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非线性特征值问题平移对称幂法的收敛率估计

唐耀宗1,2, 杨庆之1,2   

  1. 1. 喀什大学数学与统计学院, 喀什 844000;
    2. 南开大学数学科学学院, 天津 300071
  • 收稿日期:2020-07-24 出版日期:2021-11-14 发布日期:2021-11-12
  • 基金资助:
    国家自然科学基金项目(12071234,11671217);新疆维吾尔自治区自然科学基金面上项目(2018D01A01)资助.

唐耀宗, 杨庆之. 非线性特征值问题平移对称幂法的收敛率估计[J]. 计算数学, 2021, 43(4): 529-538.

Tang Yaozong, Yang Qingzhi. CONVERGENCE RATE ESTIMATION ON SS-HOPM FOR NONLINEAR EIGENVALUE PROBLEMS[J]. Mathematica Numerica Sinica, 2021, 43(4): 529-538.

CONVERGENCE RATE ESTIMATION ON SS-HOPM FOR NONLINEAR EIGENVALUE PROBLEMS

Tang Yaozong1,2, Yang Qingzhi1,2   

  1. 1. School of Mathematics and Statistics, Kashi University, Kashi 844000, China;
    2. School of Mathematical Sciences, Nankai University, Tianjin 300071, China
  • Received:2020-07-24 Online:2021-11-14 Published:2021-11-12
平移对称幂法(SS-HOPM)在求解源自玻色-爱因斯坦凝聚态的非线性特征值问题时,不仅具有较高的计算效率,而且具有点列收敛性,但其收敛率尚未得到有效估计.本文通过将多项式Kurdyka-Łojasiewicz(K-Ł)指数界的相关结果应用到所涉及优化问题的Lagrange函数上,得到了平移对称幂法的次线性收敛率估计,从理论上解释了平移对称幂法的计算效率.
In solving the nonlinear eigenvalue problems originated from Bose-Einstein Condensation, the shifted symmetric higher-order power method (SS-HOPM for short) not only has high computational efficiency, but also has point-wise convergence. However, the convergence rate of SS-HOPM has not been given. We apply the bound of the Kurdyka-Lojasiewicz (K-L) exponent of polynomial to the Lagrange function of the optimization problem involved in this paper, then we obtain sublinear convergence rate of the SS-HOPM, which can explain the calculation efficiency of the algorithm theoretically.

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