Convergence of the Arnoldi Iteration for Estimating Extreme Eigenvalues

Abstract

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.