi.org/10.1007/s12190-009-0236-7
  8. I. K. Argyros, A semilocal convergence analysis for directional Newton methods, Math. Comp. 80 (2011), no. 273, 327-343. https://doi.org/10.1090/S0025-5718-2010-02398-1
  9. I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical Methods for Equations and its Applications, CRC Press, Taylor and Francis, New York, 2012.
  10. I. K. Argyros and S. Hilout, Sreffensen methods for solving generalized equations, Serdica Math. J. 34 (2008), 1001-1012.
  11. I. K. Argyros and S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complexity 28 (2012), no. 3, 364-387. https://doi.org/10.1016/j.jco.2011.12.003
  12. B. A. Bel'tjukov, On a certain method of solution of nonlinear functional equations, Z. Vycisl. Mat. i Mat. Fiz. 5 (1965), 927-931.
  13. K.-W. Chen, Generalization of Steffensen's method for operator equations, Comment. Math. Univ. Carolinae 5 (1964), no. 2, 47-77.
  14. L. B. Ciric, Generalized contractions and fixed-point theorems, Publ. Inst. Math. (Beograd) 12(26) (1971), 19-26.
  15. L. B. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267-273.
  16. L. B. Ciric, On fixed point theorems in Banach spaces, Publ. Inst. Math. 19(33) (1975), 43-50.
  17. L. B. Ciric, Fixed Point Theory, Contraction Mapping Principle, FME Press, Beograd, 2003.
  18. L. B. Ciric and J. S. Ume, Iterative processes with errors for nonlinear equations, Bull. Austral. Math. Soc. 69 (2004), no. 2, 177-189. https://doi.org/10.1017/S0004972700035929
  19. A. Cordero, J. R. Torregrosa, and M. P. Vasileva, Increasing the order of convergence of iterative schemes for solving nonlinear systems, J. Comput. Appl. Math. 253 (2013), 86-94.
  20. J. A. Ezquerro and M. A. Hernandez, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000), no. 2, 227-236. https://doi.org/10.1007/s002459911012
  21. L. W. Johnson and D. R. Scholz, On Steffensen's method, SIAM J. Numer. Anal. 5 (1968), 296-302. https://doi.org/10.1137/0705026
  22. L. V. Kantorovich and G. P. Akilov, Functional analysis in normed spaces, The MacMillan Company, New York, 1964.
  23. A. A. Magrenan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248 (2014), 215-224.
  24. J .W. Schmidt, Eine Ubertragung der Regulae Falsi aur Gleichungen in Banachraumen, I. II, Z. Angew. Math. Mech. 43 (1963), 1-8 https://doi.org/10.1002/zamm.19630430102
  25. J .W. Schmidt, Eine Ubertragung der Regulae Falsi aur Gleichungen in Banachraumen, I. II, Z. Angew. Math. Mech. 43 (1963), 97-110. https://doi.org/10.1002/zamm.19630430302
  26. S. Ul'm, A generalization of Steffensen's method for solving non-linear operator equations, Z. Vycisl. Mat. i Mat. Fiz. 4 (1964), 1093-1097.
  27. L. Wegge, On a discrete version of the Newton-Raphson method, SIAM J. Numer. Anal. 3 (1966), no. 1, 134-142. https://doi.org/10.1137/0703009