Minimization Principles for the Linear Response Eigenvalue Problem III: General Case
نویسندگان
چکیده
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 response eigenvalue problem for the 2n×2n matrix [ 0 K M 0 ] 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 the 2n×2n matrix pencil [ 0 K M 0 ] −λ [ E+ 0 0 E− ] is considered, where K and M are as before and E± are n×n nonsingular real matrices such that E + = 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. 2000 Mathematics Subject Classification. Primary 65L15. Secondary 15A18, 81Q15
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