The extended Krylov subspace method and orthogonal Laurent polynomials

Abstract

Dedicated to Henk van der Vorst on the occasion of his 65th birthday.Abstract. The need to evaluate expressions of the form f (A)v, where A is a large sparse or structured symmetric matrix, v is a vector, and f is a nonlinear function, arises in many applications. The extended Krylov subspace method can be an attractive scheme for computing approximations of such expressions. This method projects the approximation problem onto an extended Krylov subspace K ℓ,m (A) = span{A −ℓ+1 v, . . . , A −1 v, v, Av, . . . , A m−1 v} of fairly small dimension, and then solves the small approximation problem so obtained. We review available results for the extended Krylov subspace method and relate them to properties of Laurent polynomials. The structure of the projected problem receives particular attention. We are concerned with the situations when m = ℓ and m = 2ℓ. . 1 e.g., by determining the spectral or Cholesky factorizations of T m . For instance, m steps of the Lanczos process applied to A with initial vector v yields the decompositionis symmetric and tridiagonal, g m ∈ R n , and V T m g m = 0. Here and below e j = [0, . . . , 0, 1, 0, . . . , 0] T denotes the jth axis vector and · the Euclidean vector norm. We tacitly assume that m is chosen small enough so that a decomposition of the form (1.3) exists. The columns of V m form an orthonormal basis for the Krylov subspaceThe expression (1.1) now can be approximated by(1.5) see, e.g., [4,10,14,21] for discussions on this approach. Indeed, if g m = 0, then w m = w. Moreover, let P m−1 denote the set of all polynomials of degree at most m − 1. Then f ∈ P m−1 implies that w m = w; see, e.g., [10] or [22, Proposition 6.3].The decomposition (1.3) and the fact that range(V m ) = K m (A, v) show that:i) The columns v j of V m satisfy a three-term recurrence relation. This follows from the fact that T m is tridiagonal. The vectors v j therefore are quite inexpensive to compute; only one matrix vector-product evaluation with A and a few vector operations are required to compute v j+1 from v j and v j−1 . ii) The columns v j can be expressed as

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).…”

“…We can reformulate the expressions in (2.2) and (2.3) to see the similarities with the extended Lanczos method (see, e.g., [13]) with starting vector v 0 and matrix A T A, so that for k ≥ 1: This way of representing the procedure will be convenient in the next sections where we will investigate the structure of the generated matrices and introduce Laurent polynomials.…”

“…The matrices in the reformulated expressions (2.5) also have a special structure, just as the matrices formed in the extended Lanczos method in [13]. The four symmetric matrices generated in this extended Lanczos process, for k ≥ 1, are given by (3.5)…”

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

“…, v k }. The interplay between orthogonalization coefficients and T k , for the extended Krylov subspace methods, are discussed by Simoncini [42] and Jagels and Reichel [34,33].…”

“…5.1], which we instantiate with the same data as in the original paper, corresponding to four levels of refinement of the mesh, and thus the matrices described in Table 4. We keep the notation of the original paper, so that the matrices are of size n 2 × n 2 rather n × n. We compute the solution of the linear system (33) H α y = v, with a random right hand side v ∈ R corresponding to a = 0, 2, 4.…”