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.