Uniform Error Estimates for the Lanczos Method

The Lanczos method is one of the most powerful and fundamental techniques for solving an extremal symmetric eigenvalue problem. Convergence-based error estimates depend heavily on the eigenvalue gap. 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 explicit upper and lower uniform error estimates for the Lanczos method. These lower bounds, paired with numerical results, imply that the maximum error of $m$ iterations of the Lanczos method over all $n \times n$ symmetric matrices does indeed depend on the dimension $n$ in practice. The improved bounds for extremal eigenvalues translate immediately to error estimates for the condition number of a symmetric positive definite matrix. 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.