Statistical Rounding Error Analysis for the Computation of Random Vectors and Matrices

The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for random vectors and matrices computation. By assuming the relative errors are independent random variables, we derive the approximate closed-form expressions for the expectation and variance of the rounding errors in various key computations for vectors and random matrices. 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.