可对角化矩阵特征值分解扰动问题的快速求解方法  

A Fast Computation Method for Eigenvalue Decomposition Perturbation Problems of Diagonalizable Matrix

作　　者：胡志祥[1] 杨其东 黄潇 贺文宇 HU Zhixiang;YANG Qidong;HUANG Xiao;HE Wenyu(College of Civil Engineering,Hefei University of Technology,Hefei 230009,China;College of Civil Engineering,Anhui Jianzhu University,Hefei 230601,China)

机构地区：[1]合肥工业大学土木与水利工程学院,安徽合肥230009 [2]安徽建筑大学土木工程学院,安徽合肥230601

出　　处：《湖南大学学报（自然科学版）》2024年第7期119-126,共8页Journal of Hunan University:Natural Sciences

基　　金：国家自然科学基金资助项目(52178283,52378298);安徽省杰出青年基金项目(2208085J20)。

摘　　要：针对特征值扰动计算的传统方法收敛速度慢的问题,提出了一种求解特征值扰动问题的快速迭代算法.首先,通过矩阵变换将初始矩阵的特征值扰动问题转化为对角矩阵的特征值扰动问题.然后,提出了一种快速迭代算法求解扰动参数,同时对算法的收敛性进行分析,并将其与基于摄动级数展开法导出的方法进行对比.再次,采用逐一求解特征值并进行矩阵降阶的策略,有效降低运算量.最后,通过2个算例分别展示算法的计算过程及其在结构模态参数追踪方面的应用效果.A fast iterative algorithm for solving the eigenvalue perturbation problem is proposed in this paper for solving the problem of slow convergence of traditional methods for eigenvalue perturbation calculation.Firstly,the eigenvalue perturbation problem of the initial matrix is transformed into the eigenvalue perturbation problem of the diagonal matrix by matrix transformation.Then,a fast iterative algorithm is proposed to solve the perturbation parameter.The convergence of the algorithm is analyzed and compared with the method derived based on the perturbation series expansion method,and the strategy of solving the eigenvalues one by one and reducing the order of the matrix is adopted to effectively reduce the computation cost.Finally,two examples are used to show the calculation process of the algorithm and its application in the tracking of modal parameters of vibration structures.

关 键 词：特征值分解 特征值扰动 摄动级数展开法 可对角化矩阵 收敛性分析 

分 类 号：O324[理学—一般力学与力学基础]