Minimization Principles for the Linear Response Eigenvalue Problem, III: General Case
Zhaojun Bai, Ren‐Cang Li · 2013
Previously a minimization principle for the sum of the first few smallest eigenvalues with the positive sign and Cauchy-like interlacing inequalities [ ] for the standard linear 0 K response eigenvalue problem for the 2n×2n matrix were obtained by the authors, where K and M are n×n real symmetric positive semi-definite matrices and one of them is definite. In this [ paper, a] more [ general linear] response eigenvalue problem for 0 K E+ 0 the 2n × 2n matrix pencil − λ is considered, where K and M are M 0 0 E− as before and E ± are n × n nonsingular real matrices such that E T + = E−. In a similar way as we have done for the standard linear response eigenvalue problem, we develop minimization principles and Cauchy-like interlacing inequalities. Subsequently, we investigate the best eigenvalue approximations through a structure-preserving subspace projection and conjugate gradient-like algorithms for simultaneously computing the first few smallest eigenvalues with the positive sign and their associated eigenvectors. Finally, we present some numerical results to illustrate essential convergence behaviors of the proposed conjugate gradient-like methods.