AN INVERSE ITERATION METHOD FOR EIGENVALUE PROBLEMS WITH EIGENVECTOR NONLINEARITIES

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.