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A block Lanczos method for the linear response eigenvalue problem. (English) Zbl 1392.65102

Summary: In the linear response eigenvalue problem arising from computational quantum chemistry and physics one needs to compute a small portion of eigenvalues around zero together with the associated eigenvectors. Lanczos-type methods are particularly suitable for such a task. However, single-vector Lanczos methods can only find one copy of any multiple eigenvalue and can be very slow when the desired eigenvalues form a cluster. In this paper, we propose a block Lanczos-type implementation for the linear response eigenvalue problem, which is able to compute a cluster of eigenvalues much faster and more efficiently than the single-vector version. Convergence results are established and reveal the accuracy of the approximations of eigenvalues in a cluster and of the eigenspace. A practical thick-restart procedure is introduced to alleviate the increasing numerical difficulties of the block Lanczos method in computational costs, memory demands, and numerical stability. Numerical examples are presented to support the effectiveness of the thick-restart block Lanczos method.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
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[1] Z. B AI AND R.-C. L I, Minimization principles for the linear response eigenvalue problem I: Theory, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 1075-1100. · Zbl 1263.65078
[2] , Minimization principles for the linear response eigenvalue problem II: Computation, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 392-416. · Zbl 1311.65102
[3] , Minimization principles and computation for the generalized linear response eigenvalue problem, BIT, 54 (2014), pp. 31-54. · Zbl 1293.65053
[4] Z. B AI, R.-C. L I,AND W.-W. L IN, Linear response eigenvalue problem solved by extended locally optimal preconditioned conjugate gradient methods, Sci. China Math., 59 (2016), pp. 1443-1460. · Zbl 1358.65021
[5] R. B HATIA, Matrix Analysis, Springer, New York, 1997.
[6] T. A. D AVIS AND Y. H U, The University of Florida sparse matrix collection, ACM Trans. Math. Software, 38 (2011), Art. 1, 25 pages. · Zbl 1365.65123
[7] P. G IANNOZZI ET AL., QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter, 21 (2009), Art. 395502, 19 pages.
[8] M. G RÜNING, A. M ARINI,AND X. G ONZE, Exciton-plasmon states in nanoscale materials: breakdown of the Tamm-Dancoff approximation, Nano Lett., 9 (2009), pp. 2820-2824.
[9] R. B. L EHOUCQ AND D. C. S ORENSEN, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 789-821. ETNA Kent State University and Johann Radon Institute (RICAM) BLOCK LANCZOS FOR THE LINEAR RESPONSE EIGENVALUE PROBLEM 523 · Zbl 0863.65016
[10] R.-C. L I AND L.-H. Z HANG, Convergence of the block Lanczos method for eigenvalue clusters, Numer. Math., 131 (2015), pp. 83-113. · Zbl 1334.65073
[11] M. T. L USK AND A. E. M ATTSSON, High-performance computing for materials design to advance energy science, MRS Bull. 36 (2011), pp. 169-174.
[12] P. P APAKONSTANTINOU, Reduction of the RPA eigenvalue problem and a generalized Cholesky decomposition for real-symmetric matrices, Europhys. Lett. EPL, 78 (2007), Art. 12001, 5 pages. · Zbl 1244.65054
[13] P. R ING, Z.-Y. M A, N. V. G IAI, D. V RETENAR, A. W ANDELT,AND L.-G. G AO, The time-dependent relativistic mean-field theory and the random phase approximation, Nuclear Phys. A, 694 (2001), pp. 249- 268. · Zbl 0971.81586
[14] D. R OCCA, Time-Dependent Density Functional Perturbation Theory: New algorithms with applications to molecular spectra, PhD. Thesis, International School for Advanced Studies, Trieste, Italy, 2007.
[15] D. R OCCA, D. L U,AND G. G ALLI, Ab initio calculations of optical absorpation spectra: solution of the Bethe Salpeter equation within density matrix perturbation theory, J. Chem. Phys., 133 (2010), Art. 164109, 10 pages.
[16] D. C. S ORENSEN, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357-385. · Zbl 0763.65025
[17] G. W. S TEWART, A Krylov-Schur algorithm for large eigenproblems, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 601-614. · Zbl 1003.65045
[18] R. E. S TRATMANN, G. E. S CUSERIA,AND M. J. F RISCH, An efficient implementation of time-dependent density-functional theory for the calculation of excitation of large molecules, J. Chem. Phys., 109 (1998), pp. 8218-8824.
[19] Z. T ENG AND R.-C. L I, Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem, J. Comput. Appl. Math., 247 (2013), pp. 17-33. · Zbl 1279.65040
[20] Z. T ENG, L. L U,AND R.-C. L I, Perturbation of partitioned linear response eigenvalue problems, Electron. Trans. Numer. Anal., 44 (2015), pp. 624-638. http://etna.ricam.oeaw.ac.at/vol.44.2015/pp624-638.dir/pp624-638.pdf · Zbl 1330.65060
[21] Z. T ENG, Y. Z HOU,AND R.-C. L I, A block Chebyshev-Davidson method for linear response eigenvalue problems, Adv. Comput. Math., 42 (2016), pp. 1103-1128. · Zbl 1357.65046
[22] E. V. T SIPER, Variational procedure and generalized Lanczos recursion for small-amplitude classical oscilla tions, JETP Lett., 70 (1999), pp. 751-755.
[23] , A classical mechanics technique for quantum linear response, J. Phys. B, 34 (2001), pp. L401-L407.
[24] W.-G. W ANG, L.-H. Z HANG,AND R.-C. L I, Error bounds for approximate deflating subspaces for linear response eigenvalue problems, Linear Algebra Appl., 528 (2017), pp. 273-289. · Zbl 1398.15016
[25] K. W U AND H. S IMON, Thick-restart Lanczos method for large symmetric eigenvalue problems, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 602-616. · Zbl 0969.65030
[26] I. Y AMAZAKI, Z. B AI, H. S IMON, L.-W. W ANG,AND K. W U, Adaptive projection subspace dimension for the thick-restart Lanczos method, ACM Trans. Math. Software, 37 (2010), Art. 27, 18 pages. · Zbl 1364.65089
[27] L.-H. Z HANG, W.-W. L IN,AND R.-C. L I, Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem, BIT, 55 (2015), pp. 869-896. · Zbl 1350.65081
[28] L.-H. Z HANG, J. X UE,AND R.-C. L I, Rayleigh-Ritz approximation for the linear response eigenvalue problem, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 765-782. · Zbl 1305.65126
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