Abstract
Recently, contour integral-based methods have been actively studied for solving interior eigenvalue problems that find all eigenvalues located in a certain region and their corresponding eigenvectors. In this paper, we reconsider the algorithms of the five typical contour integral-based eigensolvers from the viewpoint of projection methods, and then map the relationships among these methods. From the analysis, we conclude that all contour integral-based eigensolvers can be regarded as projection methods and can be categorized based on their subspace used, the type of projection and the problem to which they are applied implicitly.
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References
Asakura, J., Sakurai, T., Tadano, H., Ikegami, T., Kimura, K.: A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 1, 52–55 (2009)
Asakura, J., Sakurai, T., Tadano, H., Ikegami, T., Kimura, K.: A numerical method for polynomial eigenvalue problems using contour integral. Japan J. Ind. Appl. Math. 27, 73–90 (2010)
Austin, A.P., Kravanja, P., Trefethen, L.N.: Numerical algorithms based on analytic function values at roots of unity. SIAM J. Numer. Anal. 52, 1795–1821 (2014)
Austin, A.P., Trefethen, L.N.: Computing eigenvalues of real symmetric matrices with rational filters in real arithmetic. SIAM J. Sci. Comput. 37, A1365–A1387 (2015)
van Barel, M., Kravanja, P.: Nonlinear eigenvalue problems and contour integrals. J. Comput. Appl. Math. 292, 526–540 (2016)
Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. 436, 3839–3863 (2012)
Fang, H., Saad, Y.: A filtered Lanczos procedure for extreme and interior eigenvalue problems. SIAM J. Sci. Comput. 34, A2220–A2246 (2012)
Gutknecht, M.H.: Block Krylov space methods for linear systems with multiple right-hand sides: an introduction. In: Siddiqi, A.H., Duff, I.S., Christensen, O. (eds.) Proceedings of Modern Mathematical Models, Methods and Algorithms for Real World Systems, pp. 420–447. Anamaya Publishers, New Delhi (2007)
Güttel, S., Polizzi, E., Tang, T., Viaud, G.: Zolotarev quadrature rules and load balancing for the FEAST eigensolver. ar**v:1407.8078
Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai–Sugiura projection method, Technical Report of Department of Computer Science, University of Tsukuba (CS-TR), CS-TR-08-13 (2008)
Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai–Sugiura projection method. J. Comput. Appl. Math. 233, 1927–1936 (2010)
Ikegami, T., Sakurai, T.: Contour integral eigensolver for non-Hermitian systems: a Rayleigh–Ritz-type approach. Taiwan. J. Math. 14, 825–837 (2010)
Imakura, A., Du, L., Sakurai, T.: A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems. Appl. Math. Lett. 32, 22–27 (2014)
Imakura, A., Du, L., Sakurai, T.: Communication-avoiding Arnoldi-type contour integral-based eigensolver (in Japanese). In: Proceedings of Annual Meeting of JSIAM (2014)
Imakura, A., Du, L., Sakurai, T.: Error bounds of Rayleigh–Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems. Numer. Algorithms 71, 103–120 (2016)
Kravanja, P., Sakurai, T., van Barel, M.: On locating clusters of zeros of analytic functions. BIT 39, 646–682 (1999)
Polizzi, E.: A density matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79, 115112 (2009)
Saad, Y.: Numerical Methods for Large Eigenvalue Problems, 2nd edn. SIAM, Philadelphia (2011)
Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)
Sakurai, T., Tadano, H.: CIRR: a Rayleigh–Ritz type method with counter integral for generalized eigenvalue problems. Hokkaido Math. J. 36, 745–757 (2007)
Sakurai, T., Futamura, Y., Tadano, H.: Efficient parameter estimation and implementation of a contour integral-based eigensolver. J. Algorithms Comput. Technol. 7, 249–269 (2014)
Schofield, G., Chelikowsky, J.R., Saad, Y.: A spectrum slicing method for the Kohn–Sham problem. Comput. Phys. Commun. 183, 497–505 (2012)
Tang, P.T.P., Polizzi, E.: FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J. Matrix Anal. Appl. 35, 354–390 (2014)
Yin, G., Chan, R.H., Yeung, M.-C.: A FEAST algorithm for generalized non-Hermitian eigenvalue problems. ar**v:1404.1768
Yokota, S., Sakurai, T.: A projection method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 5, 41–44 (2013)
Zhou, Y., Saad, Y., Tiago, M.L., Chelikowsky, J.R.: Self-consistent-field calculations using Chebyshev-filtered subspace iteration. J. Comput. Phys. 219, 172–184 (2006)
Acknowledgements
The authors would like to thank Dr. Kensuke Aishima, The University of Tokyo for his valuable comments. The authors are also grateful to an anonymous referee for useful comments.
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This research was supported partly by Interdisciplinary Computational Science Program in CCS, University of Tsukuba, Strategic Programs for Innovative Research (SPIRE) Field 5 “The origin of matter and the universe”, JST/CREST and KAKENHI (Grant Nos. 25286097, 25870099), the Fundamental Research Funds for the Central Universities (No: DUT16LK05) and the National Natural Science Foundation of China (No: 11501079).
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Imakura, A., Du, L. & Sakurai, T. Relationships among contour integral-based methods for solving generalized eigenvalue problems. Japan J. Indust. Appl. Math. 33, 721–750 (2016). https://doi.org/10.1007/s13160-016-0224-x
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DOI: https://doi.org/10.1007/s13160-016-0224-x