A linear eigenvalue algorithm for the nonlinear eigenvalue problem
2012 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 122, no 1, 169-195 p.Article in journal (Refereed) Published
The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted . We consider the Arnoldi method for the operator and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.
Place, publisher, year, edition, pages
2012. Vol. 122, no 1, 169-195 p.
Arnoldi Method, Polynomial Eigenproblems, Inverse Iteration, Rayleigh-Ritz
Computer and Information Science
IdentifiersURN: urn:nbn:se:kth:diva-53365DOI: 10.1007/s00211-012-0453-0ISI: 000307532900005ScopusID: 2-s2.0-84865645381OAI: oai:DiVA.org:kth-53365DiVA: diva2:469994
FunderSwedish e‐Science Research Center
QC 20120914. Updated from submitted to published.2011-12-272011-12-272013-04-08Bibliographically approved