Abstract
In this paper, we consider the semilocal convergence of multi-point improved super-Halley-type methods in Banach space. Different from the results of super-Halley method studied in reference Gutiérrez, J.M. and Hernández, M.A. (Comput. Math. Appl. 36,1–8, 1998) these methods do not require second derivative of an operator, the R-order is improved and the convergence condition is also relaxed. We prove a convergence theorem to show existence and uniqueness of the solution.
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References
Ostrowski, A.M. Solution of Equations in Euclidean and Banach Spaces, 3rd edn. Academic Press, New York (1973)
Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev-Halley type methods in banach spaces. Bull. Aust. Math. Soc 55, 113–130 (1997)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: the Halley method. Computing 44, 169–184 (1990)
Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-type methods. Appl. Math. Optim 41(2), 227–236 (2000)
Chen, D., Argyros, I.K., Qian, Q.S.: A note on the Halley method in Banach spaces. App. Math. Comput 58, 215–224 (1993)
Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Appl 36, 1–8 (1998)
Babajee, D.K.R., Dauhoo, M.Z., Darvishi, M.T., Karami, A., Barati, A.: Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. J. Comput. Appl. Math 233, 2002–2012 (2010)
Ezquerro, J.A., Hernández, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl 303, 591–601 (2005)
Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA. J. Numer. Anal 11, 21–31 (1991)
Ezquerro, J.A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer Math 49, 325–342 (2009)
Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci 32, 257–264 (1977)
Kou, J., Li, Y.: A family of modified super-Halley methods with fourth-order convergence. Appl Math. Comput 189, 366–370 (2007)
Hernández, M.A.: Second-derivative-free variant of the Chebyshev method for nonlinear equations. J. Optim. Theory Appl 104(3), 501–515 (2000)
Powell, M.J.D.: On the convergence of trust region algorithms for unconstrained minimization without derivatives. Comput Optim Appl 53, 527–555 (2012)
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Wang, X., Kou, J. Semilocal convergence of multi-point improved super-Halley-type methods without the second derivative under generalized weak condition. Numer Algor 71, 567–584 (2016). https://doi.org/10.1007/s11075-015-0010-x
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DOI: https://doi.org/10.1007/s11075-015-0010-x