Year: 2025
Type: article
Abstract: . The quality of numerical computations can be measured through their forward error, for which finding good error bounds is challenging in general. For several algorithms and using stochastic rounding (SR), probabilistic analysis has been shown to be an effective alternative for obtaining tight error bounds. This analysis considers the distribution of errors and evaluates the algorithm’s performance on average. Using martingales and the Azuma–Hoeffding inequality, it provides error bounds that are valid with a certain probability and in [Formula: see text] instead of deterministic worst-case bounds in [Formula: see text], where [Formula: see text] is the number of operations and [Formula: see text] is the unit roundoff. In this paper, we present a general method that automatically constructs a martingale for any computation scheme with multilinear errors based on additions, subtractions, and multiplications. We apply this generalization to algorithms previously studied with SR, such as pairwise summation and the Horner algorithm, and prove equivalent results. We also analyze a previously unstudied algorithm, Karatsuba polynomial multiplication, which illustrates that the method can handle reused intermediate computations. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/verificarlo/sr-karatsuba and in the supplementary materials ( sr-karatsuba-main.zip [6.87KB]). [Formula: see text]
DOI: 10.1137/24m1710966
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
OpenAlex ID: W4416844416