Author:

Keywords:

Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, nonlinear eigenvalue problems, inverse iteration, Gross-Pitaevskii equation, convergence factors, GROUND-STATE SOLUTION, GRADIENT FLOW, CONVERGENCE, ALGORITHMS, cs.NA, math.NA, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, 0802 Computation Theory and Mathematics, Numerical & Computational Mathematics, 4901 Applied mathematics, 4903 Numerical and computational mathematics

Abstract:

© 2014 Society for Industrial and Applied Mathematics. Consider a symmetric matrix A(v) ∈ ℝnxn depending on a vector v ∈ ℝn and satisfying the property A(αv) = A(v) for any α ∈ ℝ\{0}. We will here study the problem of finding (λ, v) ∈ ℝ x ℝn\{0} such that (ℝ, 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 Schrödinger 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.