Strictly balancing matrices in polynomial time using osborne's iteration

Rafail Ostrovsky, Yuval Rabani, Arman Yousefi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

Osborne's iteration is a method for balancing n × n matrices which is widely used in linear algebra packages, as balancing preserves eigenvalues and stabilizes their numeral computation. The iteration can be implemented in any norm over Rn, but it is normally used in the L2 norm. The choice of norm not only a ects the desired balance condition, but also defines the iterated balancing step itself. In this paper we focus on Osborne's iteration in any Lp norm, where p < ∞. We design a specific implementation of Osborne's iteration in any Lp norm that converges to a strictly - balanced matrix in O( 2n9K) iterations, where K measures, roughly, the number of bits required to represent the entries of the input matrix. This is the first result that proves a variant of Osborne's iteration in the L2 norm (or any Lp norm, p < ∞) strictly balances matrices in polynomial time. This is a substantial improvement over our recent result (in SODA 2017) that showed weak balancing in Lp norms. Previously, Schulman and Sinclair (STOC 2015) showed strict balancing of another variant of Osborne's iteration in the L norm. Their result does not imply any bounds on strict balancing in other norms.

Original languageAmerican English
Title of host publication45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
EditorsChristos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770767
DOIs
StatePublished - 1 Jul 2018
Event45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: 9 Jul 201813 Jul 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume107
ISSN (Print)1868-8969

Conference

Conference45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Country/TerritoryCzech Republic
CityPrague
Period9/07/1813/07/18

Bibliographical note

Publisher Copyright:
© 2018 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

Keywords

  • Numerical linear algebra
  • Optimization

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