Dynamical strategies using Discrete Stochastic Arithmetic for approximation methods

Let us consider the converging sequence generated by successively dividing by two the step size used in an approximation method. With an appropriate stopping criterion, we show that in the last approximation obtained, the significant bits which are not affected by round-off errors are in common with the exact result, up to one. This strategy has been successfully applied to several composite quadrature methods. Other strategies, which are not based on “step halving”, are also proposed. For approximation methods of a relatively high order, these alternative strategies may sometimes be less costly.