Automated parameter selection for rational Arnoldi approximation of Markov functions

Abstract

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present a heuristic for the automated pole selection when the function to be approximated is of Markov type, such as the matrix square root. The performance of this approach is demonstrated at several numerical examples.

Computing the Weighted Geometric Mean of Two Large-Scale Matrices and Its Inverse Times a Vector

“…We observe that Quad2 converges much faster than Quad1 as M/m grows, as predicted in (24). Regarding the Krylov subspace methods, we observe linear convergence which is very slow for the Arnoldi method and it is quite fast when the adaptive strategy is used in the rational Krylov method.…”

“…In our implementation, when the matrices are larger than 1000 × 1000, we get the poles by running rkfit on a surrogate problem of size 1000 × 1000 whose setup requires a rough estimate of the extrema of the spectrum of A −1 B. In the case of rational Krylov methods, in order to obtain an approximation of Af (A −1 B)v, we use the estimate (32), even when the last pole is not at infinity, as done by Güttel and Knizhnermann [24], with good results.…”

“…For the rational Krylov methods, the poles are chosen according to either the adaptive strategy by Güttel and Knizhnerman [24] or the function rkfit from the rktoolbox [5], based on an algorithm by Berljafa and Güttel [6,7]. In our implementation, when the matrices are larger than 1000 × 1000, we get the poles by running rkfit on a surrogate problem of size 1000 × 1000 whose setup requires a rough estimate of the extrema of the spectrum of A −1 B.…”

“…The extended Krylov subspace method [18] (Extended); 3. A rational Krylov subspace method, with poles chosen according to the adaptive strategy of Güttel and Knizhnermann [24] (RatAdapt); 4. A rational Krylov subspace method, where the choice of the poles is based on the solution of the best rational approximation of an auxiliary problem [6] (RatFit); 5.…”

Limited‐memory polynomial methods for large‐scale matrix functions

“…Such functions are a subclass of Markov functions [54]. For more information about properties of these functions, see the text by Henrici [59] or the introductions to Ilić et al [69] and Schweitzer's thesis [89].…”