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An Inverse Iteration Method for Eigenvalue Problems with Eigenvector Nonlinearities
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0001-9443-8772
2014 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, no 4, A1978-A2001 p.Article in journal (Refereed) Published
Abstract [en]

Consider a symmetric matrix A(v) is an element of R-nxn depending on a vector v is an element of R-n and satisfying the property A(alpha v) = A(v) for any alpha is an element of R\{0}. We will here study the problem of finding (lambda,v) is an element of R x R-n\{0} such that (lambda,v) is an eigenpair of the matrix A(v) and we propose a generalization of inverse iteration for eigenvalue problems with this type of eigenvector nonlinearity. The convergence of the proposed method is studied and several convergence properties are shown to be analogous to inverse iteration for standard eigenvalue problems, including local convergence properties. The algorithm is also shown to be equivalent to a particular discretization of an associated ordinary differential equation, if the shift is chosen in a particular way. The algorithm is adapted to a variant of the Schrodinger equation known as the Gross-Pitaevskii equation. We use numerical simulations to illustrate the convergence properties, as well as the efficiency of the algorithm and the adaption.

Place, publisher, year, edition, pages
2014. Vol. 36, no 4, A1978-A2001 p.
Keyword [en]
nonlinear eigenvalue problems, inverse iteration, Gross-Pitaevskii equation, convergence factors
National Category
Computer and Information Science Mathematics
URN: urn:nbn:se:kth:diva-157582DOI: 10.1137/130910014ISI: 000344743800027OAI: diva2:771258

QC 20141212

Available from: 2014-12-12 Created: 2014-12-11 Last updated: 2014-12-12Bibliographically approved

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Jarlebring, Elias
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Numerical Analysis, NA
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