Abstract
We propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theories 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 even have infinite 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.
Highlights
The condition number of a computational problem measures the sensitivity of an output with respect to perturbations in the input
To remedy the shortcomings of the stochastic condition number as defined in 2.2, we propose a change in focus from the expected value to tail bounds and quantiles, and the key concept for that purpose is the directional sensitivity
We illustrate how the weak condition number of the problem of computing a simple eigenvalue of a singular matrix polynomial can be estimated in practice
Summary
The condition number of a computational problem measures the sensitivity of an output with respect to perturbations in the input. In other words: in Wilkinson’s experience, the likelihood of seeing the admittedly terrifying worst case appeared to be very small, and Wilkinson believed that being afraid of the potential catastrophic instability of Gaussian elimination is an irrational attitude Based on this experience, Weiss et al [48] and Stewart [41] proposed to study the effect of perturbations on average, as opposed to worst case; see [29, §2.8] for more references on work addressing the stochastic analysis of roundoff errors. We will get back to the matrix pencil (1) in Example 5.3, where we show that our new theory does explain why this problem is solved to high accuracy using standard backward stable algorithms. A precise probabilistic analysis of the sensitivity of the problem of computing simple eigenvalues of singular matrix polynomials 1. a new species of “weak” condition numbers, which we call the weak worst-case condition number and the weak stochastic condition number that give a more accurate description of the perturbation behaviour of a computational map (Sect. 2); 2. a precise probabilistic analysis of the sensitivity of the problem of computing simple eigenvalues of singular matrix polynomials (Sects. 4 and 5); 3. an illustration of the advantages of the new concept by demonstrating that, unlike both classical and stochastic condition numbers, the weak condition numbers are able to explain why the apparently uncomputable eigenvalues of singular matrix polynomials, such as the eigenvalue 1 in the example above, can be computed with remarkable accuracy (Example 5.3); 4. a concrete method for bounding the weak condition numbers for the eigenvalues of singular matrix polynomials (Sect. 6)
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