A preconditioned conjugate gradient method for computing eigenvector derivatives with distinct and repeated eigenvalues

A preconditioned conjugate gradient method is proposed for computing eigenvector derivatives with distinct and repeated eigenvalues in the real symmetric eigensystems. In view of singular character of the coefficient matrices of the governing equations for particular solutions of eigenvector derivatives, a modified governing equation for the complementary part of the computed modal contribution excluding those of the repeated modes is introduced, and its coefficient matrix is symmetric and positive definite. The existing factored (shifted) stiffness matrix from an iterative eigensolution such as Lanczos or Subspace Iteration is then utilized as preconditioner. High accurate approximations to particular solutions of eigenvector derivatives can be provided with a few iterations. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner, and can be integrated into a coupled eigensolver/derivative software module. It is especially suitable for the large sparse matrices that arise in industrial-size finite element models. Finally, two numerical examples are used to demonstrate the superior efficiency and fast convergence of the present method.