Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format

Abstract

The computation of eigenvalues is one of the core topics of numerical mathematics. We will discuss an eigenvalue algorithm for the computation of inner eigenvalues of a large, symmetric, and positive definite matrix M based on the preconditioned inverse iteration $$\begin{array}{rcl} x_{i+1} = x_{i} - {B}^{-1}\left (Mx_{ i} - \mu (x_{i})x_{i}\right ),& & \\ \end{array}$$ and the folded spectrum method (replace M by \({(M - \sigma I)}^{2}\)). We assume that M is given in the tensor train matrix format and use the TT-toolbox from I.V. Oseledets (see http://spring.inm.ras.ru/osel/) for the numerical computations. We will present first numerical results and discuss the numerical difficulties.