Eigenvalues of the real generalized eigenvalue equation perturbed by a low-rank perturbation

The low-rank perturbation (LRP) method solves the perturbed eigenvalue equation (B +V) _k = _k (C +P) _k , where the eigenvalues and the eigenstates of the related unperturbed eigenvalue equationB _i = _i C _i are known. The method is designed for arbitraryn-by-n matricesB, V, C, andP, with the only restriction that the eigenstates _i of the unperturbed equation should form a complete set. We consider here a real LRP problem where all matrices are Hermitian, and where in addition matricesC and (C +P) are positive definite. These conditions guarantee reality of the eigenvalues _k and _i . In the original formulation of the LRP method, each eigenvalue _k is obtained iteratively, starting from some approximate eigenvalue _k . If this approximate eigenvalue is not well chosen, the iteration may sometimes diverge. It is shown that in the case of a real LRP problem, this danger can be completely eliminated. If the rank of the generalized perturbation {V, P} is small with respect ton, then one can easily bracket and hence locate to any desirable accuracy the eigenvalues _k (k = 1, ...,n) of the perturbed equation. The calculation of alln eigenvalues requiresO(² n ²) operations. In addition, if the perturbation (V, P) is local with the localizabilityl p, then onlyO(² n) operations are required for a derivation of a single eigenvalue.