Near-Optimality Guarantees for Approximating Rational Matrix Functions by the Lanczos Method

We study the Lanczos method for approximating the action of a symmetric matrix function $f(\mathbf{A})$ on a vector $\mathbf{b}$ (Lanczos-FA). For the function $\mathbf{A}^{-1}$, it is known that the error of Lanczos-FA after $k$ iterations matches the error of the best approximation from the Krylov subspace of degree $k$ when $\vec{A}$ is positive definite. We prove that the same holds, up to a multiplicative approximation factor, when $f$ is a rational function with no poles in the interval containing $\mathbf{A}$'s eigenvalues. The approximation factor depends the degree of $f$'s denominator and the condition number of $\mathbf{A}$, but not on the number of iterations $k$. Experiments confirm that our bound accurately predicts the convergence of Lanczos-FA. Moreover, we believe that our result provides strong theoretical justification for the excellent practical performance that has long by observed of the Lanczos method, both for approximating rational functions and functions like $\mathbf{A}^{-1/2}\mathbf{b}$ that are well approximated by rationals.