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

Subject Areas : International Journal of Industrial Mathematics

M. A.  Fariborzi ‎Araghi‎ ¹ (Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran)
E.  Zarei‎ ² (Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran.)

Received: 2017-05-08 Accepted : 2017-05-08 Published : 2017-04-01

Keywords: Stochastic arithmetic, CESTAC method, CADNA library, Two-point methods, Nonlinear ‎equations,

Abstract :

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$^1$\footnote{$^1$Controle et Estimation Stochastique des Arrondis de Calculs} method and CADNA$^2$\footnote{$^2$Control 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.‎