Matrix functions via linear systems built from continued fractions

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 − 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.

A New ParaDiag Time-Parallel Time Integration Method

“…Since K is SPD, \lambda min (K) can be cheaply computed by, e.g., the inverse power method. 5 Therefore, it is easy to check the magnitude of \tau \lambda min (K). On the other hand, one may want to select a time grid such that the latter value is big enough to guarantee a well-conditioned J \ell .…”

“…Recently, the matrix equation formulation (1.2) has been used to design new solution procedures. In particular, low-rank solvers can be very successful in solving 1 An anonymous reviewer pointed out that this corresponds to approximating a partial fraction expansion of the rational approximation for the matrix exponential defined by the numerical scheme (which is equivalent to a Weierstrass normal form [5], here essentially the Jordan canonical form that was discovered independently [18]), by imposing periodicity to obtain a normal form which is diagonal.…”

“…Though better insight is provided by the entire eigenvalue distribution 5. The implementation of such a method must make use of the properties of K to be efficient.…”