The conventional probabilistic rounding error analysis in numerical linear algebra provides worst-case bounds with an associated failure probability, which can still be pessimistic. In this paper, we develop a new probabilistic rounding error analysis from a statistical perspective. By assuming both the data and the relative error are independent random variables, we derive the approximate closed-form expressions for the expectation and variance of the rounding errors in various key computational kernels. Our analytical expressions have three notable characteristics: they are statistical and do not involve a failure probability; they are sharper than other deterministic and probabilistic bounds, using mean square error as the metric; they are correct to all orders of unit roundoff and valid for any dimension. Furthermore, 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. We also discuss a scenario involving inner products where the underlying assumptions are invalid, i.e., input data are dependent, rendering the analytical expressions inapplicable.

翻译：数值线性代数中的传统概率舍入误差分析提供了带有失效概率的最坏情况界，但这类界仍可能过于悲观。本文从统计学视角提出了一种新的概率舍入误差分析方法。通过假设数据与相对误差均为独立随机变量，我们推导了多种关键计算核中舍入误差期望与方差的近似闭式表达式。这些解析表达式具有三个显著特征：采用统计形式且不涉及失效概率；以均方误差为度量，其精度优于其他确定性与概率性界；在任意维度下均对所有阶的单位舍入误差成立且有效。数值实验验证了推导的准确性，并以内积为例表明，我们的解析表达式通常比备选最坏情况界至少紧两个数量级。此外，我们讨论了一种内积场景（即输入数据存在依赖性导致基本假设失效），在此情况下解析表达式不再适用。