Implicit application of polynomial filters in a k-step Arnoldi method. The author describes and analyses a new implementation of the Arnoldi method for computing a few eigenvalues and the corresponding eigenvectors of a large general square matrix (which reduces to the Lanczos method in the symmetric case). Using a truncated variant of the implicitly shifted QR-iteration, the author applies a polynomial filter to the Arnoldi (Lanczos) vector on each iteration. This approach generalizes explicit restart methods. Advantages of the method are discussed and some preliminary computational results using parallel and vector computers are given

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  1. Camps, Daan; Meerbergen, Karl; Vandebril, Raf: An implicit filter for rational Krylov using core transformations (2019)
  2. Embree, Mark: Unstable modes in projection-based reduced-order models: how many can there be, and what do they tell you? (2019)
  3. Hou, Thomas Y.; Huang, De; Lam, Ka Chun; Zhang, Ziyun: A fast hierarchically preconditioned eigensolver based on multiresolution matrix decomposition (2019)
  4. Hack, M. J. Philipp; Moin, Parviz: Coherent instability in wall-bounded shear (2018)
  5. Krämer, Lukas; Lang, Bruno: Convergence of integration-based methods for the solution of standard and generalized Hermitian eigenvalue problems (2018)
  6. Ma, Lingling; Jiang, Lijian: Convergence analysis for GMsFEM approximation of elliptic eigenvalue problems (2018)
  7. Mastronardi, Nicola; van Dooren, Paul: The (QR) steps with perfect shifts (2018)
  8. Morgan, Ronald B.; Yang, Zhao: Two-grid and multiple-grid Arnoldi for eigenvalues (2018)
  9. Niu, Datian; Meng, Jiana; Li, Hongying: A new shift strategy for the implicitly restarted refined harmonic Lanczos method (2018)
  10. Patanè, G.: Laplacian spectral basis functions (2018)
  11. Rostami, Minghao W.; Xue, Fei: Robust linear stability analysis and a new method for computing the action of the matrix exponential (2018)
  12. Aishima, Kensuke: On convergence of iterative projection methods for symmetric eigenvalue problems (2017)
  13. Bosch, Jessica; Greif, Chen: Numerical solution of linear eigenvalue problems (2017)
  14. Dax, Achiya: The numerical rank of Krylov matrices (2017)
  15. Dong, Yongxin; Gu, Chuanqing; Chen, Zhibing: An Arnoldi-Inout method accelerated with a two-stage matrix splitting iteration for computing PageRank (2017)
  16. Gu, Chuanqing; Wang, Wenwen: An Arnoldi-Inout algorithm for computing PageRank problems (2017)
  17. Hou, Thomas Y.; Li, Qin; Zhang, Pengchuan: A sparse decomposition of low rank symmetric positive semidefinite matrices (2017)
  18. Kao, Chiu-Yen; Lai, Rongjie; Osting, Braxton: Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces (2017)
  19. Lloyd, David J. B.; Scheel, Arnd: Continuation and bifurcation of grain boundaries in the Swift-Hohenberg equation (2017)
  20. Miao, Cun-Qiang: A filtered-Davidson method for large symmetric eigenvalue problems (2017)

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