Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-28T19:41:55.462Z Has data issue: false hasContentIssue false

A family of Chebyshev-Halley type methods in Banach spaces

Published online by Cambridge University Press:  17 April 2009

J.M. Gutiérrez
Affiliation:
Dpt Matemáticas y Computación, Universidad de La Rioja, 26004 Logron¯o, Spain, e-mail: jmguti@siur.unirioja.es, mahernan@siur.unirioja.es
M.A. Hernández
Affiliation:
Dpt Matemáticas y Computación, Universidad de La Rioja, 26004 Logron¯o, Spain, e-mail: jmguti@siur.unirioja.es, mahernan@siur.unirioja.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A family of third-order iterative processes (that includes Chebyshev and Halley's methods) is studied in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates. This study allows us to compare the most famous third-order iterative processes.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Alefeld, G., ‘On the convergence of Halley's method’, Amer. Math. Monthly 88 (1981), 530536.CrossRefGoogle Scholar
[2]Altman, M., ‘Concerning the method of tangent hyperbolas for operator equations’, Bull. Acad. Pol. Sci., Ser. Sci. Math., Ast. et Phys. 9 (1961), 633637.Google Scholar
[3]Argyros, I.K. and Chen, D., ‘Results on the Chebyshev method in Banach spaces’, Proyecciones 12 (1993), 119128.Google Scholar
[4]Candela, V. and Marquina, A., ‘Recurrence relations for rational cubic methods I: The Halley method’, Computing 44 (1990), 169184.Google Scholar
[5]Candela, V. and Marquina, A., ‘Recurrence relations for rational cubic methods II: The Chebyshev method’, Computing 45 (1990), 355367.Google Scholar
[6]Chen, D., ‘Ostrowski–Kantorovich theorem and S–order of convergence of Halley method in Banach spaces’, Comment. Math. Univ. Carolin. 34 (1993), 153163.Google Scholar
[7]Chen, D., Argyros, I.K. and Qian, Q.S., ‘A note on the Halley method in Banach spaces’, Appl. Math. Comput. Sci. 58 (1993), 215224.CrossRefGoogle Scholar
[8]Chen, D., Argyros, I.K. and Qian, Q.S., ‘A local convergence theorem for the Super–Halley method in a Banach space’, Appl. Math. Lett. 7 (1994), 4952.Google Scholar
[9]Döring, B., ‘Einige Sätze über das verfahren der tangierenden hyperbeln in Banach–Räumen’, Aplikace Mat. 15 (1970), 418464.Google Scholar
[10]Gander, W., ‘On Halley's iteration method’, Amer. Math. Monthly 92 (1985), 131134.CrossRefGoogle Scholar
[11]Gutiérrez, J.M., Hernández, M.A. and Salanova, M.A., ‘Accesibility of solutions by Newton's method’, Intern. J. Computer Math. 57 (1995), 239247.Google Scholar
[12]Gutiérrez, J.M., Newton's method in Banach spaces, Ph.D. Thesis (University of La Rioja, Logrono, 1995).Google Scholar
[13]Hernández, M.A., ‘A note on Halley's method’, Numer. Math. 59 (1991), 273276.Google Scholar
[14]Hernández, M.A., ‘Newton–Raphson's method and convexity’, Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 22 (1993), 159166.Google Scholar
[15]Hernández, M.A. and Salanova, M.A., ‘A family of Chebyshev–Halley type methods’, Intern. J. Computer Math. 47 (1993), 5963.Google Scholar
[16]Kantorovich, L.V. and Akilov, G.P., Functional analysis (Pergamon Press, Oxford, 1982).Google Scholar
[17]Ostrowski, A.M., Solution of equations in Euclidean and Banach spaces (Academic Press, New York, 1943).Google Scholar
[18]Rall, L.B., Computational solution of nonlinear operator equations (Robert E. Krieger Publishing Company, Inc., New York, 1979).Google Scholar
[19]Rheinboldt, W.C., ‘A unified convergence theory for a class of iterative process’, SIAM J. Numer. Anal. 5 (1968), 4263.CrossRefGoogle Scholar
[20]Yamamoto, T., ‘On the method of tangent hyperbolas in Banach spaces’, J. Comput. Appl. Math. 21 (1988), 7586.CrossRefGoogle Scholar