Dynamic control of a bicgstab algorithm Original Research Article

In many scientific fields, krylov-type methods prove to be very powerful. Nevertheless, they are prone to rounding errors that slow down their convergence rate. Some of these methods, like the bicgstab method, are also subject to breakdowns and near-breakdowns, situations where a denominator is zero or very affected by rounding errors. Generally, the last case is detected by means of the classical inequality |x|<ε in floating-point arithmetic but it is very sensitive to ε values. The discrete stochastic arithmetic enables us to estimate the accuracy of intermediate results and to avoid us from these a priori choices. Look ahead techniques allow to carry on computations, but sometimes, it is necessary or at least more advantageous to restart the algorithm. The aim of this paper is to present a dynamic strategy which allows to detect breakdowns and near-breakdowns and to choose between a look ahead technique and a restart.