论文标题
计算上的不相等总和及其括号形式
Computationally Inequivalent Summations and Their Parenthetic Forms
论文作者
论文摘要
有限前置的机器上的浮点添加不是关联的，因此并非所有数学上等效的总和都是计算等效的。做出此假设可能会导致计算中的数值错误。正确的订购和括号是在浮点求和中减轻此错误的低空方法。 有序和括号的总和属于等效类。我们描述了这些类别，这些类别中的括号形式的总结所示。我们为在其他情况下已知的序列提供了与求和相关的解释，并为序列提供了新的递归和封闭公式，以前与以前与求和相关的序列。 我们还引入了一个数据结构，以促进对这些对象的理解，并使用它来考虑默认情况下使用广泛使用的计算机语言使用的某些形式的求和。最后，我们将这些数据结构与数学分析和算法分析领域的其他数学构建体相关联。
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.
