Eigenvalue and eigenvector derivatives of nonlinear eigenproblems

Abstract

Introduction T HE derivatives of eigenvalues and eigenvectors with respect to design parameters are useful, if not essential, for design sensitivity and structural optimization studies. The computational problem of the derivatives of eigenvalues and eigenvectors of linear eigenvalue problems has been treated adequately in the past.' However, recent work on the nonlinear case has shown that the computational problem of the eigenvalue and eigenvector derivatives of nonlinear eigenproblems with respect to design parameters remains a dilemma to be studied. Though this problem has been addressed in Ref. 5 and the six theorems of the derivatives of eigenvalues and eigenvectors of nonlinear eigenproblems have been derived, the first four theorems are not correct and the assumptions stated there are not often satisfied, as pointed out in Ref. 6. In this Note, the explicit formulas for the first-order derivatives of distinct eigenvalues are derived for self-adjoint nonlinear eigenproblems and the three computational methods for the derivatives of eigenvectors proposed.