Precision-aware Deterministic and Probabilistic Error Bounds for Floating Point Summation

TL;DR

This paper develops explicit deterministic and probabilistic error bounds for floating point summation, especially in low precision or large-scale computations, improving understanding and accuracy of numerical summation methods.

Contribution

It introduces a systematic recurrence-based approach to derive new error bounds for mono-precision and mixed-precision summation algorithms, including the first probabilistic bounds for FABsum.

Findings

Probabilistic bounds are accurate in numerical experiments.

Compensated summation generally yields the most accurate results.

Minimizing intermediate partial sums improves mixed-precision summation accuracy.

Abstract

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 without asymptotic 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.