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. Embree, Mark; Loe, Jennifer A.; Morgan, Ronald: Polynomial preconditioned Arnoldi with stability control (2021)
  2. Hu, Qian-Ying; Wen, Chun; Huang, Ting-Zhu; Shen, Zhao-Li; Gu, Xian-Ming: A variant of the Power-Arnoldi algorithm for computing PageRank (2021)
  3. Aishima, Kensuke: Convergence proof of the harmonic Ritz pairs of iterative projection methods with restart strategies for symmetric eigenvalue problems (2020)
  4. Dax, Achiya: A cross-product approach for low-rank approximations of large matrices (2020)
  5. Miao, Cun-Qiang: On Chebyshev-Davidson method for symmetric generalized eigenvalue problems (2020)
  6. Polizzi, Eric; Saad, Yousef: Computational materials science and engineering (2020)
  7. Wang, Qing-Wen; Wang, Xiang-Xiang: Arnoldi method for large quaternion right eigenvalue problem (2020)
  8. Camps, Daan; Meerbergen, Karl; Vandebril, Raf: A rational QZ method (2019)
  9. Camps, Daan; Meerbergen, Karl; Vandebril, Raf: An implicit filter for rational Krylov using core transformations (2019)
  10. Choi, Young-Geun; Lim, Johan; Roy, Anindya; Park, Junyong: Fixed support positive-definite modification of covariance matrix estimators via linear shrinkage (2019)
  11. Dax, Achiya: Computing the smallest singular triplets of a large matrix (2019)
  12. Dong, Yiqiu; Hansen, Per Christian; Hochstenbach, Michiel E.; Brogaard Riis, Nicolai André: Fixing nonconvergence of algebraic iterative reconstruction with an unmatched backprojector (2019)
  13. Elman, Howard C.; Su, Tengfei: Low-rank solution methods for stochastic eigenvalue problems (2019)
  14. Embree, Mark: Unstable modes in projection-based reduced-order models: how many can there be, and what do they tell you? (2019)
  15. Hou, Thomas Y.; Huang, De; Lam, Ka Chun; Zhang, Ziyun: A fast hierarchically preconditioned eigensolver based on multiresolution matrix decomposition (2019)
  16. Jia, Zhigang; Ng, Michael K.; Song, Guang-Jing: Lanczos method for large-scale quaternion singular value decomposition (2019)
  17. Miao, Cun-Qiang: Filtered Krylov-like sequence method for symmetric eigenvalue problems (2019)
  18. Wu, Lingfei; Xue, Fei; Stathopoulos, Andreas: TRPL+K: thick-restart preconditioned Lanczos+K method for large symmetric eigenvalue problems (2019)
  19. Zadeh, Najmeh Azizi; Tajaddini, Azita; Wu, Gang: Weighted and deflated global GMRES algorithms for solving large Sylvester matrix equations (2019)
  20. Bucci, M. A.; Puckert, D. K.; Andriano, C.; Loiseau, J.-C.; Cherubini, S.; Robinet, J.-C.; Rist, U.: Roughness-induced transition by quasi-resonance of a varicose global mode (2018)

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