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Approximating the action of a matrix function $$f(\vec{A})$$ on a vector $$\vec{b}$$ is an increasingly important primitive in machine learning, data science, and statistics, with applications such as sampling high dimensional Gaussians, Gaussian process regression and Bayesian inference, principle component analysis, and approximating Hessian spectral densities. Over the past decade, a number of algorithms enjoying strong theoretical guarantees have been proposed for this task. Many of the most successful belong to a family of algorithms called \emph{Krylov subspace methods}. Remarkably, a classic Krylov subspace method, called the Lanczos method for matrix functions (Lanczos-FA), frequently outperforms newer methods in practice. Our main result is a theoretical justification for this finding: we show that, for a natural class of \emph{rational functions}, Lanczos-FA matches the error of the best possible Krylov subspace method up to a multiplicative approximation factor. The approximation factor depends on the degree of $f(x)$'s denominator and the condition number of $$\vec{A}$$, but not on the number of iterations $$k$$. Our result provides a strong justification for the excellent performance of Lanczos-FA, especially on functions that are well approximated by rationals, such as the matrix square root. 

We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted $N\times <\#comment/>N$matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the $N\to <\#comment/>\mathrm{\infty <\#comment/>}$behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix’s conjecture. 

Abstract. We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function’s denominator, and it can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature-based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.