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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]

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]