Evaluating the action of a matrix function on a vector, that is x= f(M) v, is an ubiquitous task in applications. When 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 M is a positive definite matrix. Relying on the same tools used to analyze the generic situation, we then focus on the case M= I⊗ A- BT⊗ 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.
Bibliographical noteFunding Information:
The work of Stefano Massei has been partially supported by the SNSF research project Fast algorithms from low-rank updates, Grant Number: 200020_178806, and by the INdAM/GNCS project “Analisi di matrici sparse e data-sparse: metodi numerici ed applicazioni”. The work of Leonardo Robol has been partially supported by a GNCS/INdAM project “Giovani Ricercatori” 2018.
© 2020, The Author(s).
Copyright 2021 Elsevier B.V., All rights reserved.
- Function of matrices
- Kronecker sum
- Pole selection
- Rational Krylov
- Stieltjes functions
- Zolotarev problem