Convergence of the Arnoldi Iteration for Estimating Extreme Eigenvalues

Krylov subspace methods, like the Arnoldi iteration, are a powerful tool for efficiently solving high-dimensional linear algebra problems. In this work, we analyze the convergence of Krylov methods for estimating the numerical range of a matrix. Prior bounds on approximation error often depend on eigenvalue gaps of the matrix, which lead to weaker bounds than observed in practice, specifically in applications where these gaps are small. Instead, we extend a line of work proving gap-independent bounds for the Lanczos method, which depend only on the matrix dimensions and number of iterations, to the more general Arnoldi case.

Name thesis:degree_name

Master

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2025

Author dc:creator

• Chen, Cecilia

Advisor dc:contributor.advisor

• Urschel, John

Statement dc:rights

• Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
• Copyright retained by author(s)

Licence dc:rights.uri

• https://creativecommons.org/licenses/by-nc-nd/4.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/159091

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/159091