SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1541-A1557, January 2020.
The need to sum floating-point numbers is ubiquitous in scientific computing. Standard recursive summation of $n$ summands, often implemented in a blocked form, has a backward error bound proportional to $nu$, where $u$ is the unit roundoff. With the growing interest in low precision floating-point arithmetic and ever increasing $n$ in applications, computed sums are more likely to have insufficient accuracy. We propose a class of summation algorithms called FABsum (for “fast and accurate block summation'') that applies a fast summation algorithm (such as recursive summation) blockwise and then sums the partial sums using an accurate summation algorithm (such as compensated summation, or recursive summation in higher precision). We give a rounding error analysis to show that FABsum with a fixed block size $b$ has a backward error bound $(b+1)u + O(u^2)$, which is independent of $n$ to first order. Our computational experiments show that with a suitable choice of $b$ (independent of $n$) FABsum can deliver substantially more accurate results than blocked recursive summation, with only a modest drop in performance. FABsum is especially attractive for low precisions, where it can provide correct digits for much larger $n$ than recursive summation.

中文翻译：

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一类快速准确的求和算法

SIAM科学计算杂志，第42卷，第3期，第A1541-A1557页，2020年1月。
在科学计算中，普遍存在对浮点数求和的需求。$ n $加法标准递归求和，通常以阻塞形式实现，后向误差范围与$ nu $成比例，其中$ u $是单位舍入。随着人们对低精度浮点算术的兴趣日益浓厚，并且在应用程序中的价值不断增加，计算出的总和更可能具有不足的精度。我们提出了一类称为FABsum（用于“快速准确的块求和”）的求和算法，该算法以块为单位应用快速求和算法（例如递归求和），然后使用精确求和算法（例如补偿求和，或以更高的精度进行递归求和）。我们给出了舍入误差分析，显示出具有固定块大小$ b $的FABsum具有向后误差范围$（b + 1）u + O（u ^ 2）$，该误差独立于$ n $一阶。我们的计算实验表明，与$ B $的适当选择（独立的$ N $）FABsum可以提供比阻断递归求和实质上更精确的结果，只有在性能的适度下降。FABsum对于低精度特别有吸引力，因为它可以提供比递归求和大得多的$ n $的正确数字。