Low-Memory Krylov Subspace Methods for Optimal Rational Matrix Function Approximation

Tyler Chen, Anne Greenbaum, Cameron N Musco, Christopher P Musco

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 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.