Abstract
In this paper, the estimate of the radius of the convergence ball of the modified Halley’s method for finding multiple zeros of nonlinear equations is provided under the hypotheses that the derivative f (m + 1) of function f is Hölder continuous, and f (m + 1) is bounded. The uniqueness ball of solution is also established. Finally, some examples are provided to show applications of our theorem.
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Schröer, E.: Üer unendlich viele Algorithmen zur Auflöung der Gleichungen. Math. Ann. 2, 317–365 (1870)
Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977)
Traub, J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1977)
Dong, C.: A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation. Math. Numer. Sinica 11, 445–450 (1982)
Victory, H.D., Neta, B.: A higher order method for multiple zeros of nonlinear functions. Int. J. Comput. Math. 12, 329–335 (1983)
Osada, N.: An optimal multiple root-finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994)
Frontini, M., Sormani, E.: Modified Newton’s method with third-order convergence and multiple roots. J. Comput. Appl. Math. 156, 345–354 (2003)
Neta, B.: New third order nonlinear solvers for multiple roots. Appl. Math. Comput. 202, 162–170 (2008)
Neta, B., Johnson, A.N.: High order nonlinear solver for multiple roots. Comput. Math. Appl. 55, 2012–2017 (2008)
Chun, C., Neta, B.: A third-order modification of Newton’s method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)
Shengguo, L., Housen, L., Lizhi, C.: Some second-derivative-free variants of Halley’s method for multiple roots. Appl. Math. Comput. 215, 2192–2198 (2009)
Homeier, H.H.H.: On Newton-type methods for multiple roots with cubic convergence. J. Comput. Appl. Math. 231, 249–254 (2009)
Traub, J.F., Wozniakowski, H.: Convergence and complexity of Newton iteration for operator equation. J. Assoc. Comput. Mech. 26, 250–258 (1979)
Wang, X.H.: The convergence ball on Newton’s method. Chin. Sci. Bull. (A Special Issue of Mathematics, Physics, Chemistry) 25, 36–37 (1980) (in Chinese)
Huang, Z.D.: The convergence ball of Newton’s method and the uniqueness ball of equations under Hölder-type continuous derivatives. Comput. Math. Appl. 5, 247–251 (2004)
Guo, X.P.: Convergence ball of iterations with one parameter. Appl. Math. J. Chinese Univ. Ser. B 20, 462–468 (2005)
Luo, X.F., Ye, X.T., Convergence of the family of the deformed Euler–Halley iterations with parameters under the Hölder condition of the derivative. Numer. Math. J. Chinese Univ. 27, 185–192 (2005)
Ren, H.M., Wu, Q.B.: The convergence ball of the Secant method under Hölder continuous divided differences. J. Comput. Appl. Math. 194, 284–293 (2006)
Wu, Q.B., Ren, H.M.: Convergence ball of a modified secant method for finding zero of derivatives. Appl. Math. Comput. 174, 24–33 (2006)
Wu, Q.B., Ren, H.M., Bi, W.H.: Convergence ball and error analysis of Müller’s method. Appl. Math. Comput. 184, 464–470 (2007)
Ren, H.M., Wu, Q.B.: Convergence ball and error analysis of a family of iterative methods with cubic convergence. Appl. Math. Comput. 209, 369–378 (2009)
Wu, Q.B., Ren, H.M., Bi, W.H.: The convergence ball of Wang’s method for finding a zero of a derivative. Math. Comput. Modell. 49, 740–744 (2009)
Safiev, R.A.: On some iterative processes, Ž. Vyčcisl. Mat. Fiz. 4m 139–143 (1964). Translated into English by L.B. Rall as MRC Technical Summary Report, No. 649, Univ. Wisconsin-Madison, 1966
Yamamoto, T.: On the method of tangent hyperbolas in Banach spaces. J. Comput. Appl. Math. 21, 75–86 (1988)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I. The Halley method. Computing 44, 169–184 (1990)
Chen, D., Argyros, I.K., Qian, Q.S.: A note on the Halley method in Banach spaces. Appl. Math. Comput. 58, 215–224 (1993)
Han, D.F., Wang, X.H.: The error estimates of Halley’s method. Numer. Math. J. Chinese Univ. (English Ser.) 6, 231–239 (1997)
Wang, X.H.: Convergence on the iteration of Halley’s family in weak condition. Chin. Sci. Bull. 42, 119–121 (1997)
Argyros, I.K.: The Halley method in Banach spaces and the Ptak error estimates. Rev. Acad. Cienc. Zaragoza 52, 31–41 (1997)
Wang, X.H.: Convergence of the iteration of Halley’s family and Smale operator class in Banach space. Sci. China Ser. A 41, 700–709 (1998)
Hernández, M.A., Salanova, M.A.: Indices of convexity and concavity: application to Halley method. Appl. Math. Comput. 103 27–49 (1999)
Ezquerro, J.A., Hernández, M.A.: On the R-order of the Halley mthod. J. Math. Anal. Appl. 303, 591–601 (2005)
Ezquerro, J.A., Hernandez, M.A.: Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57, 354–360 (2007)
Ye, X.T., Li, C.: Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative. J. Comput. Appl. Math. 194, 294–308 (2006)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, New York (1980)
Xu, L., Wang, X.: Topics on Methods and Examples of Mathematical Analysis. High Education Press (1983) (in Chinese)
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Bi, W., Ren, H. & Wu, Q. Convergence of the modified Halley’s method for multiple zeros under Hölder continuous derivative. Numer Algor 58, 497–512 (2011). https://doi.org/10.1007/s11075-011-9466-5
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DOI: https://doi.org/10.1007/s11075-011-9466-5