Rational Krylov for Stieltjes matrix functions: convergence and pole selection

Abstract

AbstractEvaluating the action of a matrix function on a vector, that is $$x=f({\mathcal {M}})v$$ x = f ( M ) v , is an ubiquitous task in applications. When $${\mathcal {M}}$$ M  is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating x when f(z) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent, completely monotonic) and $${\mathcal {M}}$$ M  is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case $${\mathcal {M}}=I \otimes A - B^T \otimes I$$ M = I ⊗ A - B T ⊗ I , and v obtained vectorizing a low-rank matrix; this finds application, for instance, in solving fractional diffusion equation on two-dimensional tensor grids. We see how to leverage tensorized Krylov subspaces to exploit the Kronecker structure and we introduce an error analysis for the numerical approximation of x. Pole selection strategies with explicit convergence bounds are given also in this case.