Matrix functions via linear systems built from continued fractions
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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- τ I for the various poles τ 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 Padé 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.
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