Resumen de A reliable algorithm to check the accuracy of iterative schemes for solving nonlinear equations: an application of the CESTAC method

Mohammad Ali Fariborzi Araghi

• The aim of this study is to apply the discrete stochastic arithmetic (DSA) to validate the class of muli-step iterative methods and find the optimal numerical solution of nonlinear equations. To this end, the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are applied. By using this approach, the optimal number of iteration and the optimal solution with its accuracy are found. In this case, the usual stopping termination in the iterative procedure is replaced by a new criterion which is independent of the given tolerance (ϵ) such that the optimal results are evaluated computationally. A main theorem is proved which shows the accuracy of the iterative schemes by means of the concept of common significant digits. The numerical results are presented to illustrate the efficiency and importance of using the DSA in place of the floating-point arithmetic (FPA).