A note on block diagonal and block triangular preconditioners for complex symmetric linear systems

  • Li-Dan Liao
  • Guo-Feng Zhang
Original Paper


In this note, the additive block diagonal preconditioner (Bai et al., Numer. Algorithms 62, 655–675 2013) and the block triangular preconditioner (Pearson and Wathen, Numer. Linear Algebra Appl. 19, 816–829 2012) are further studied and optimized, respectively. The eigenvalue properties of these two preconditioned matrices are analyzed by new way and an expression of the quasi-optimal parameter is derived. Particularly, when WT or TW is symmetric positive semidefinite, the exact eigenvalue bounds are obtained which are tighter than the state of the art. At last, numerical experiments are presented to show the effectiveness of the two proposed optimized preconditioners.


Preconditioner Complex symmetric problem Optimal parameter Condition number 

Mathematics Subject Classification (2010)

65F10 65F50 65W05 


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The authors are very thankful 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.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Key Laboratory of Applied Mathematics and Complex SystemsLanzhouPeople’s Republic of China

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