Escaping the Krylov space during finite precision Lanczos

TL;DR

This paper investigates the numerical issues in the Lanczos algorithm, revealing that finite precision causes the basis vectors to escape the true vector space, impacting interpretations related to Krylov complexity.

Contribution

It demonstrates that loss of orthogonality and reorthogonalization are insufficient, showing instead that eigenvectors escape the true vector space in finite precision.

Findings

Eigenvectors escape the true vector space in finite precision

Reorthogonalization does not fully address numerical issues

Implications for Krylov complexity interpretations

Abstract

The Lanczos algorithm, introduced by Cornelius Lanczos, has been known for a long time and is widely used in computational physics. While often employed to approximate extreme eigenvalues and eigenvectores of an operator, recently interest in the sequence of basis vectors produced by the algorithm rose in the context of Krylov complexity. Although it is generally accepted and partially proven that the procedure is numerically stable for approximating the eigenvalues, there are numerical problems when investigating the Krylov basis constructed via the Lanczos procedure. In this paper, we show that loss of orthogonality and the attempt of reorthoganalization fall short of understanding and addressing the problem. Instead, the numerical sequence of eigenvectors in finite precision arithmetic escapes the true vector space spanned by the exact Lanczos vectors. This poses the real threat to an interpretation in view of the operator growth hypothesis.