Efficient approximation of functions of some large matrices by partial fraction expansions

ABSTRACT

Some important applicative problems require the evaluation of functions Ψ of large and sparse and/or localized matrices A. Popular and interesting techniques for computing $\mathrm{\Psi }\left(A\right)$ and $\mathrm{\Psi }\left(A\right)\mathbf{v}$, where $\mathbf{v}$ is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from A by a complex multiple of the identity matrix I for computing $\mathrm{\Psi }\left(A\right)\mathbf{v}$ or require inverting sequences of matrices with the same characteristics for computing $\mathrm{\Psi }\left(A\right)$. Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that $A^{-1}$ shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests.

KEYWORDS:

• Matrix functions
• partial fraction expansions
• large linear systems
• incomplete factorizations

AMS CLASSIFICATIONS:

• 65F60
• 65F08
• 15A23

Acknowledgments

We wish to thank two anonymous referees for their constructive comments which have improved the readability of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by Istituto Nazionale di Alta Matematica “Francesco Severi” INDAM-GNCS 2018 projects ‘Tecniche innovative per problemi di algebra lineare’ and ‘Risoluzione numerica di equazioni di evoluzione integrali e differenziali con memoria’. The first author gratefully acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006 and the Tor Vergata University ‘MISSION: SUSTAINABILITY’ project ‘NUMnoSIDS’, CUP E86C18000530005.