Minisymposium Presentation

Classical probabilistic rounding error analysis is well suited to stochastic rounding (SR), yielding strong results for floating-point algorithms relying on summation. For many numerical linear algebra algorithms, one can prove probabilistic error bounds that grow as $\mathcal{O}(\sqrt{n}u)$, where $n$ is the problem size and $u$ is the unit roundoff. These bounds are asymptotically tighter than the worst-case ones, which grow as $\mathcal{O}(nu)$. For certain algorithms, SR is unbiased. However, all these results were derived under the assumption that SR is implemented exactly, which requires a number of random bits that is too large to be suitable for practical implementations. We investigate the effect of using $r$ random bits in probabilistic SR error analysis. To this end, we introduce a new rounding mode, limited-precision SR. Considering the $r$ used, this new rounding mode accurately matches hardware implementations, unlike the ideal SR generally used in the literature. We show that this new rounding mode is biased and that the bias is a function of $r$. As $r$ approaches infinity, however, the bias disappears, and limited-precision SR converges to the ideal SR. We develop a novel model for probabilistic error analysis of algorithms employing SR. Several numerical examples corroborate our theoretical findings.