Probabilistic estimation of the accuracy of inner products and application to stochastic validation

Abstract Numerical validation is concerned with certifying the correctness of scientific computations in finite precision arithmetic. This requires tools to reliably estimate the accuracy of floating-point computations. Dynamic approaches based on stochastic arithmetic are such tools, but they require each elementary operation to be stochastically rounded, which prevents the use of optimized linear algebra kernels and leads to a heavy performance penalty. In this article, we develop a new probabilistic method to estimate the accuracy of inner products. This new approach only requires to randomize the input of the computation, and thus allows for the actual computation to be efficiently performed with optimized inner product kernel implementations using classical arithmetic. We carry out a probabilistic error analysis of this method that proves that it is able to reliably estimate the accuracy of inner products with high probability. We then present its implementation within the CADNA numerical validation library. Our experimental results confirm that this new method can be used as an equally reliable, yet much more efficient estimator of the accuracy of inner products.