CIRR: a Rayleigh-Ritz method with contour integral for generalized eigenvalue problems. We consider a Rayleigh-Ritz type eigensolver for finding a limited set of eigenvalues and their corresponding eigenvectors in a certain region of generalized eigenvalue problems. When the matrices are very large, iterative methods are used to generate an invariant subspace that contains the desired eigenvectors. Approximations are extracted from the subspace through a Rayleigh-Ritz projection. In this paper, we present a Rayleigh-Ritz type method with a contour integral (CIRR method). In this method, numerical integration along a circle that contains relatively small number of eigenvalues is used to construct a subspace. Since the process to derive the subspace can be performed in parallel, the presented method is suitable for master-worker programming models. Numerical experiments illustrate the property of the proposed method.

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  1. Huang, Tsung-Ming; Liao, Weichien; Lin, Wen-Wei; Wang, Weichung: An efficient contour integral based eigensolver for 3D dispersive photonic crystal (2021)
  2. Kalantzis, Vassilis; Xi, Yuanzhe; Horesh, Lior: Fast randomized non-Hermitian eigensolvers based on rational filtering and matrix partitioning (2021)
  3. Kollnig, Konrad; Bientinesi, Paolo; Di Napoli, Edoardo A.: Rational spectral filters with optimal convergence rate (2021)
  4. Morikuni, Keiichi: Projection method for eigenvalue problems of linear nonsquare matrix pencils (2021)
  5. Nedamani, F. Abbasi; Sheikhani, A. H. Refahi; Najafi, H. Saberi: A new algorithm for solving large-scale generalized eigenvalue problem based on projection methods (2020)
  6. Polizzi, Eric; Saad, Yousef: Computational materials science and engineering (2020)
  7. Alvermann, Andreas; Basermann, Achim; Bungartz, Hans-Joachim; Carbogno, Christian; Ernst, Dominik; Fehske, Holger; Futamura, Yasunori; Galgon, Martin; Hager, Georg; Huber, Sarah; Huckle, Thomas; Ida, Akihiro; Imakura, Akira; Kawai, Masatoshi; Köcher, Simone; Kreutzer, Moritz; Kus, Pavel; Lang, Bruno; Lederer, Hermann; Manin, Valeriy; Marek, Andreas; Nakajima, Kengo; Nemec, Lydia; Reuter, Karsten; Rippl, Michael; Röhrig-Zöllner, Melven; Sakurai, Tetsuya; Scheffler, Matthias; Scheurer, Christoph; Shahzad, Faisal; Simoes Brambila, Danilo; Thies, Jonas; Wellein, Gerhard: Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects (2019)
  8. Li, Ruipeng; Xi, Yuanzhe; Erlandson, Lucas; Saad, Yousef: The eigenvalues slicing library (EVSL): algorithms, implementation, and software (2019)
  9. Manguoğlu, Murat; Mehrmann, Volker: A robust iterative scheme for symmetric indefinite systems (2019)
  10. Murakami, Hiroshi: Filters consist of a few resolvents to solve real symmetric definite generalized eigenproblems (2019)
  11. Yin, Guojian: A contour-integral based method with Schur-Rayleigh-Ritz procedure for generalized eigenvalue problems (2019)
  12. Yin, Guojian: On the non-Hermitian FEAST algorithms with oblique projection for eigenvalue problems (2019)
  13. Yin, Guojian: A harmonic FEAST algorithm for non-Hermitian generalized eigenvalue problems (2019)
  14. Yin, Guojian: A contour-integral based method for counting the eigenvalues inside a region (2019)
  15. Ruipeng Li, Yuanzhe Xi, Lucas Erlandson, Yousef Saad: The Eigenvalues Slicing Library (EVSL): Algorithms, Implementation, and Software (2018) arXiv
  16. Imakura, Akira; Sakurai, Tetsuya: Block Krylov-type complex moment-based eigensolvers for solving generalized eigenvalue problems (2017)
  17. Xi, Yuanzhe; Saad, Yousef: A rational function preconditioner for indefinite sparse linear systems (2017)
  18. Ye, Xin; Xia, Jianlin; Chan, Raymond H.; Cauley, Stephen; Balakrishnan, Venkataramanan: A fast contour-integral eigensolver for non-Hermitian matrices (2017)
  19. Yin, Guojian; Chan, Raymond H.; Yeung, Man-Chung: A FEAST algorithm with oblique projection for generalized eigenvalue problems. (2017)
  20. Chen, Xiao-Ping; Dai, Hua: Stability analysis of time-delay systems using a contour integral method (2016)

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