A Refined Probabilistic Error Bound for Sums

TL;DR

This paper introduces an improved probabilistic error bound for the sum of real numbers in floating-point computations, accounting for all error orders and providing more accurate bounds for larger problems.

Contribution

It presents a refined probabilistic error bound that considers all orders of roundoff errors, enhancing accuracy for large-scale floating-point summations.

Findings

Provides a probabilistic bound that holds to all error orders.

Offers more informative bounds for larger problem sizes.

Improves upon existing probabilistic error bounds.

Abstract

This paper considers a probabilistic model for floating-point computation in which the roundoff errors are represented by bounded random variables with mean zero. Using this model, a probabilistic bound is derived for the forward error of the computed sum of n real numbers. This work improves upon existing probabilistic bounds by holding to all orders, and as a result provides informative bounds for larger problem sizes.