A new investigation of the extended Krylov subspace method for matrix function evaluations

Abstract

Abstract. For large square matrices A and functions f , the numerical approximation of the action of f (A) to a vector v has received considerable attention in the last two decades. In this paper we investigate the Extended Krylov subspace method, a technique that was recently proposed to approximate f (A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques.

Efficient and Stable Arnoldi Restarts for Matrix Functions Based on Quadrature

“…The first one, not further considered in this paper, is to use other subspaces with superior approximation properties, like extended Krylov subspaces [13,32], shift-and-invert Krylov subspaces [35,47], both special cases of general rational Krylov subspaces [4,[22][23][24], with the aim to reach a targeted accuracy within significantly fewer iterations. However, rational Krylov methods typically involve linear system solves with (shifted versions of) the matrix A at each iteration.…”

Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization

“…Extended Krylov subspace methods have been studied in the last 15 years by various authors [3,13,14,16,20]. The second contribution of this paper is that we obtain simultaneously a lower bound for σ 1 and an upper bound for σ n , which leads to a lower bound of good quality for κ(A).…”

A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions

“…In this section we derive some practical estimates for the approximation error f (A)v − f n . Similar techniques can be found in [14,29,24]. The error estimators are compared in Figure 4.1 for a simple test matrix.…”

“…A similar estimator has been successfully applied in [14,29]. One starts by assuming the ideal equalities…”

“…Although restarted variants of the polynomial Arnoldi method for f (A)v have been proposed, which prevent the dimension of the Krylov space to grow above memory limit (see [20,1,21]), the use of finite poles ξ j typically is a worthwhile alternative if linear systems with shifted versions of A can be solved efficiently. If the poles alternate between ξ 2j = ∞ and ξ 2j+1 = 0, we obtain the so-called extended Krylov subspace method with convergence (see [15] and [29,Theorem 3.4 A computational advantage of the extended Krylov subspace method is that only the actions of A and A −1 on vectors are required. In particular, if a direct solver is applicable, only one factorization of A needs to be computed.…”