Matrix functions via linear systems built from continued fractions

TL;DR

This paper introduces a novel method for computing matrix functions using linear systems derived from continued fraction representations, offering advantages when partial fraction expansions are unavailable or ill-conditioned.

Contribution

It develops the CF-matrix approach from continued fractions for matrix functions, providing a new computational framework and initial theoretical insights.

Findings

The CF-matrix is a block tridiagonal matrix constructed from continued fractions.

Solving a single linear system with the CF-matrix computes the matrix function action.

Numerical experiments show effective convergence with standard preconditioners.

Abstract

A widely used approach to compute the action $f(A)v$ of a matrix function $f(A)$ on a vector $v$ is to use a rational approximation $r$ for $f$ and compute $r(A)v$ instead. If $r$ is not computed adaptively as in rational Krylov methods, this is usually done using the partial fraction expansion of $r$ and solving linear systems with matrices $A- \tau I$ for the various poles $\tau$ of $r$. Here we investigate an alternative approach for the case that a continued fraction representation for the rational function is known rather than a partial fraction expansion. This is typically the case, for example, for Pad\'e approximations. From the continued fraction, we first construct a matrix pencil from which we then obtain what we call the CF-matrix (continued fraction matrix), a block tridiagonal matrix whose blocks consist of polynomials of $A$ with degree bounded by 1 for many continued fractions. We show that one can evaluate $r(A)v$ by solving a single linear system with the CF-matrix and present a number of first theoretical results as a basis for an analysis of future, specific solution methods for the large linear system. While the CF-matrix approach is of principal interest on its own as a new way to compute $f(A)v$, it can in particular be beneficial when a partial fraction expansion is not known beforehand and computing its parameters is ill-conditioned. We report some numerical experiments which show that with standard preconditioners we can achieve fast convergence in the iterative solution of the large linear system.