Approximation of the matrix exponential for matrices with a skinny field of values
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Approximation of the matrix exponential for matrices with a skinny field of values Marco Caliari1 · Fabio Cassini2 · Franco Zivcovich1 Received: 22 May 2019 / Accepted: 9 April 2020 © Springer Nature B.V. 2020
Abstract The backward error analysis is a great tool which allows selecting in an effective way the scaling parameter s and the polynomial degree of approximation m when the action of the matrix exponential exp(A)v has to be approximated by s pm (s −1 A) v = exp(A + ΔA)v. We propose here a rigorous bound for the relative backward error ΔA2 / A2 , which is of particular interest for matrices whose field of values is skinny, such as the discretization of the advection–diffusion or the Schrödinger operators. The numerical results confirm the superiority of the new approach with respect to methods based on the classical power series expansion of the backward error for the matrices of our interest, both in terms of computational cost and achieved accuracy. Keywords Backward error analysis · Action of matrix exponential · Leja–Hermite interpolation · Skinny field of values Mathematics Subject Classification 65D05 · 65F30 · 65F60
Communicated by Christian Lubich.
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Marco Caliari [email protected] Fabio Cassini [email protected] Franco Zivcovich [email protected]
1
University of Verona, Verona, Italy
2
University of Trento, Trento, Italy
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M. Caliari et al.
1 Introduction In the recent years, the problem of approximating the action of the matrix exponential on a vector exp(A)v has attracted an increasing amount of attentions. Among polynomial methods we recall the recent implementations of the Krylov method (see [8,9,20]), the truncated Taylor series expansion [2] and polynomial interpolation methods (e.g., [5,6]). Among rational methods we recall instead the rational Krylov methods (see [10,19,23]) and the Carathéodory–Fejér approach used in [21]. For a survey on these and other methods we refer to [18] and [13, § 10 and § 13]. This interest is mainly due to the several applications where the action of the matrix exponential plays a fundamental role. Prominent examples are the exponential integrators [15], which constitute effective methods for the time integration of large stiff or oscillatory systems of differential equations. These very practical applications led the authors of this manuscript into refining existing techniques to achieve better accuracy and performances over a fairly specific class of matrices, that is the family of matrices having a skinny field of values. In fact, when it comes to the real applications, very often the spectrum of the matrices of interest is not just a scattered bunch of points on the complex plane. It is, on the contrary, contained in a skinny rectangle centered at the origin of the complex plane, after a proper shift of the matrix. We just mention the spatial discretization of diffusion, advection–diffusion, advection, and Schrödinger operators, among others. The goal of this paper is to outline an algorithm for the computation of the act
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