Probabilistic Error Analysis for Inner Products
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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 √(n) 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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