A simple stability analysis of the Lanczos algorithm in finite precision arithmetic

We give a self-contained finite-precision analysis of the symmetric Lanczos algorithm without reorthogonalization. In particular, we derive the perturbed three-term recurrence, Paige's loss-of-orthogonality identity, containment of all computed Ritz values, and localization of stabilized Ritz values. We then prove a Greenbaum-type backward stability result, exhibiting a nearby problem on which exact Lanczos produces the computed tridiagonal matrix. Our proofs simplify those of Paige and Greenbaum, at the cost of hiding polynomial factors in the iteration count.

Source: arXiv:2608.21268v1 - http://arxiv.org/abs/2608.21268v1 PDF: https://arxiv.org/pdf/2608.21268v1 Original Link: http://arxiv.org/abs/2608.21268v1