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. Imakura, Akira; Du, Lei; Sakurai, Tetsuya: Error bounds of Rayleigh-Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems (2016)
  2. Van Barel, Marc: Designing rational filter functions for solving eigenvalue problems by contour integration (2016)
  3. Van Barel, Marc; Kravanja, Peter: Nonlinear eigenvalue problems and contour integrals (2016)
  4. Austin, Anthony P.; Trefethen, Lloyd N.: Computing eigenvalues of real symmetric matrices with rational filters in real arithmetic (2015)
  5. Austin, Anthony P.; Kravanja, Peter; Trefethen, Lloyd N.: Numerical algorithms based on analytic function values at roots of unity (2014)
  6. Imakura, Akira; Du, Lei; Sakurai, Tetsuya: A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems (2014)
  7. Ikegami, Tsutomu; Sakurai Tetsuya: Contour integral eigensolver for non-Hermitian systems: A Rayleigh-Ritz-type approach (2010)
  8. Ohno, Hiroshi; Kuramashi, Yoshinobu; Sakurai, Tetsuya; Tadano, Hiroto: A quadrature-based eigensolver with a Krylov subspace method for shifted linear systems for Hermitian eigenproblems in lattice QCD (2010)
  9. Sakurai, Tetsuya; Tadano, Hiroto; Ikegami, Tsutomu; Nagashima, Umpei: A parallel eigensolver using contour integration for generalized eigenvalue problems in molecular simulation (2010)
  10. Senzaki, Kenta; Tadano, Hiroto; Sakurai, Tetsuya; Bai, Zhaojun: A method for profiling the distribution of eigenvalues using the AS method (2010)
  11. Sakurai, Tetsuya; Asakura, Junko; Tadano, Hiroto; Ikegami, Tsutomu: Error analysis for a matrix pencil of Hankel matrices with perturbed complex moments (2009)
  12. Sakurai, Tetsuya; Tadano, Hiroto: CIRR: a Rayleigh-Ritz method with contour integral for generalized eigenvalue problems (2007)