浮点求和中停滞现象的分析与检测
Analysis and Detection of Stagnation in Floating-Point Summation
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中文总结 AI 辅助
本文首次严格定义浮点求和的停滞现象，在MATLAB和Julia中构建检测工具，应用于多类场景，可帮助用户定位停滞区域并采取优化措施以提升浮点运算精度。
中文摘要 AI 辅助
浮点数求和中的停滞现象指，尽管加数大于零，但运行和（即累加器）的值不发生变化。本文定义了不同类型的停滞——通常称为淹没或吸收，即加法中的一个操作数在浮点求和中被近似或完全舍掉——并提出了在使用浮点运算的数值代码中检测该现象的算法。虽然已有研究探讨了浮点运算中的抵消，例如Lam等人在《动态浮点抵消检测》中的工作，但停滞现象尚未被用作检测和报告浮点运算中精度损失的属性。我们已在MATLAB和Julia两种语言中构建了动态停滞检测工具原型，并将其应用于从向量基本求和、矩阵乘法到流体流动模拟模型等各类应用。借助该工具，用户可识别其浮点代码中发生停滞的区域，在这些代码区域中，可能需要切换到更高精度的数字格式、应用随机舍入，或使用不同的求和算法（如补偿求和）。据我们所知，本文首次对浮点求和中的停滞现象给出了严格定义。
英文摘要
Stagnation in the summation of floating-point numbers occurs when the running sum, or the accumulator, does not change its value despite addends being greater than zero. Here we define different types of stagnation - commonly referred to as swamping or absorption, where one operand in an addition is nearly or completely rounded off---in floating-point summation, and propose algorithms for detecting it in numerical code that uses floating-point arithmetic. Whilst cancellation in floating-point arithmetic has been studied, for example by Lam et al. in `Dynamic floating point cancellation detection', stagnation has not been exploited as a property for detecting and reporting accuracy loss in floating-point operations. We have built prototype dynamic stagnation detection tools in two languages, MATLAB and Julia, and applied them to applications ranging from basic summation of a vector, to matrix multiplication, to a fluid flow simulation model. With this tool, users are able to identify areas of their floating-point code where stagnation occurs - in those areas of the code one may need to switch to a higher precision number format, apply stochastic rounding, or use different summation algorithms such as compensated summation. To the best of our knowledge we provide a first rigorous definition of stagnation in floating-point summation.
发表机构
- School of Computer Science, University of Leeds（利兹大学计算机科学学院）
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