Homotopy Method for the Singular Symmetric Tridiagonal Eigenproblem

Abstract

In this paper, a homotopy algorithm for finding some or all finite eigenvalues and corresponding eigenvectors of a real symmetric matrix pencil $(A,B)$ is presented, where $A$ is a symmetric tridiagonal matrix and $B$ is a diagonal matrix with $b_i \ge 0$, $i = 1, 2, \ldots , n$. It is shown that there are exactly $m$ ($m$ is the number of finite eigenvalues of $(A,B)$) disjoint, smooth homotopy paths connecting the trivial eigenpairs to the desired eigenpairs. And the eigenvalue curves are monotonic and easy to follow. The performance of the parallel version of our algorithm is presented.