Lower eigenvalue bounds for pencils of matrices

A procedure is set up for obtaining lower eigenvalue bounds for pencils of matrices A—vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of locally perturbed factorization iterative schemes. Using these results and a formerly developed approach for estimating upper bounds, we widely confirm Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations. In so doing, we however keep local perturbations, thereby enlarging the number of applications where their sufficiency is proven; their necessity remains, on the other hand, an open question.