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This paper aims at providing new modifications of Steffensen's scheme for solving nonlinear equations. The new general approach is discussed in detail theoretically. It is observed that a super fast R-order along with an interesting computational efficiency index will be achieved. Dynamical behaviors of the schemes are given to show an enlarged radii of convergence. Some experiments are also brought forward to support the given theory. [ABSTRACT FROM AUTHOR]

In this work, two-step methods with memory by applying three self-accelerator parameters are proposed and analyzed. In fact, we hit the high bound $7.77200^{\frac {1}{3}}\approx 1.98082$ as the efficiency index. Theoretical results are then supported by numerical examples. [ABSTRACT FROM AUTHOR]

We derive new iterative methods with memory for approximating a simple zero of a nonlinear single variable function. To this end, we first consider several modifications on some existing optimal classes without memory in such a way that their extensions to cases with memory could obtain the higher efficiency index 12 1 4 ≈ 1.86120 . Furthermore, we construct our main method with memory using three self-accelerators. It is demonstrated that this new scheme possesses the very high efficiency index 7.23814 1 3 ≈ 1.93438 . [ABSTRACT FROM AUTHOR]

Some new iterative methods without memory are constructed to reach the optimal convergence order four using only three function evaluations to solve nonlinear equations. Per full cycle, each method from the derived classes is derivative-free. We further extend the derived classes of methods to contribute some schemes with memory, without any additional evaluation of the function. Hence, with a same cost, the contributed methods with memory possess higher R-orders and better efficiency indices. In order to attest the efficiency of the obtained methods, we employ numerical comparisons. [ABSTRACT FROM AUTHOR]

In this work, we prove a third and fourth convergence order result for a family of iterative methods for solving nonlinear systems in Banach spaces. We analyze the semilocal convergence by using recurrence relations, giving the existence and uniqueness theorem that establishes the R-order of the method and the priori error bounds. Finally, we apply the methods to two examples in order to illustrate the presented theory. [ABSTRACT FROM AUTHOR]