Two bi-accelerator improved with memory schemes for solving nonlinear equations (Q1723575)

Summary: The present paper is devoted to the improvement of the \(R\)-order convergence of with memory derivative free methods presented by \textit{T. Lotfi} et al. [ibid. 2014, Article ID 912796, 9 p. (2014; Zbl 1422.65077)] without doing any new evaluation. To achieve this aim one more self-accelerating parameter is inserted, which is calculated with the help of Newton's interpolatory polynomial. First theoretically it is proved that the \(R\)-order of convergence of the proposed schemes is increased from 6 to 7 and 12 to 14, respectively, without adding any extra evaluation. Smooth as well as nonsmooth examples are discussed to confirm theoretical result and superiority of the proposed schemes.