A note on the Rayleigh quotient iteration for symmetric eigenvalue problems

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• 作者：Kensuke Aishima (1)
  
• 关键词：Numerical linear algebra ; Eigensolver ; 65F15
• 刊名：Japan Journal of Industrial and Applied Mathematics
• 出版年：2014
• 出版时间：November 2014
• 年：2014
• 卷：31
• 期：3
• 页码：575-581
• 全文大小：122 KB
• 参考文献：1. Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H.: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000) CrossRef
  2. Ostrowski, A.M.: On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, II. Arch. Ration. Mech. Anal. 1 233鈥?41 (1958)
  3. Ostrowski, A.M.: On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, II. Arch. Ration. Mech. Anal. 2 423鈥?28 (1958/1959)
  4. Parlett, B.N.: The Rayleigh quotient iteration and some generalizations for nonnormal matrices. Math. Comp. 28, 679鈥?93 (1974) CrossRef
  5. Parlett, B.N.: The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, New Jersey, 1980. SIAM, Philadelphia (1998)
  6. Parlett, B.N., Poole Jr, W.G.: A geometric theory for the QR, LU and power iterations. SIAM J. Numer. Anal. 10, 389鈥?12 (1973) CrossRef
  7. Parlett, B.N., Kahan, W.: On the convergence of a practical QR algorithm. In: Information Processing 68 (Proceedings of IFIP Congress, Edinburgh, 1968), Mathematics, Software, vol. 1, pp. 114鈥?18. North-Holland, Amsterdam
  8. Stewart, G.W.: Matrix Algorithms Volume II: Eigensystems. SIAM, Philadelphia (2001) CrossRef
  9. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)
  10. Wilkinson, J.H.: Global convergence of tridiagonal QR algorithm with origin shifts. Linear Algebra Appl. 1, 409鈥?20 (1968) CrossRef
  
• 作者单位：Kensuke Aishima (1)
  
  1. The University of Tokyo, Hongo7-3-1, Bunkyo-ku, Tokyo, Japan
  
• ISSN：1868-937X

文摘

The Rayleigh quotient iteration is a famous algorithm for solving symmetric eigenvalue problems but suffers a serious limitation: it does not converge in a few peculiar cases. In the present study we show that the Rayleigh quotient iteration always converges when the iterative vector is replaced. The main benefit of our proposed algorithm is that, unlike the existing modification that also guarantees convergence, it admits a direct convergence proof.