Computing eigenpairs of Hermitian matrices in perfect Krylov subspaces

  • Zhong-Zhi BaiEmail author
  • Cun-Qiang Miao
Original Paper


For computing the smallest eigenvalue and the corresponding eigenvector of a Hermitian matrix, by introducing a concept of perfect Krylov subspace, we propose a class of perfect Krylov subspace methods. For these methods, we prove their local, semilocal, and global convergence properties, and discuss their inexact implementations and preconditioning strategies. In addition, we use numerical experiments to demonstrate the convergence properties and exhibit the competitiveness of these methods with a few state-of-the art iteration methods such as Lanczos, rational Krylov sequence, and Jacobi-Davidson, when they are employed to solve large and sparse Hermitian eigenvalue problems.


Hermitian eigenproblem Krylov subspace method Inexact iteration Convergence property 

Mathematics Subject Classification (2010)

65F10, 65F15, 15A18 CR: G1.3 


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The authors are very much indebted to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript of this paper.


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Authors and Affiliations

  1. 1.State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China

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