Computationally Inequivalent Summations and Their Parenthetic Forms

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.