The Library

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

Tools

Lotz, Martin and Noferini, Vanni (2020) Wilkinson's bus : weak condition numbers, with an application to singular polynomial eigenproblems. Foundations of Computational Mathematics, 20 . pp. 1439-1473. doi:10.1007/s10208-020-09455-y ISSN 1615-3375.

Preview

PDF
WRAP-Wilkinsons-bus-weak-numbers-application-singular-Lotz-2020.pdf - Published Version - Requires a PDF viewer.
Available under License Creative Commons Attribution 4.0.
Download (663Kb) | Preview

PDF
WRAP-Wilkinsons-bus-weak-numbers-application-singular-Lotz-2020.pdf - Accepted Version
Embargoed item. Restricted access to Repository staff only - Requires a PDF viewer.
Download (1136Kb)

Official URL: https://doi.org/10.1007/s10208-020-09455-y

Request Changes to record.

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.

Item Type:

Journal Article

Alternative Title:

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science, Engineering and Medicine > Science > Mathematics

Library of Congress Subject Headings (LCSH):

Stochastic processes, Stochastic analysis, Polynomials, Algebras, Linear, Perturbation (Mathematics)

Journal or Publication Title:

Foundations of Computational Mathematics

Publisher:

Springer

ISSN:

1615-3375

Official Date:

10 March 2020

Dates:

Date	Event
10 March 2020	Published
4 February 2020	Accepted

Volume:

Page Range:

pp. 1439-1473

DOI:

10.1007/s10208-020-09455-y

Status:

Peer Reviewed

Publication Status:

Published

Reuse Statement (publisher, data, author rights):

This is a post-peer-review, pre-copyedit version of an article published in Foundations of Computational Mathematics. The final authenticated version is available online at: http://dx.doi.org/[insert DOI]”.

Access rights to Published version:

Open Access (Creative Commons)

Date of first compliant deposit:

12 February 2020

Date of first compliant Open Access:

27 April 2020

RIOXX Funder/Project Grant:

Project/Grant ID	RIOXX Funder Name	Funder ID
UNSPECIFIED	Aalto-yliopisto	http://dx.doi.org/10.13039/501100002666
UNSPECIFIED	University Of Manchester	http://dx.doi.org/10.13039/501100000770
UNSPECIFIED	Isaac Newton Institute for Mathematical Sciences	http://dx.doi.org/10.13039/100012112