  We precisely quantify the impact of statistical error in the quality of a
numerical approximation to a random matrix eigendecomposition, and under mild
conditions, we use this to introduce an optimal numerical tolerance for
residual error in spectral decompositions of random matrices. We demonstrate
that terminating an eigendecomposition algorithm when the numerical error and
statistical error are of the same order results in computational savings with
no loss of accuracy. We also repair a flaw in a ubiquitous termination
condition, one in wide employ in several computational linear algebra
implementations. We illustrate the practical consequences of our stopping
criterion with an analysis of simulated and real networks. Our theoretical
results and real-data examples establish that the tradeoff between statistical
and numerical error is of significant import for data science.
