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Journal Articles ACM Transactions on Mathematical Software Year : 2023

Accurate calculation of Euclidean Norms using Double-word arithmetic

Abstract

We consider the computation of the Euclidean (or L2) norm of an n-dimensional vector in floating-point arithmetic. We review the classical solutions used to avoid spurious overflow or underflow and/or to obtain very accurate results. We modify a recently published algorithm (that uses double-word arithmetic) to allow for a very accurate solution, free of spurious overflows and underflows. To that purpose, we use a double-word square-root algorithm of which we provide a tight error analysis. The returned L2 norm will be within very slightly more than 0.5 ulp from the exact result, which means that we will almost always provide correct rounding.
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Dates and versions

hal-03482567 , version 1 (16-12-2021)
hal-03482567 , version 2 (07-10-2022)

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Vincent Lefèvre, Nicolas Louvet, Jean-Michel Muller, Joris Picot, Laurence Rideau. Accurate calculation of Euclidean Norms using Double-word arithmetic. ACM Transactions on Mathematical Software, 2023, 49 (1), pp.1-34. ⟨10.1145/3568672⟩. ⟨hal-03482567v2⟩
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