Improved Calculation of Eigenvector Sensitivities Using Matrix Perturbation Analysis

Abstract

Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. Based on the concept of matrix perturbation, this paper presents a new method for the improved calculation of eigenvector derivatives in the case where only few of the lower modes of a system under study have been computed. By using this new proposed method, considerable improvement on the accuracy of the estimation of eigenvector derivatives can be achieved at the expense of very tiny extra computational effort since only few matrix vector operations are required. Convergency criterion of the method has been established and the required accuracy can be controlled by including more higher order terms. Numerical results from practical finite element model have demonstrated the practicality of the proposed method. Further, the proposed method can be easily incorporated into commercial finite element packages to improve the accuracy of eigenderivatives needed for practical applications.