A fully parallel method for the singular eigenvalue problem

Abstract

In this paper, a fully parallel method for finding some or all finite eigenvalues of a real symmetric matrix pencil ( A, B) is presented, where A is a symmetric tridiagonal matrix and B is a diagonal matrix with b 1 > 0 and b i ≥ 0, i = 2,3,…, n. The method is based on the homotopy continuation with rank 2 perturbation. It is shown that there are exactly m disjoint, smooth homotopy paths connecting the trivial eigenvalues to the desired eigenvalues, where m is the number of finite eigenvalues of ( A, B). It is also shown that the homotopy curves are monotonic and easy to follow.