The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for random vectors and matrices computation. By assuming the relative errors are independent random variables, we derive the approximate closed-form expressions for the expectation and variance of the rounding errors in various key computations for vectors and random matrices. Numerical experiments validate the accuracy of our derivations and demonstrate that our analytical expressions are generally at least two orders of magnitude tighter than alternative worst-case bounds, exemplified through the inner products.

翻译：传统的舍入误差分析提供的是具有相关失败概率的最坏情况界，忽略了舍入误差的统计特性。本文针对随机向量与矩阵的计算，提出了一种新的统计舍入误差分析方法。通过假设相对误差是独立的随机变量，我们推导了向量和随机矩阵在各种关键计算中舍入误差的期望与方差的近似闭式表达式。数值实验验证了我们推导结果的准确性，并通过内积计算等示例证明，我们的解析表达式通常比替代的最坏情况界至少紧致两个数量级。