Computationally Inequivalent Summations and Their Parenthetic Forms

TL;DR

This paper explores how different parenthetic arrangements of floating-point summations affect numerical accuracy, classifies these arrangements into equivalence classes, and introduces a data structure to analyze and optimize summation orderings.

Contribution

It introduces a new classification of summation parenthetic forms, provides recursive and closed formulas for related sequences, and develops a data structure to analyze summation equivalence classes.

Findings

Ordered and parenthesized summations form distinct equivalence classes.

New formulas relate summation sequences to known mathematical contexts.

The data structure aids in understanding and optimizing floating-point summation orderings.

Abstract

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.