Orthogonal Eigenvectors and Relative Gaps
Inderjit S. Dhillon, Beresford Ν. Parlett · SIAM Journal on Matrix Analysis and Applications · 2003
This paper presents and analyzes a new algorithm for computing eigenvectors of symmetric tridiagonal matrices factored as LDL t , 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 LDL t 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 × 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 × n symmetric tridiagonal at O(n 2 ) cost, a great improvement over existing algorithms.