Recent Developments in Algorithms for Solving Dense Eigenproblems (II) : Multishift QR Algorithms(Survey,Algorithms for Matrix/Eigenvalue Problems and their Application,<Special Issue> "Joint Symposium of JSIAM Activity Groups 2006")

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  • Yamamoto Yusaku
    Department of Computational Science & Engineering, Nagoya University

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Other Title
  • 密行列固有値解法の最近の発展(II) : マルチシフトQR法(サーベイ,行列・固有値問題の解法とその応用,<特集>平成18年研究部会連合発表会)
  • 密行列固有値解法の最近の発展(2)マルチシフトQR法
  • ミツギョウレツ コユウチカイホウ ノ サイキン ノ ハッテン 2 マルチシフト QRホウ

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Abstract

The QR algorithm is one of the most reliable and widely used methods to compute the eigenvalues of symmetric and nonsymmetric matrices. However, it is not straightforward to execute the QR algorithm efficiently on modern architectures such as processors with hierarchical memory or parallel computers because of its inherent sequential nature and low data reference locality. To overcome this difficulty, Bai & Demmel proposed the multishift QR algorithm in 1989 and this idea has been greatly expanded since then. In this paper, we introduce the basic theory of the multishift QR algorithm and review recent developments to improve its efficiency, such as the two-tone QR algorithm, aggressive early deflation and the fully-pipelined multishift QR algorithm. Directions for future research are also discussed.

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