Stochastic Vs Worst-case Condition Numbers

TL;DR

This paper compares stochastic and worst-case condition numbers, providing bounds on their ratios and differences, which helps understand the stability and precision loss in computational problems under different perturbation measures.

Contribution

It introduces bounds relating stochastic and worst-case condition numbers and loss of precision, applicable to norm-wise and componentwise perturbations.

Findings

Upper bound of O(sqrt m) for ratio of worst-case to stochastic condition numbers.

Upper bound of O(ln m) for difference in loss of precision.

Results hold for both norm-wise and componentwise perturbations.

Abstract

We compare Stochastic and Worst-case condition numbers and loss of precision for general computational problems. We show an upper bound for the ratio of Worst-case condition number to the Stochastic condition number of order O(sqrt m). We show an upper bound for the difference between the Worst-case loss of precision and the Stochastic loss of precision of order O(ln m). The results hold if the perturbations are measured norm-wise or componentwise.