PROBABILISTIC ERROR ANALYSIS FOR INNER PRODUCTS.
PROBABILISTIC ERROR ANALYSIS FOR INNER PRODUCTS.
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DOI：
10.1137/19m1270434
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Zhou H
中科院分区：
文献类型：
--
作者：
Ipsen ICF;Zhou H
提出了一种概率模型来限定两个实数n向量之间的数值计算内积（点积、标量积）的前向误差。我们导出了内积顺序累积的概率扰动界和概率舍入误差界。这些边界是非渐近的，明确的，具有最小的假设，并且在失效概率和相对误差之间具有明确的关系。舍入值表示为有界的零均值随机变量，这些随机变量是独立的或具有条件独立的均值。我们的概率边界基于Azuma不等式及其相关的鞅，它反映了计算的顺序顺序。“从第一性原理”推导前向误差界限的优点是可以产生为概率界限定制的条件数。数值实验证实，我们的边界比传统的确定性边界提供了更多的信息，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，概率舍入误差界限是n的函数，而不是n的函数，从而定量地证实了威尔金森的直觉。本文最后对概率方法进行了批判性的评估。
Probabilistic models are proposed for bounding the forward error in the numerically computed inner product (dot product, scalar product) between 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 nonasymptotic, explicit, with minimal assumptions, and with a clear relationship between failure probability and relative error. The roundoffs are represented as bounded, zero-mean random variables that are independent or have conditionally independent means. Our probabilistic bounds are based on Azuma’s inequality and its associated martingale, which mirrors the sequential order of computations. The derivation of forward error bounds “from first principles” has the advantage of producing condition numbers that are customized for the probabilistic bounds. 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 rather than n, thus giving a quantitative confirmation of Wilkinson’s intuition. The paper concludes with a critical assessment of the probabilistic approach.
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影响因子：
3.7
作者：
BARLOW, JL;BAREISS, EH
通讯作者：
BAREISS, EH
影响因子：
2
作者：
CALVETTI, D
通讯作者：
CALVETTI, D
影响因子：
2
作者：
CALVETTI, D
通讯作者：
CALVETTI, D
影响因子：
22.7
作者：
HULL, TE;SWENSON, JR
通讯作者：
SWENSON, JR
影响因子：
3.7
作者：
BARLOW, JL;BAREISS, EH
通讯作者：
BAREISS, EH