Why are numerical algorithms accurate at large scale and low precisions?

Standard worst-case rounding error bounds of most numerical linear algebra algorithms grow linearly with the problem size and the machine precision. These bounds suggest that numerical algorithms could be inaccurate at large scale and/or at low precisions, but fortunately they are pessimistic. We will review recent advances in probabilistic rounding error analyses, which have attracted renewed interest due to the emergence of low precisions on modern hardware as well as the rise of stochastic rounding.

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