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Improving the order and rates of convergence for the Super-Halley method in Banach spaces

  • Ioannis K. Argyros
Article

Abstract

In this study we are concerned with the problem of approximating a locally unique solution of an equation on a Banach space. A semilocal convergence theorem is given for the Super-Halley method in Banach spaces. Earlier results have shown that the order of convergence is four for a certain class of operators [4], [5], [8]. These results were not given in affine invariant form, and made use of a real quadratic majorizing polynomial. Here, we provide our results in affine invariant form showing that the order of convergence is at least four. In cases that it is exactly four the rate of convergence is improved. We achieve these results by using a cubic majorizing polynomial. Some numerical examples are given to show that our error bounds are better than earlier ones.

AMS Mathematics Subject Classification

65J15 65B05 47H17 49D15 

Key word and phrases

Super-Halley method Banach space majorizing sequence order of convergence rate of convergence 

References

  1. 1.
    Argyros, I.K.Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. Austral. Math. Soc. 32, (1985), 275–292.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Argyros, I.K.On the convergence of an Euler-Chebysheff-type method under Newton-Kantorovich hypotheses, Pure Math. Appl. Vol. 4, No. 3, (1993), 369–373.MATHMathSciNetGoogle Scholar
  3. 3.
    Argyros, I.K.Convergence results for the Super-Halley method using divided differences, Funct. et. Approx. Comment. Math. XXIII, (1994), 109–122.Google Scholar
  4. 4.
    Argyros, I.K., Chen, D. and Qian, Q.S.A note on the Halley method in Banach spaces, Appl. Math. Comp. 58, (1993), 215–224.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Argyros, I.K. and Szidarovszky, F.The Theory and Applications of Iteration Methods, C.R.C. Press, Inc., Boca Raton, Florida, U.S.A., 1993.MATHGoogle Scholar
  6. 6.
    Deuflhard, P. and Heindl, G.A.Affine invariant convergence theorems for Newton’s method and extension to related methods, SIAM J. Numer. Anal. 16, 1, (1979), 1–10.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gutierez, J.M., Hernandez, M.A. and Salanova, M.A.Accessibility of solutions by Newton’s method, Intern. J. Computer. Math. 57, (1995), 239–247.CrossRefGoogle Scholar
  8. 8.
    Gutierez, J.M., Hernandez, M.A. and Salanova,Resolution of quadratic equations in Banach spaces, Numer. Funct. Anal. Optimiz. 17 (1 and 2), (1996), 113–121.CrossRefGoogle Scholar
  9. 9.
    Gutierez, J.M. and Hernandez, M.A.Convex acceleration of Newton’s method, Pub. Sem. G. Galdeano, Serie 11, Seccion 1, 8, Univ. Zaragoza, 1995.Google Scholar
  10. 10.
    Hernandez, M.A.A note on Halley’s method, Num. Math. 59, 3, (1991), 273–276.MATHCrossRefGoogle Scholar
  11. 11.
    Hernandez, M.A. and Salanova, M.A.A family of Chebyshev-Haley type methods, Inter. J. Computer. Math. 47, (1993), 59–63.MATHCrossRefGoogle Scholar
  12. 12.
    Kantorovich, L.V. and Akilov, G.P.Functional Analysis, Pergamon Press, New York, U.S.A., 1982.MATHGoogle Scholar
  13. 13.
    Mertvecova, M.A.An analog of the process of tangent hyperbolas for general functional equations (Russian), Dokl. Akad. Nauk SSSP, 88, (1953), 611–614.MATHMathSciNetGoogle Scholar
  14. 14.
    Necepurenko, M.T.On Chebysheff’s method for functional equations (Russian), Usephi Mat. Nauk. 9, (1954), 163–170.MATHMathSciNetGoogle Scholar
  15. 15.
    Potra, F.A.Sharp error bounds for Newton’s process, Numer. Math. 34, (1980), 63–72.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Potra, F.A.An iterative algorithm of order 1.839... for solving nonlinear equations, Numer. Funct. Anal. Optimiz. 7, 1, (1984–85), 75–106.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 1998

Authors and Affiliations

  1. 1.Department of MathematicsCameron UniversityLawtonUSA

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