A semilocal convergence analysis for the method of tangent parabolas

Ioannis K. Argyros

doi:10.33993/jnaat341-786

Authors

Ioannis K. Argyros Cameron of University, Lawton, USA

DOI:

https://doi.org/10.33993/jnaat341-786

Keywords:

Banach space, tangent parabola, Euler-Chebyshev method, majorizing sequence, Fréchet-derivative, Lipschitz-center conditions

Abstract views: 226

Abstract

We present a semilocal convergence analysis for the method of tangent parabolas (Euler-Chebyshev) using a combination of Lipschitz and center Lipschitz conditions on the Fréchet derivatives involved. This way we produce a majorizing sequence which converges under weaker conditions than before. The error bounds obtained are more precise and the information of the location of the solution better than in earlier results.

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References

Argyros, I.K., On the convergence of a Chebysheff-Halley-type method under Newton-Kantorovich hypotheses, Appl. Math. Letters, 6, no. 5, pp. 71-74, 1993, https://doi.org/10.1016/0893-9659(93)90104-u DOI: https://doi.org/10.1016/0893-9659(93)90104-U

Argyros, I.K., A note on the Halley method in Banach spaces, Appl. Math. and Comp., 58, pp. 215-224, 1993, https://doi.org/10.1016/0096-3003(93)90137-4 DOI: https://doi.org/10.1016/0096-3003(93)90137-4

Argyros, I.K., Advances in the Efficiency of Computational Methods and Applications, World Scientific Publ. Co., River Edge, NJ, 2000, https://doi.org/10.1142/4448 DOI: https://doi.org/10.1142/4448

Ezquerro, J.A. and Hernandez, M.A., A modification of the super-Halley method under mild differentiability conditions, J. Comput. Appl. Math., 114, pp. 405-409, 2000, https://doi.org/10.1016/s0377-0427(99)00348-9 DOI: https://doi.org/10.1016/S0377-0427(99)00348-9

Gutiérrez, J.M. and Hernandez, M.A., A family of Chebyshev-Halley type methods in Banach spaces, Bull. Austral. Math. Soc., 55, pp. 113-130, 1997, https://doi.org/10.1017/s0004972700030586 DOI: https://doi.org/10.1017/S0004972700030586

Kanno, S., Convergence theorems for the method of tangent hyperbolas, Math. Japonica, 37 no. 4, pp. 711-722, 1992.

Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982. DOI: https://doi.org/10.1016/B978-0-08-023036-8.50010-2

Mertvecova, M.A., An analog of the process of tangent hyperbolas for general functional equations, Dokl. Akad. Nauk SSSR, 88, pp. 611-614, 1953.

Necepurenko, M.T., On Chebysheff's method for functional equations, Usephi, Mat. Nauk., 9, pp. 163-170, 1954.

Safiev, R.A., The method of tangent hyperbolas, Sov. Math. Dokl., 4, pp. 482-485, 1963.

Schwetlick, H., Numerische Losung Nichtlinearer Gleichungen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979.

Yamamoto, T., On the method of tangent hyperbolas in Banach spaces, J. Comput. Appl. Math., 21, pp. 75-86, 1988, https://doi.org/10.1016/0377-0427(88)90389-5 DOI: https://doi.org/10.1016/0377-0427(88)90389-5