Dynamical control of converging sequences computation

Abstract

Under some assumptions on the speed of convergence of a sequence, the significant digits of one of its iterates in common with the exact limit can be determined by comparing this iterate with the next one. Using a finite precision arithmetic, if computations are performed until the difference between two successive iterates is insignificant, the global error on the last iterate is minimal. Furthermore, for sequences converging at least linearly, we can determine in the result obtained which exact significant digits, i.e., not affected by round-off errors, are in common with the exact limit. This strategy can be used for the computation of integrals with the trapezoidal or Simpson's rule. A sequence is then generated by halving the step value at each iteration, while the difference between two successive iterates is a significant value. The exact significant digits of the last iterate are in common with the exact value of the integral, up to one bit. This kind of strategy is then extended to numerical algorithms involving several sequences, such as the approximation of integrals on an infinite interval.