Numerical Validation of Compensated Summation Algorithms with Stochastic Arithmetic

Abstract

International audienceCompensated summation algorithms are designed to improve the accuracy of ill-conditioned sums. They are based on algorithms, such as FastTwoSum, which are proved to provide, with rounding to nearest, the sum of two floating-point numbers and the associated rounding error. Discrete stochastic arithmetic enables one to estimate rounding error propagation in numerical codes. It requires a random rounding mode which consists in rounding each computed result toward −∞ or +∞ with the same probability. In this paper we analyse the impact of this random rounding mode on compensated summations based on the FastTwoSum algorithm. We show the accuracy improvement obtained using such compensated summations in numerical simulations controlled with discrete stochastic arithmetic

Tight Interval Inclusions with Compensated Algorithms

“…With directed rounding, Algorithm 1 (FastTwoSum) is not an error-free transformation. The error generated by Algorithm 6 with directed rounding is given in [13] and is recalled in Proposition 4.4.…”

“…In this paper we show that EFTs executed with directed rounding provide guaranteed bounds on the results of additions and multiplications. We complete results established in [13], [14] on the behaviour with directed rounding of compensated algorithms based on these EFTs. Then we show that, thanks to compensated algorithms executed with directed rounding, tight interval inclusions can be computed for summation, dot product, and polynomial evaluation with Horner scheme.…”

Formal methods for statistical software

“…Graillat et al analyze [11] the impact of random rounding mode on compensated algorithms. They show the increase in accuracy of computations when an error-free transformation is used to "enable the computation of rounding errors".…”

On the Robustness of the 2Sum and Fast2Sum Algorithms

“…1 Demmel and Nguyen show that if 4ulp(a) ≤ |b| ≤ a then Algorithm 1 returns the error of the floating-point addition of a and b when directed rounding functions are used. Graillat, Jézéquel, and Picot [Graillat et al 2015] give an error bound on the value returned by Algorithm 1 when directed rounding functions are used. We will improve on their bound, showing that the algorithm always returns the best possible result, namely a floating-point number t closer to the error of the floating-point addition of a and b than any other floating-point number.…”

“…Hence, with directed roundings, even if we cannot always obtain the "exact" error of floating-point addition, it would still be useful to obtain a value close to that error. This problem was partly dealt with by Demmel and Nguyen [Demmel and Nguyen 2013], and later on by Graillat, Jézéquel, and Picot [Graillat et al 2015] for the Fast2Sum algorithm, and by Martin-Dorel et al [Martin-Dorel et al 2013] in the case of "double roundings". We aim at tackling this issue in a more general context, and we wish to study the behaviour of 2Sum and Fast2Sum just assuming "general" rounding functions (see definition 2.1 below).…”