Iterative refinement for symmetric eigenvalue decomposition II: clustered eigenvalues

Abstract

We are concerned with accurate eigenvalue decomposition of a real symmetric matrix A. In the previous paper (Ogita and Aishima in Jpn J Ind Appl Math 35(3): 1007-1035, 2018), we proposed an efficient refinement algorithm for improving the accuracy of all eigenvectors, which converges quadratically if a sufficiently accurate initial guess is given. However, since the accuracy of eigenvectors depends on the eigenvalue gap, it is difficult to provide such an initial guess to the algorithm in the case where A has clustered eigenvalues. To overcome this problem, we propose a novel algorithm that can refine approximate eigenvectors corresponding to clustered eigenvalues on the basis of the algorithm proposed in the previous paper. Numerical results are presented showing excellent performance of the proposed algorithm in terms of convergence rate and overall computational cost and illustrating an application to a quantum materials simulation. In [17], we proposed a refinement algorithm for the eigenvalue decomposition of A, which works not for an individual eigenvector but for all eigenvectors. Since the algorithm is based on Newton's method, it converges quadratically, provided that an initial guess is sufficiently accurate. In practice, although the algorithm refines computed eigenvectors corresponding to sufficiently separated simple eigenvalues, it cannot refine computed eigenvectors corresponding to "nearly" multiple eigenvalues. This is because it is difficult for standard numerical algorithms in floating-point arithmetic to provide sufficiently accurate initial approximate eigenvectors corresponding to nearly multiple eigenvalues as shown in (2). The purpose of this paper is to remedy this problem, i.e., we aim to develop a refinement algorithm for the eigenvalue decomposition of a symmetric matrix with clustered eigenvalues.We briefly explain the idea of our proposed algorithm. We focus on the so-called sin θ theorem by Davis-Kahan [5, Section 2] as follows. For an index set J with |J | = < n, let X J ∈ R n× denote the eigenvector matrix comprising x ( j) for all j ∈ J . For 1 ≤ k ≤ , let μ k denote the Ritz values for the subspace spanned by some given vectors with μ 1 ≤ · · · ≤ μ , and let z k be the corresponding normalized Ritz vectors. Assume that the eigenvalues λ i for all i / ∈ J are entirely outside of [μ 1 , μ ]. Let Gap denote the smallest difference between the Ritz values μ k for all k, 1 ≤ k ≤ , and the eigenvalues λ i for all i / ∈ J , i.e., Gap := min{|μ k −λ i | : 1 ≤ k ≤ , i / ∈ J }. Moreover, let Z J := [z 1 , . . . , z ] ∈ R n× . Then, we obtainThis indicates that the subspace spanned by eigenvectors associated with the clustered eigenvalues is not very sensitive to perturbations, provided that the gap between the clustered eigenvalues and the others is sufficiently large. That means backward stable algorithms can provide a sufficiently accurate initial guess of the "subspace" corresponding to the clustered eigenvalues. To extract eigenvectors from the subspace correctly, relatively larger ga...

An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations

“…The refinement procedure will be crucial, in particular, when lower-precision arithmetic, such as half-precision or single-precision arithmetic, is used for calculating an approximate solution as an initial guess. For example, refinement algorithms for the symmetric eigenvalue problem have recently been proposed in [32,33], which are based on matrix multiplications. Such refinement algorithms enhance application researchers to use lower-precision arithmetic with satisfactory reliability of the computed results, which will be of great importance in next-generation architecture that is optimized for lower-precision arithmetic.…”

Fixed-point analysis of Ogita-Aishima's symmetric eigendecomposition refinement algorithm for multiple eigenvalues

“…Note that D is also an unknown matrix to be determined from (2) and (3). Apparently, its diagonal elements are the eigenvalues of A.…”

“…Note that the present paper is focused on multiple eigenvalues and not on clustered eigenvalues, which pose a more subtle problem. Consult [2,3] for recent advances on this topic.…”

Finding the Maximal Eigenpair for a Large, Dense, Symmetric Matrix Based on Mufa Chen's Algorithm

“…Example 4.4. This example comes from [15]. We generate real symmetric and positive definite matrices using the MATLAB function randsvd.…”

“…The reason is that the convergence rate of the power method depends on the ratio of the second largest eigenvalue over the largest one. In Section 4, our numerical example taken from [15] shows that p pw can be as big as O(N). Thus, for the cases with clustered eigenvalues, our proposed method has some obvious advantage.…”