An incomplete modal method for eigenvector derivatives of polynomial eigenvalue problems

Abstract

Eigenvector derivatives of eigenproblems analytically dependent on parameters are required in diverse technical fields such as structural design, system identification. Modal methods, a kind of popular methods in engineering problems, calculate eigenvector derivatives by expressing them as linear combinations of eigenvectors of eigenproblems. In this paper, an incomplete modal method is presented to compute eigenvector derivatives of polynomial eigenproblems analytically dependent on parameters. The proposed method only uses a small part of eigenvectors of polynomial eigenproblems. The contributions of other eigenvectors to eigenvector derivatives are approximated by an iterative formula. Our method can simultaneously compute the eigenvector derivatives corresponding to several eigenvalues, and it is effective whether the differentiated eigenvectors correspond to simple or semisimple eigenvalues, and whether the leading coefficient matrix is singular or not. Numerical examples are given to test the efficiency of the proposed method.