Probabilistic models are proposed for bounding the forward error in the numerically computed inner product (dot product, scalar product) between of two real $n$-vectors. We derive probabilistic perturbation bounds, as well as probabilistic roundoff error bounds for the sequential accumulation of the inner product. These bounds are non-asymptotic, explicit, and make minimal assumptions on perturbations and roundoffs. The perturbations are represented as independent, bounded, zero-mean random variables, and the probabilistic perturbation bound is based on Azuma's inequality. The roundoffs are also represented as bounded, zero-mean random variables. The first probabilistic bound assumes that the roundoffs are independent, while the second one does not. For the latter, we construct a Martingale that mirrors the sequential order of computations. Numerical experiments confirm that our bounds are more informative, often by several orders of magnitude, than traditional deterministic bounds -- even for small vector dimensions~$n$ and very stringent success probabilities. In particular the probabilistic roundoff error bounds are functions of $\sqrt{n}$ rather than~$n$, thus giving a quantitative confirmation of Wilkinson's intuition. The paper concludes with a critical assessment of the probabilistic approach.

中文翻译：

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内积的概率误差分析

提出了概率模型来限制两个实数 $n$-向量之间的数值计算内积（点积、标量积）中的前向误差。我们推导出概率扰动界限，以及内积顺序累加的概率舍入误差界限。这些界限是非渐近的、明确的，并且对扰动和舍入做出最小的假设。扰动表示为独立的、有界的、零均值随机变量，概率扰动界限基于 Azuma 不等式。舍入也表示为有界的零均值随机变量。第一个概率界限假设四舍五入是独立的，而第二个则不是。对于后者，我们构建了一个反映计算顺序的 Martingale。数值实验证实，我们的界限比传统的确定性界限更具信息量，通常是几个数量级——即使对于小向量维度 ~$n$ 和非常严格的成功概率也是如此。特别是概率舍入误差范围是 $\sqrt{n}$ 而不是~$n$ 的函数，因此对威尔金森的直觉进行了定量确认。本文最后对概率方法进行了批判性评估。的直觉。本文最后对概率方法进行了批判性评估。的直觉。本文最后对概率方法进行了批判性评估。