On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems

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• 作者：Tsung-Ming Huang (1)
  Zhongxiao Jia (2)
  Wen-Wei Lin (3)
  
• 关键词：Rayleigh ; Ritz method ; Ritz value ; Ritz vector ; Refined Ritz vector ; Convergence ; 15A18 ; 65F15 ; 65F50
• 刊名：BIT Numerical Mathematics
• 出版年：2013
• 出版时间：December 2013
• 年：2013
• 卷：53
• 期：4
• 页码：941-958
• 全文大小：
• 作者单位：Tsung-Ming Huang (1)
  Zhongxiao Jia (2)
  Wen-Wei Lin (3)
  
  1. Department of Mathematics, National Taiwan Normal University, Taipei, 116, Taiwan
  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
  3. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan
  
• ISSN：1572-9125

文摘

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.