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Three-step iterative methods with eighth-order convergence for solving nonlinear equations. (English) Zbl 1161.65039

For solving a nonlinear scalar equation, the authors propose a one parameter family of three-step iterative methods of eight-order. The new methods are based on the classical Newton method, on King’s methods [see R. F. King, SIAM J. Numer. Anal. 10, 876–879 (1973; Zbl 0266.65040)], and on the methods of C. Chun and Y. Ham [Appl. Math. Comp. 193, 389–394 (2007)].

The iterations are expressed by means of divided difference of orders two and three and by means of a given real-valued function. To apply these methods for solving equations, three evaluations of the function from the left hand side of the equation and one evaluation of its first derivative are required. Using the definition of efficiency index, the family of derived methods has the index 1.682, better than the index of Newton’s method, of King’s methods and of Chun’s methods. Numerical examples are performed and comparison of various iterative methods under the same total number of function evaluations are made.

65H05Single nonlinear equations (numerical methods)
65Y20Complexity and performance of numerical algorithms