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An Arnoldi method with structured starting vectors for the delay eigenvalue problem
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.ORCID iD: 0000-0001-9443-8772
2010 (English)In: Proceedings of the 9th IFAC Workshop on Time Delay Systems, 2010Conference paper (Refereed)
Abstract [en]

 The method called Arnoldi is currently a very popular method to solve largescale eigenvalue problems. The general purpose of this paper is to generalize Arnoldi to the characteristic equation of a time-delay system, here called adelay eigenvalue problem . The presented generalization is mathematically equivalent to Arnoldi applied to the problem corresponding to a Taylor approximation of the exponential. Even though the derivation of the result is with a Taylor approximation, the constructed method can be implemented in such a way that it is independent of the Taylor truncation paramater N . This is achieved by exploiting properties of vectors with a special structure, the vectorization of a rank one matrix plus the vectorization of a matrix which right-most columns are zero. It turns out that this set of vectors is closed under the matrix vector product as well as orthogonalization. Moreover, both operations can be efficiently computed. Since Arnoldi only consists of these operations, if Arnoldi is started with the special vector structure, the method can be efficiently executed. The presented numerical experiments indicate that the method is very efficient in comparison to methods in the literature.

Place, publisher, year, edition, pages
Keyword [en]
Arnoldi, time-delay systems, nonlinear eigenvalue problems, characteristic roots, companion linearization, polynomial eigenvalue problems, stability
National Category
Computer and Information Science
URN: urn:nbn:se:kth:diva-53372OAI: diva2:470007
9th IFAC Workshop on Time Delay Systems
QC 20120207Available from: 2011-12-27 Created: 2011-12-27 Last updated: 2012-02-07Bibliographically approved

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Jarlebring, Elias
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Numerical Analysis, NA
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