# Wilkinson's bus: Weak condition numbers, with an application to singular polynomial eigenproblems

## TL;DR

This paper introduces a new weak condition number framework that better predicts the accuracy of solutions in problems where classical and stochastic theories fail, with applications to singular polynomial eigenproblems.

## Contribution

It develops a novel weak condition number theory that improves prediction of computational accuracy in challenging problems and demonstrates practical estimation methods.

## Key findings

- Weak condition numbers outperform classical ones in predicting accuracy.
- Application to singular polynomial eigenproblems shows improved insights.
- Practical estimation methods for weak condition numbers are provided.

## Abstract

We propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theory fail to predict the observed accuracy of computed solutions. To motivate our ideas, we present examples of problems that are discontinuous at a given input and have infinite classical and stochastic condition number, but where the solution is still computed to machine precision without relying on structured algorithms. Stimulated by the failure of classical and stochastic perturbation theory in capturing such phenomena, we define and analyse a weak worst-case and a weak stochastic condition number. This new theory is a more powerful predictor of the accuracy of computations than existing tools, especially when the worst-case and the expected sensitivity of a problem to perturbations of the input is not finite. We apply our analysis to the computation of simple eigenvalues of matrix polynomials, including the more difficult case of singular matrix polynomials. In addition, we show how the weak condition numbers can be estimated in practice.