Year: 2026
Type: preprint
Abstract: We give a self-contained analysis of the symmetric Lanczos algorithm without reorthogonalization when computations are performed inexactly. We show the computed Lanczos vectors are well-behaved locally. From this we (i) prove Paige's loss-of-orthogonality identity, and (ii) construct a Greenbaum-type backward stability result, exhibiting a nearby problem on which exact Lanczos produces the computed tridiagonal matrix. This allows us to derive several facts about the computed Ritz values at the original eigenvalues. Our analysis simplifies the analyses of Paige and Greenbaum, at the cost of hiding polynomial factors in the iteration count.
Source: arXiv (Cornell University)
Author: Chen, Tyler
Institution: New York University Shanghai
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204079892
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