The Laguerre Iteration in Solving the Symmetric Tridiagonal Eigenproblem, Revisited

This paper presents an algorithm for the eigenvalue problem of symmetric tridiagonal matrices. The algorithm employs the determinant evaluation, split-and-merge strategy, and the Laguerre iteration. The method directly evaluates eigenvalues and uses inverse iteration as an option when eigenvectors are needed. This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection, and Cuppen’s divide-and-conquer method. It is fully parallel and competitive in speed with the most efficient QR algorithm in serial mode. On the other hand, the algorithm is as accurate as any standard algorithm for the symmetric tridiagonal eigenproblem and enjoys the flexibility in evaluating partial spectrum.