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In this work we focus on iterative splitting methods that are one of the most important techniques to solve large systems of ordinary and partial differential equations. Serial versions of these methods have been deeply studied in published works, see [1] and [3]. Our main aim in this paper is to extend iterative splitting methods to parallel iterative splitting methods based on the idea of multi-splitting approaches improving the approximated solution obtained. Moreover, the parallel splitting versions guarantee the reduction of the computational time needed preserving a high accuracy in the whole process and finally giving a very efficient alternative for solving these applied problems. So, we recall that parallel iterative splitting methods are important to solve large scale problems, which decompose large problems into simple manageable subproblems and solve them independently by using different processors. The principal notion it is the use of multi-splitting methods, see [2], [10] and [15], we improve these methods designing new parallel iterative splitting methods which are solved with waveform-relaxation (WR) methods, see [9]. We analyze the convergence results of the parallel iterative splitting methods by using known partial convergence results of each individual WR methods. Finally, we perform numerical tests in order to corroborate the theoretical results obtained, so we give convergence results of the serial and parallel iterative splitting methods and show the competitiveness of these parallel versions. [ABSTRACT FROM AUTHOR]

Second and third-order Steffensen-type methods have been developed in order to estimate the solution of nonlinear systems of equations without the evaluation of any jacobian matrix, being the efficiency indices of the resulting schemes much higher than their counterparts. [ABSTRACT FROM AUTHOR]

In this work, some families of derivative-free methods, with optimal and non-optimal order of convergence, for solving nonlinear equations are suggested. In the proposed methods, polynomials of different degrees are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. [ABSTRACT FROM AUTHOR]