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In this paper, we consider two-step fourth-order and three-step sixth-order derivative free iterative methods and study their convergence in Banach spaces to approximate a locally-unique solution of nonlinear equations. Study of convergence analysis provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions using higher order derivatives. Furthermore, in quest of fast algorithms, a generalized q-step scheme with increasing convergence order 2q + 2 is proposed and analyzed. Novelty of the q-step algorithm is that, in each step, order of convergence is increased by an amount of two at the cost of only one additional function evaluation. To maximize the computational efficiency, the optimal number of steps is calculated. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation. [ABSTRACT FROM AUTHOR]

Based on a two-step Newton-like scheme, we propose a three-step scheme of convergence order p + 2 (p ≥ 3) for solving systems of nonlinear equations. Furthermore, on the basis of this scheme a generalized k+2-step scheme with increasing convergence order p + 2k is presented. Local convergence analysis, including radius of convergence and uniqueness results of the methods, is presented. Computational efciency in the general form is discussed. Theoretical results are verifed through numerical experimentation. Finally, performance is demonstrated by application of the methods on some nonlinear systems of equations. [ABSTRACT FROM AUTHOR]

We propose a derivative-free iterative method with fifth order of convergence for solving systems of nonlinear equations. The scheme is composed of three steps, of which the first two steps are that of third order Traub-Steffensen-type method and the last is derivative-free modification of Chebyshev's method. Computational efficiency is examined and comparison between the efficiencies of presented technique with existing techniques is performed. It is proved that, in general, the new method is more efficient. Numerical problems, including those resulting from practical problems viz. integral equations and boundary value problems, are considered to compare the performance of the proposed method with existing methods. Calculation of computational order of convergence shows that the order of convergence of the new method is preserved in all the numerical examples, which is not so in the case of some of the existing higher order methods. Moreover, the numerical results, including the CPU-time consumed in the execution of program, confirm the accurate and efficient behavior of the new technique. [ABSTRACT FROM AUTHOR]