The infinite bi-Lanczos method for nonlinear eigenvalue problems

TL;DR

This paper introduces an infinite bi-Lanczos method for nonlinear eigenvalue problems that efficiently approximates both left and right eigenvectors using finite computations within an infinite-dimensional framework.

Contribution

It presents a novel two-sided Lanczos algorithm for NEPs that works with infinite-dimensional vectors but is implemented with finite matrices, including a new representation for infinite vectors.

Findings

Efficient approximation of eigenvectors for NEPs.

Reduced computational cost and storage through short recurrences.

Effective handling of infinite-dimensional vectors in finite computations.

Abstract

We propose a two-sided Lanczos method for the nonlinear eigenvalue problem (NEP). This two-sided approach provides approximations to both the right and left eigenvectors of the eigenvalues of interest. The method implicitly works with matrices and vectors with infinite size, but because particular (starting) vectors are used, all computations can be carried out efficiently with finite matrices and vectors. We specifically introduce a new way to represent infinite vectors that span the subspace corresponding to the conjugate transpose operation for approximating the left eigenvectors. Furthermore, we show that also in this infinite-dimensional interpretation the short recurrences inherent to the Lanczos procedure offer an efficient algorithm regarding both the computational cost and the storage.