Dynamical ‎C‎ontrol of Computations Using the Family of Optimal Two-point Methods to Solve Nonlinear ‎Equations

One of the considerable discussions for solving the nonlinear equations is to find the optimal iteration, and to use a proper termination criterion which is able to obtain a high accuracy for the numerical solution. In this paper, for a certain class of the family of optimal two-point methods, we propose a new scheme based on the stochastic arithmetic to find the optimal number of iterations in the given iterative solution and obtain the optimal solution with its accuracy. For this purpose, a theorem is proved to illustrate the accuracy of the iterative method and the CESTAC^۱\footnote{^۱Controle et Estimation Stochastique des Arrondis de Calculs} method and CADNA^۲\footnote{^۲Control of Accuracy and Debugging for Numerical Application} library are applied which allows us to estimate the round-off error effect on any computed result. The classical criterion to terminate the iterative procedure is replaced by a criterion independent of the given accuracy (\epsilon) such that the best solution is evaluated numerically, which is able to stop the process as soon as a satisfactory informatical solution is obtained. Some numerical examples are given to validate the results and show the efficiency and importance of using the stochastic arithmetic in place of the floating-point ‎arithmetic.‎