Universality in numerical computations with random data

Significance Universal fluctuations are shown to exist when well-known and widely used numerical algorithms are applied with random data. Similar universal behavior is shown in stochastic algorithms and also in an algorithm that models neural computation. The question of whether universality is present in all, or nearly all, computation is raised. The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time—i.e., the histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components—the sample average and the sample variance—the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms as well as a model of neural computation and decision-making. A link to relevant software is provided for readers who would like to do computations of their own.