Uniform Error Estimates for the Lanczos Method

by   John C. Urschel, et al.

The Lanczos method is one of the most powerful and fundamental techniques for solving an extremal symmetric eigenvalue problem. Convergence-based error estimates are well studied, with the estimate depending heavily on the eigenvalue gap. However, in practice, this gap is often relatively small, resulting in significant overestimates of error. One way to avoid this issue is through the use of uniform error estimates, namely, bounds that depend only on the dimension of the matrix and the number of iterations. In this work, we prove a number of upper and lower uniform error estimates for the Lanczos method. These results include the first known lower bounds for error in the Lanczos method and significantly improved upper bounds for error measured in the p-norm, p>1. These lower bounds imply that the maximum error of m iterations of the Lanczos method over all n × n symmetric matrices does indeed depend on the dimension n. In addition, we prove more specific results for matrices that possess some level of eigenvalue regularity or whose eigenvalues converge to some limiting empirical spectral distribution. Through numerical experiments, we show that the theoretical estimates of this paper do apply to practical computations for reasonably sized matrices.


page 1

page 2

page 3

page 4


A priori and a posteriori error estimates for the quad-curl eigenvalue problem

In this paper, we propose a new family of H(curl^2)-conforming elements ...

Lower bounds for eigenvalues of the Steklov eigenvalue problem with variable coefficients

In this paper, using new correction to the Crouzeix-Raviart finite eleme...

Largest and Least H-Eigenvalues of Symmetric Tensors and Hypergraphs

In tensor eigenvalue problems, one is likely to be more interested in H-...

Eigenvalues of symmetric tridiagonal interval matrices revisited

In this short note, we present a novel method for computing exact lower ...

Subspace method for multiparameter-eigenvalue problems based on tensor-train representations

In this paper we solve m-parameter eigenvalue problems (mEPs), with m an...

Hidden Positivity and a New Approach to Numerical Computation of Hausdorff Dimension: Higher Order Methods

In [14], the authors developed a new approach to the computation of the ...

Random Schreier graphs of the general linear group over finite fields and expanders

In this paper we discuss potentially practical ways to produce expander ...

Please sign up or login with your details

Forgot password? Click here to reset