Johnson, Chris; Kennedy, A. D. Numerical determination of partial spectrum of Hermitian matrices using a Lánczos method with selective reorthogonalization. (English) Zbl 1302.65088 Comput. Phys. Commun. 184, No. 3, 689-697 (2013). Summary: We introduce a new algorithm for finding the eigenvalues and eigenvectors of Hermitian matrices within a specified region, based upon the LANSO algorithm of Parlett and Scott. It uses selective reorthogonalization to avoid the duplication of eigenpairs in finite-precision arithmetic, but uses a new bound to decide when such reorthogonalization is required, and only reorthogonalizes with respect to eigenpairs within the region of interest. We investigate its performance for the Hermitian Wilson-Dirac operator \(\gamma_5 D\) in lattice quantum chromodynamics, and compare it with previous methods. MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis Keywords:spectrum; Lanczos; Krylov; eigenvalue; eigenvector; Hermitian; LANSO; lattice Software:Chroma; LAPACK PDFBibTeX XMLCite \textit{C. Johnson} and \textit{A. D. Kennedy}, Comput. Phys. Commun. 184, No. 3, 689--697 (2013; Zbl 1302.65088) Full Text: DOI arXiv References: [1] Kennedy, A. D., Algorithms for Dynamical Fermions (2006), World Scientific, pp. 15-82 (Chapter 2) [2] Parlett, B. N.; Scott, D. S., The Lánczos algorithm with selective orthogonalization, Math. Comp., 33, 145, 217-238 (1979), URL: ftp://ftp.math.utah.edu/pub/tex/bib/gvl.bib, ftp://ftp.math.utah.edu/pub/tex/bib/mathcomp1970.bib · Zbl 0405.65015 [3] Edwards, R. G.; Joo, B., The Chroma software system for lattice QCD, Nuclear Phys. B Proc. Suppl., 140, 832 (2005) [4] Kalkreuter, T.; Simma, H., An accelerated conjugate gradient algorithm to compute low lying eigenvalues: A study for the Dirac operator in SU(2) lattice QCD, Comput. Phys. Comm., 93, 33-47 (1996) · Zbl 0917.65036 [5] Bunk, B., Computing the lowest eigenvalues of the fermion matrix by subspace iterations, Nuclear Phys. B Proc. Suppl., 53, 987-989 (1997) [6] Golub, G. H.; van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press · Zbl 0865.65009 [7] Simon, H. D., The Lánczos algorithm with partial reorthogonalization, Math. Comp., 42, 165, 115-142 (1984), URL: http://www.jstor.org/stable/2007563 · Zbl 0546.65017 [8] Parlett, B. N., The Symmetric Eigenvalue Problem (1998), Prentice-Hall, Inc.: Prentice-Hall, Inc. Upper Saddle River, NJ, USA · Zbl 0885.65039 [10] Kaniel, S., Estimates for some computational techniques in linear algebra, Math. Comp., 369-378 (1966) · Zbl 0156.16202 [12] Saad, Y., On the rates of convergence of the Lánczos and the block-Lánczos methods, SIAM J. Numer. Anal., 17, 5, 687-706 (1980) · Zbl 0456.65016 [15] Anderson, E.; Bai, Z.; Dongarra, J.; Greenbaum, A.; McKenney, A.; Du Croz, J.; Hammerling, S.; Demmel, J.; Bischof, C.; Sorensen, D., Lapack: a portable linear algebra library for high-performance computers, (Proceedings of the 1990 ACM/IEEE Conference on Supercomputing, Supercomputing’90 (1990), IEEE Computer Society Press: IEEE Computer Society Press Los Alamitos, CA, USA), 2-11, URL: http://dl.acm.org/citation.cfm?id=110382.110385 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.