Efficient and Stable Arnoldi Restarts for Matrix Functions Based on Quadrature

Abstract

Abstract. When using the Arnoldi method for approximating f (A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a-priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numerical stability by various numerical experiments.

Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices

“…This result is contained in [16,Thm 3.2] for the case when µ is differentiable and thus f is of the form (1.8).…”

“…A first result in this direction was given in [10], characterizing the restart function e (1) m (z) as the mth order divided difference [6] of f (z) with respect to the Ritz values, i.e., the eigenvalues of H m ; see also [23]. For functions representable by a "Cauchy-type" integral, an integral representation of the restart function instead of a representation using divided differences was given in [16], which was then used to develop a numerically stable restart procedure. We rephrase this result for Stieltjes functions.…”

“…3) * In [16] the integration interval is (−∞, 0], i.e., the integration variable t is transformed as t → −t.…”

“…We refer the reader to [16] for a detailed description of the method when the restart function from (2.2) is evaluated via numerical quadrature, or to [2] for a different approach using a rational approximation of f . …”

“…To overcome this problem a number of restarting approaches have been proposed in the literature, where-similarly to the techniques for linear systems-after a certain number of iterations the Arnoldi basis is discarded and a new Arnoldi cycle is started to approximate the error of the last iterate, cf. [1,2,10,11,16,23,35]. While much work has been devoted to tuning the methods towards numerical stability and efficiency, there are only few theoretical results concerning the convergence of these methods.…”