  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.
