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• W3197038406 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 Pade 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." @default.
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• W3197038406 date "2021-09-08" @default.
• W3197038406 modified "2023-09-27" @default.
• W3197038406 title "Matrix functions via linear systems built from continued fractions." @default.
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