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. 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)
  2. Li, Ruipeng; Xi, Yuanzhe; Erlandson, Lucas; Saad, Yousef: The eigenvalues slicing library (EVSL): algorithms, implementation, and software (2019)
  3. Manguoğlu, Murat; Mehrmann, Volker: A robust iterative scheme for symmetric indefinite systems (2019)
  4. Murakami, Hiroshi: Filters consist of a few resolvents to solve real symmetric definite generalized eigenproblems (2019)
  5. Yin, Guojian: On the non-Hermitian FEAST algorithms with oblique projection for eigenvalue problems (2019)
  6. Yin, Guojian: A harmonic FEAST algorithm for non-Hermitian generalized eigenvalue problems (2019)
  7. Yin, Guojian: A contour-integral based method with Schur-Rayleigh-Ritz procedure for generalized eigenvalue problems (2019)
  8. Yin, Guojian: A contour-integral based method for counting the eigenvalues inside a region (2019)
  9. Ruipeng Li, Yuanzhe Xi, Lucas Erlandson, Yousef Saad: The Eigenvalues Slicing Library (EVSL): Algorithms, Implementation, and Software (2018) arXiv
  10. Xi, Yuanzhe; Saad, Yousef: A rational function preconditioner for indefinite sparse linear systems (2017)
  11. Ye, Xin; Xia, Jianlin; Chan, Raymond H.; Cauley, Stephen; Balakrishnan, Venkataramanan: A fast contour-integral eigensolver for non-Hermitian matrices (2017)
  12. Chen, Xiao-Ping; Dai, Hua: Stability analysis of time-delay systems using a contour integral method (2016)
  13. Imakura, Akira; Du, Lei; Sakurai, Tetsuya: Error bounds of Rayleigh-Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems (2016)
  14. Imakura, Akira; Du, Lei; Sakurai, Tetsuya: Relationships among contour integral-based methods for solving generalized eigenvalue problems (2016)
  15. Kestyn, James; Polizzi, Eric; Tang, Ping Tak Peter: Feast eigensolver for non-Hermitian problems (2016)
  16. Pieper, Andreas; Kreutzer, Moritz; Alvermann, Andreas; Galgon, Martin; Fehske, Holger; Hager, Georg; Lang, Bruno; Wellein, Gerhard: High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations (2016)
  17. Van Barel, Marc: Designing rational filter functions for solving eigenvalue problems by contour integration (2016)
  18. Van Barel, Marc; Kravanja, Peter: Nonlinear eigenvalue problems and contour integrals (2016)
  19. Xiao, Jinyou; Meng, Shuangshuang; Zhang, Chuanzeng; Zheng, Changjun: Resolvent sampling based Rayleigh-Ritz method for large-scale nonlinear eigenvalue problems (2016)
  20. Xi, Yuanzhe; Saad, Yousef: Computing partial spectra with least-squares rational filters (2016)

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