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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 42 ›› Issue (6): 1073-1088.doi: 10.3969/j.issn.1005-3085.2025.06.007cstr: 32411.14.cjem.CN61-1269/O1.2025.06.007

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多参数特征值问题的外推 Jacobi 方法

吴家燕,  陈小山   

  1. 华南师范大学数学科学学院，广州 510631
  • 收稿日期:2023-04-26 接受日期:2023-09-11 出版日期:2025-12-15 发布日期:2026-02-15
  • 基金资助:
    国家自然科学基金 (11771159)；广东省自然科学基金 (2022A1515011123).

Extrapolation Jacobi Methods for Multiparameter Eigenvalue Problems

WU Jiayan,  CHEN Xiaoshan   

  1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631
  • Received:2023-04-26 Accepted:2023-09-11 Online:2025-12-15 Published:2026-02-15
  • Supported by:
    The National Natural Science Foundation of China (11771159); the Natural Science Foundation of Guangdong Province (2022A1515011123).

摘要：

多参数特征值问题来源于典型相关分析理论。典型相关分析理论可用于分析多组变量之间的相关性，是多元统计的重要方法之一，在计量经济学、生物统计学和信号处理等领域具有广泛的应用背景。采用外推Jacobi方法求解两类多参数特征值问题并给出了该方法的收敛性证明。数值例子与经典Jacobi方法、Gauss-Seidel方法和SOR方法等进行了数值比较，验证了外推Jacobi方法的有效性。

关键词: 多参数特征值问题, 对称正定矩阵, 外推Jacobi方法, 矩阵谱半径

Abstract:

The multiparameter eigenvalue problem originates from the theory of canonical correlation analysis, which is used to analyze the correlation between multiple sets of variables. The canonical correlation analysis is one of the important methods of multivariate statistics, and has a wide range of applications in econometrics, biostatistics and signal processing. The extrapolation Jacobi method is used to solve two kinds of multiparameter eigenvalue problems and the convergence of this method is proved. The effectiveness of the extrapolated Jacobi method is illustrated by numerical examples, and it is compared with the classical Jacobi method, Gauss-Seidel method and SOR method.

Key words: multiparameter eigenvalue problem, symmetric positive matrix, extrapolation Jacobi method, spectral matrix norm

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