Abstract
Evaluating the action of a matrix function on a vector, that is x=f({mathcal {M}})v, is an ubiquitous task in applications. When {mathcal {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}} 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, 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.
Highlights
We are concerned with the evaluation of x = f (M)v, where f (z) is a Stieltjes function, which can be expressed in integral form
We are interested in two instances of this problem; first, we consider the case M := A, where A ∈ Cn×n is Hermitian positive definite with spectrum contained in [a, b], v ∈ Cn×s is a generic vector, and a rational Krylov method [18] is used to approximate x = f (M)v
For Laplace–Stieltjes functions a direct consequence of the analysis mentioned above leads to Corollary 5; in the Cauchy case, we describe a choice of poles that enables the simultaneous solution of a set of parameter dependent Sylvester equations
Summary
We are interested in two instances of this problem; first, we consider the case M := A, where A ∈ Cn×n is Hermitian positive definite with spectrum contained in [a, b], v ∈ Cn×s is a generic (block) vector, and a rational Krylov method [18] is used to approximate x = f (M)v. Where A, −B ∈ Cn×n are Hermitian positive definite with spectra contained in [a, b], v = vec(F) ∈ Cn2 is the vectorization of a low-rank matrix F = UF VFT ∈ Cn×n, and a tensorized rational Krylov method [8] is used for computing vec(X ) = f (M)vec(F) This problem is a generalization of the solution of a Sylvester equation with a low-rank right hand side, which corresponds to evaluate the function f (z) = z−1. X − X 2, where X is the approximation obtained after steps
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