Precision-aware deterministic and probabilistic error bounds for floating point summation

We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale on a computational tree, which leads to explicit and interpre...

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Bibliographic Details
Published in:Numerische Mathematik 2023-10, Vol.155 (1-2), p.83-119
Main Authors: Hallman, Eric, Ipsen, Ilse C. F.
Format: Article
Language:English
Subjects:
Algorithms
Error analysis
Floating point arithmetic
Martingales
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Nonlinear control
Numerical Analysis
Numerical and Computational Physics
Parameters
Real numbers
Simulation
Statistical analysis
Systematic errors
Theoretical
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Summary:We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale on a computational tree, which leads to explicit and interpretable bounds with nonlinear terms controlled explicitly rather than by big-O terms. Two probability parameters strengthen the precision-awareness of our bounds: one parameter controls the first order terms in the summation error, while the second one is designed for controlling higher order terms in low precision or extreme-scale problem dimensions. Our systematic approach yields new deterministic and probabilistic error bounds for three classes of mono-precision algorithms: general summation, shifted general summation, and compensated (sequential) summation. Extension of our systematic error analysis to mixed-precision summation algorithms that allow any number of precisions yields the first probabilistic bounds for the mixed-precision FABsum algorithm. Numerical experiments illustrate that the probabilistic bounds are accurate, and that among the three classes of mono-precision algorithms, compensated summation is generally the most accurate. As for mixed precision algorithms, our recommendation is to minimize the magnitude of intermediate partial sums relative to the precision in which they are computed.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-023-01370-y