Abstract
In this paper we present a new subspace iteration for calculating eigenvalues of symmetric matrices. The method is designed to compute a cluster of k exterior eigenvalues. For example, k eigenvalues with the largest absolute values, the k algebraically largest eigenvalues, or the k algebraically smallest eigenvalues. The new iteration applies a Restarted Krylov method to collect information on the desired cluster. It is shown that the estimated eigenvalues proceed monotonically toward their limits. Another innovation regards the choice of starting points for the Krylov subspaces, which leads to fast rate of convergence. Numerical experiments illustrate the viability of the proposed ideas.
Highlights
In this paper we present a new subspace iteration for calculating a cluster of k exterior eigenvalues of a given symmetric matrix, G ∈ n×n
As with other subspace iterations, the method is best suited for handling large sparse matrices in which a matrix-vector product needs only 0(n) flops
The new method is based on a modified interlacing theorem which forces the Rayleigh-Ritz approximations to move monotonically toward their limits
Summary
In this paper we present a new subspace iteration for calculating a cluster of k exterior eigenvalues of a given symmetric matrix, G ∈ n×n. The new iteration applies a Krylov subspace method to collect information on the desired cluster. It has an additional flavor: it uses an interlacing theorem to improve the current estimates of the eigenvalues. The restriction of the target cluster to include only non-zero eigenvalues means that the target space is contained in Range(G). Uq ∈ p×k , UqTUq = I ∈ k×k , which is used to compute the related matrix of Ritz vectors, Vq+1 = X qUq. Step 2: Collecting new information. The paper ends with numerical experiments that illustrate the behavior of the proposed method
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