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We develop a class of n -point iterative methods with optimal 2 n order of convergence for solving nonlinear equations. Newton's second order and Ostrowski's fourth order methods are special cases corresponding to n = 1 and n = 2 . Eighth and sixteenth order methods that correspond to n = 3 and n = 4 of the class are special cases of the eighth and sixteenth order methods proposed by Sharma et al. [25] . The methodology is based on employing the previously obtained ( n − 1 ) -step scheme and modifying the n -th step by using rational Hermite interpolation. Unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Theoretical results are verified through numerical experimentations. The performance is also compared with already established methods in literature. It is observed that new algorithms are more accurate than existing counterparts and very effective in high precision computations. [ABSTRACT FROM AUTHOR]