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A class of Steffensen type methods with optimal order of convergence. (English) Zbl 1216.65055
Summary: A family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. H. T. Kung and J. F. Traub [J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)] conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound ${2}^{d-1}$, where $d$ is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case $d=3$. Numerical examples are presented to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare them with others.
##### MSC:
 65H05 Single nonlinear equations (numerical methods)