Order Matters: A Case Study on Reducing Floating Point Error in Sums Via Ordering and Grouping

Job, Vanessa; Grove, Terry; Fogerty, Shane; Mauney, Christopher; Neuman, Brett; Monroe, Laura; Robey, Robert W.

doi:10.1109/correctness51934.2020.00007

Cited by 1 publication

(2 citation statements)

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“…For any parenthetic form, one can always select a particular set of floating-point summands and impose an ordering in such a way that floating-point error takes place. On the other hand, optimal grouping combined with the best ordering for that parenthesization can lead to much-reduced rounding error [10]. For this reason, when we enumerate computationally inequivalent summations, we must consider both grouping and ordering.…”

Section: Grouping and Orderingmentioning

confidence: 99%

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Computationally Inequivalent Summations and Their Parenthetic Forms

Monroe,

Job

2020

Preprint

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Floating-point addition on a finite-precision machine is not associative, so not all mathematically equivalent summations are computationally equivalent. Making this assumption can lead to numerical error in computations. Proper ordering and parenthesizing is a low-overhead way of mitigating such error in a floating point summation.Ordered and parenthesized summations fall into equivalence classes. We describe these classes, and the parenthetic forms summations in these classes take. We provide summation-related interpretations for sequences known in other contexts, and give new recursive and closed formulas for sequences not previously related to summation.We also introduce a data structure that facilitates understanding of these objects, and use it to consider certain forms of summation used by default in widely used computer languages. Finally, we relate this data structure to other mathematical constructs from the fields of mathematical analysis and algorithmic analysis.

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Section: Grouping and Orderingmentioning

confidence: 99%

“…As a heuristic, the most accurate pairwise summation groups pairs of similar magnitude at each level of the summation tree. This tends to minimize rounding error that results from the use of finite precision on digital computers [10].…”

Section: A Remark On Orderings For Pairwise Summations Vs Tournamentsmentioning

confidence: 99%