Speaker
Description
We are interested in an eigenvector-nonlinear eigenvalue problem (NEPv), that is, a problem of the form $A(x)x=\lambda x$, where $A:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times n}$ is symmetric, and the eigenvector has a prescribed norm, for instance $\Vert x \Vert = 1$. In this sense, this class of problems generalize the linear eigenvalue problem. Motivated by applications such as the Gross-Pitaevskii equation (GPE) [E. Jarlebring and P. Henning. SIAM Review, 67:256-317, 2025], we further specialize this problem by considering a specific structure of $A(x)$. More precisely, we consider the case where the nonlinearity appears as a sum of products of scalar functions of the eigenvector, and rank-one matrices. We present a method based on transforming this class of problems into equivalent problems with eigenvalue nonlinearities (NEP), i.e., problems of the form $M(\lambda)x=0$. In our application, $M:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ is an algebraic function of the eigenvalue. This transformation enables us to use efficient methods for NEPs as a means of obtaining solutions to the NEPv. In particular, these methods can efficiently compute several eigenvalues, something that is rare among methods for the NEPv. We show how the transformation is constructed theoretically, and how it can be handled in practice. A numerical illustration on a large-scale problem related to a modification of the GPE shows the effectiveness of our approach in computing several eigenvalues of the original problem. This presentation is based on the preprint [E. Jarlebring and V. P. Lithell. arXiv:2506.16182, 2025].