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 cestac1§method and cadna2¶ 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 (ϵ) 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.