## An optimal Steffensen-type family for solving nonlinear equations.(English)Zbl 1227.65044

The paper is devoted to the numerical solution of nonlinear equations $$f(x) = 0$$. The authors use Newton’s iteration for the direct Newtonian interpolation of the function to construct optimal Steffensen-type methods of second-, fourth- and eighth-order, which use only two, three and four evaluations of the function, respectively. Moreover, they deduce the corresponding error equations and asymptotic convergence constants. The proposed general optimal Steffensen-type family only uses $$n$$ evaluations of $$f$$ to achieve the optimal $$2^{n-1}$$th order of convergence for solving the simple root of nonlinear functions, and the authors compare this family with Newton’s method, Steffensen’s method, Ren-Wu-Bi’s method, Kung-Traub’s method and Neta-Petković’s method for solving nonlinear equations in numerical examples.

### MSC:

 65H05 Numerical computation of solutions to single equations
Full Text:

### References:

 [1] Ortega, J.M.; Rheinboldt, W.G., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046 [2] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0121.11204 [3] Ostrowski, A.M., Solutions of equations and system of equations, (1960), Academic Press New York · Zbl 0115.11201 [4] Kung, H.T.; Traub, J.F., Optimal order of one-point and multipoint iteration, J. assoc. comput. math., 21, 634-651, (1974) · Zbl 0289.65023 [5] Ren, H.; Wu, Q.; Bi, W., A class of two-step Steffensen type methods with fourth-order convergence, Appl. math. comput., 209, 206-210, (2009) · Zbl 1166.65338 [6] Liu, Z.; Zheng, Q.; Zhao, P., A variant of steffensen’s method of fourth-order convergence and its applications, Appl. math. comput., 216, 1978-1983, (2010) · Zbl 1208.65064 [7] Petković, M.S.; Ilić, S.; Džunić, J., Derivative free two-point methods with and without memory for solving nonlinear equations, Appl. math. comput., 217, 1887-1895, (2010) · Zbl 1200.65034 [8] Zheng, Q.; Zhao, P.; Zhang, L.; Ma, W., Variants of Steffensen-secant method and applications, Appl. math. comput., 216, 3486-3496, (2010) · Zbl 1200.65036 [9] Alarcón, V.; Amat, S.; Busquier, S.; López, D.J., A steffensen’s type method in Banach spaces with applications on boundary-value problems, J. comput. appl. math., 216, 243-250, (2008) · Zbl 1139.65040 [10] Neta, B.; Petković, M.S., Construction of optimal order nonlinear solvers using inverse interpolation, Appl. math. comput., 217, 2448-2455, (2010) · Zbl 1202.65062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.