Calculation of eigenpair derivatives for symmetric quadratic eigenvalue problem with repeated eigenvalues
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Calculation of eigenpair derivatives for symmetric quadratic eigenvalue problem with repeated eigenvalues Pingxin Wang · Hua Dai
Received: 19 May 2014 / Revised: 28 June 2014 / Accepted: 11 July 2014 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2014
Abstract In this paper, we consider computing the derivatives of the semisimple eigenvalues and corresponding eigenvectors of symmetric quadratic eigenvalue problem. In the proposed method, the eigenvector derivatives of the symmetric quadratic eigenvalue problem are divided into a particular solution and a homogeneous solution; a simplified method is given to calculate the particular solution by solving a linear system with nonsingular coefficient matrix, the method is numerically stable and efficient. Two numerical examples are included to illustrate the validity of the proposed method. Keywords Sensitivity analysis · Quadratic eigenvalue problem · Eigenvalue derivatives · Eigenvector derivatives Mathematics Subject Classification
15A18 · 65F10
Communicated by Jinyun Yuan. This work was supported by Defense Basic Research Program of China (No. J152013xxx) and National Natural Science Foundation of China (No. 11071118). P. Wang · H. Dai (B) College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China e-mail: [email protected] P. Wang e-mail: [email protected] P. Wang School of Mathematics and Physics, Jiangsu University of Science and Technology, Nanjing 212003, Zhengjiang, People’s Republic of China
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P. Wang, H. Dai
1 Introduction Sensitivity analysis of eigenvalue problems plays an important role in such fields as structural optimization (Haug et al. 1986; Adhikari 2013a), finite model updating (Friswell and Mottershead 1995), structural damage detection (Messina et al. 1998) and system identification (Adhikari 2013b). The derivatives of eigenvalues and eigenvectors, which characterize the tendency of variation for frequencies and mode shapes with respect to design parameters, are widely used in many applications. Eigensensitivity analysis has received much attention over the past four decades. A number of methods for the eigenpair derivatives have been developed. However, there are two main difficulties residing in computing the eigenvector derivatives. One of the main difficulties in computation of eigenvector derivatives is the singularity issue, and another difficulty is the derivatives of eigenvectors corresponding to the repeated eigenvalues. The various methods for computing derivatives of eigenvectors for standard or generalized eigenvalue problem can be divided into three categories: modal method, Nelson’s method and the algebraic method. The modal method is first derived by Fox and Kapoor (1968). The derivatives of eigensolutions for symmetric generalized eigenvalue problem are expanded in terms of the complete eigenvectors in the modal method. Many authors have extended Fox and Kapoor’s approach to compute eigenpair derivatives for nonsymmetric matrices or nonsy
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