Orthogonal eigenvectors and relative gaps

This paper presents and analyzes a new algorithm for computing eigenvectors of symmetric tridiagonal matrices factored as LDLt, with D diagonal and L unit bidiagonal. If an eigenpair is well behaved in a certain sense with respect to the factorization, the algorithm is shown to compute an approximate eigenvector which is accurate to working precision. As a consequence, all the eigenvectors computed by the algorithm come out numerically orthogonal to each other without making use of any reorthogonalization process. The key is first running a qd-type algorithm on the factored matrix LDLt and then applying a fine-tuned version of inverse iteration especially adapted to this situation. Since the computational cost is O(n) per eigenvector for an n x n matrix, the proposed algorithm is the central step of a more ambitious algorithm which, at best (i.e., when all eigenvectors are well-conditioned), would compute all eigenvectors of an n x n symmetric tridiagonal at O(n(2)) cost, a great improvement over existing algorithms.