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.
Highlights
Let A be a real symmetric n ×n matrix
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
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
Summary
Let A be a real symmetric n ×n matrix. Since solving a standard symmetric eigenvalue problem Ax = λx, where λ ∈ R is an eigenvalue of A and x ∈ Rn is an eigenvector of A associated with λ, is ubiquitous in scientific computing, it is important to develop reliable numerical algorithms for calculating eigenvalues and eigenvectors accurately. On related work on refinement algorithms for symmetric eigenvalue decomposition, see the previous paper [17] for details. 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. That means backward stable algorithms can provide a sufficiently accurate initial guess of the “subspace” corresponding to the clustered eigenvalues. We first apply the algorithm (Algorithm 1: RefSyEv) in the previous paper [17] to the initial approximate eigenvector matrix for improving the subspace corresponding to the clustered eigenvalues. We expand eigenvalue gaps in each subproblem by using a diagonal shift and compute eigenvectors of each subproblem, which can be used for refining approximate eigenvectors corresponding to clustered eigenvalues in the entire problem. As mentioned in the previous paper [17], the discussions in this paper can be extended to generalized symmetric (Hermitian) definite eigenvalue problems
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