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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. {\it H. T. Kung} and {\it 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.

65H05Single nonlinear equations (numerical methods)
Full Text: DOI
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