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Consider a symmetric matrix $A(v)\in\RR^{n\times n}$ depending on a vector$v\in\RR^n$ and satisfying the property $A(\alpha v)=A(v)$ for any$\alpha\in\RR\backslash{0}$. We will here study the problem of finding$(\lambda,v)\in\RR\times \RR^n\backslash\{0\}$ such that $(\lambda,v)$ is aneigenpair of the matrix $A(v)$ and we propose a generalization of inverseiteration for eigenvalue problems with this type of eigenvector nonlinearity.The convergence of the proposed method is studied and several convergenceproperties are shown to be analogous to inverse iteration for standardeigenvalue problems, including local convergence properties. The algorithm isalso shown to be equivalent to a particular discretization of an associatedordinary differential equation, if the shift is chosen in a particular way. Thealgorithm is adapted to a variant of the Schr\"odinger equation known as theGross-Pitaevskii equation. We use numerical simulations toillustrate theconvergence properties, as well as the efficiency of the algorithm and theadaption.

An inverse iteration method for eigenvalue problems with eigenvector nonlinearities