Ogita, Takeshi; Aishima, Kensuke Iterative refinement for symmetric eigenvalue decomposition. II. Clustered eigenvalues. (English) Zbl 1418.65049 Japan J. Ind. Appl. Math. 36, No. 2, 435-459 (2019). Summary: We are concerned with accurate eigenvalue decomposition of a real symmetric matrix \(A\). In our previous paper [ibid. 35, No. 3, 1007–1035 (2018; Zbl 1403.65018)], 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. Cited in 8 Documents MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A18 Eigenvalues, singular values, and eigenvectors 15A23 Factorization of matrices Keywords:accurate numerical algorithm; iterative refinement; symmetric eigenvalue decomposition; clustered eigenvalues Citations:Zbl 1403.65018 Software:gmp; LBNL; LAPACK; advanpix; mctoolbox; XBLAS; k-ep; ELSES PDFBibTeX XMLCite \textit{T. Ogita} and \textit{K. Aishima}, Japan J. Ind. Appl. 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