An inverse iteration method for eigenvalue problems with eigenvector\n nonlinearities

2012/12/03 by Elias Jarlebring, Jarlebring, Elias, Simen Kvaal +3 · 8 citations
Computer Science · Physics and Astronomy · #Matrix Theory and Algorithms #Quantum optics and atomic interactions #Optical and Acousto-Optic Technologies

paper · pdf · doi:10.48550/arxiv.1212.0417

Abstract

Consider a symmetric matrix A(v)\∈ RR^{n\× n} depending on a vector\nv\∈ RRⁿ and satisfying the property A(\α v)=A(v) for any\n\α\∈ RR backslash0. We will here study the problem of finding\n(\λ,v)\∈ RR\× RRⁿ backslash 0 such that (\λ,v) is an\neigenpair of the matrix A(v) and we propose a generalization of inverse\niteration for eigenvalue problems with this type of eigenvector nonlinearity.\nThe convergence of the proposed method is studied and several convergence\nproperties are shown to be analogous to inverse iteration for standard\neigenvalue problems, including local convergence properties. The algorithm is\nalso shown to be equivalent to a particular discretization of an associated\nordinary differential equation, if the shift is chosen in a particular way. The\nalgorithm is adapted to a variant of the Schr "odinger equation known as the\nGross-Pitaevskii equation. We use numerical simulations toillustrate the\nconvergence properties, as well as the efficiency of the algorithm and the\nadaption.\n

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