Abstract
Based on Ostrowski’s fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski’s method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2n − 1. Thus, the family agrees with Kung–Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.
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References
Bi, W., Ren, H., Wu, Q.: New family of seventh-order methods for nonlinear equations. Appl. Math. Comput. 203, 408–412 (2008)
Bi, W., Ren, H., Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 225, 105–112 (2009)
Chapra, S.C., Canale, R.P.: Numerical Methods for Engineers. McGraw-Hill, New York (1988)
Chun, C., Ham, Y.: Some sixth-order variants of Ostrowski root-finding methods. Appl. Math. Comput. 193, 389–394 (2007)
Gautschi, W.: Numerical Analysis: an Introduction. Birkhäuser, Boston (1997)
Grau, M., Díaz-Barrero, J.L.: An improvement to Ostrowski root-finding method. Appl. Math. Comput. 173, 450–456 (2006)
King, R.F.: A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)
Kou, J., Li, Y., Wang, X.: Some variants of Ostrowski’s method with seventh-order convergence. J. Comput. Appl. Math. 209, 153–159 (2007)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic, New York (1970)
Ostrowski, A.M.: Solution of Equations in Euclidean and Banach Spaces. Academic, New York (1960)
Sharma, J.R., Guha, R.K.: A family of modified Ostrowski methods with accelerated sixth-order convergence. Appl. Math. Comput. 190, 111–115 (2007)
Sharma, J.R., Guha, R.K., Sharma, R.: Some variants of Hansen-Patrick method with third and fourth order convergence. Appl. Math. Comput. 214, 171–177 (2009)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third order convergence. Appl. Math. Lett. 13, 87–93 (2000)
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Sharma, J.R., Sharma, R. A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer Algor 54, 445–458 (2010). https://doi.org/10.1007/s11075-009-9345-5
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DOI: https://doi.org/10.1007/s11075-009-9345-5