Convergence of the Isometric Arnoldi Process

Abstract

It is well known that the performance of eigenvalue algorithms such as the Lanczos and the Arnoldi method depends on the distribution of eigenvalues. Under fairly general assumptions we characterize the region of good convergence for the Isometric Arnoldi Process. We also determine bounds for the rate of convergence and we prove sharpness of these bounds. The distribution of isometric Ritz values is obtained as the minimizer of an extremal problem. We use techniques from logarithmic potential theory in proving these results. Abstract. It is well known that the performance of eigenvalue algorithms such as the Lanczos and the Arnoldi method depends on the distribution of eigenvalues. Under fairly general assumptions we characterize the region of good convergence for the Isometric Arnoldi Process. We also determine bounds for the rate of convergence and we prove sharpness of these bounds. The distribution of isometric Ritz values is obtained as the minimizer of an extremal problem. We use techniques from logarithmic potential theory in proving these results.

On the Convergence of Rational Ritz Values

“…Thus the findings of [1,21,23] are related to results from the late last century about weak asymptotics of discrete polynomials due to Rakhmanov [32] and Dragnev & Saff [14], Van Assche & Kuijlaars [26], Beckermann [2], and others [10,11,25]. In the present paper we use the well-known fact [13] that rational Ritz values may be written as zeros of discrete orthogonal rational functions [9].…”

“…There is a rule of thumb proposed by Trefethen and Bau [39] that dist(λ k , Θ) is small for eigenvalues λ k lying in regions of the real line where there are "relatively few" eigenvalues. It was Kuijlaars [23] who first quantified this heuristic rule, and we also refer to the refinements given in an unpublished note [1] and some related work on isometric Ritz values [21]. Suppose that the asymptotic eigenvalue distribution is described by a finite positive Borel measure σ.…”

“…The optimal polynomials for both tasks can look quite different, see Figure 2.1 for a simple example. Therefore a precise upper bound of dist(λ k , Θ) needs to incorporate the fine structure of the spectrum, see [1,21,23]. This fine structure also explains the superlinear convergence behavior of the CG method (see [5][6][7] and the reviews [24] and [3] 3.…”

“…Statement of the main results. Following Kuijlaars and his successors [1,[5][6][7]21, 23], we will consider a sequence of Hermitian matrices A N ∈ C N ×N having a joint eigenvalue distribution described by some measure σ: write more explicitly the set Λ N = Λ(A N ) of eigenvalues λ 1,N < · · · < λ N,N of A N , and consider the normalized counting measure…”

Structured Eigenvalue Problems

“…In [54], a divide and conquer approach for unitary Hessenberg matrices was proposed, based on the methodology of Cuppen's divide and conquer approach for symmetric tridiagonal matrices [38]. Krylov subspace methods for orthogonal and unitary matrices have been developed and analyzed in [15,26,62,70,71].…”