Deterministic and Probabilistic Error Bounds for Floating Point Summation Algorithms

TL;DR

This paper provides new deterministic and probabilistic error bounds for floating point summation algorithms, including general, shifted, and compensated methods, demonstrating that compensated summation is typically most accurate.

Contribution

It introduces novel explicit error expressions and bounds for various summation algorithms, including a new first order bound for compensated summation, applicable to all summation classes.

Findings

Compensated summation generally yields the most accurate results.

Probabilistic bounds hold to all orders for general and shifted summation.

New first order bounds improve upon existing error estimates.

Abstract

We analyse the forward error in the floating point summation of real numbers, from algorithms that do not require recourse to higher precision or better hardware. We derive informative explicit expressions, and new deterministic and probabilistic bounds for errors in three classes of algorithms: general summation,shifted general summation, and compensated (sequential) summation. Our probabilistic bounds for general and shifted general summation hold to all orders. For compensated summation, we also present deterministic and probabilistic first and second order bounds, with a first order bound that differs from existing ones. Numerical experiments illustrate that the bounds are informative and that among the three algorithm classes, compensated summation is generally the most accurate method.