A note on the convergence theorem of the tridiagonal QR algorithm with Wilkinson’s shift

  • Kensuke AishimaEmail author
Original Paper Area 2


We discuss the convergence rate of the QR algorithm with Wilkinson’s shift for tridiagonal symmetric eigenvalue problems. It is well known that the convergence rate is theoretically at least quadratic, and practically better than cubic for most matrices. In an effort to derive the convergence rate, the limiting patterns of some lower right submatrices have been intensively investigated. In this paper, we first describe a new limiting pattern of the lower right 3-by-3 submatrix with a concrete example, and then prove that the convergence rate of this new pattern is strictly cubic. In addition, we stress that our analysis identifies three classes of the limiting patterns of the tridiagonal QR algorithm with Wilkinson’s shift.


Numerical linear algebra Eigensolver QR algorithm  Symmetric tridiagonal matrices Wilkinson’s shift  Convergence rate 

Mathematics Subject Classification

65F15 15A18 



The author is grateful to Professor Takayasu Matsuo and Professor Beresford Parlett for their valuable comments and suggestions. The author also thanks the anonymous reviewer for the helpful comments.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.The University of TokyoTokyoJapan

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