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A simple stability analysis of the Lanczos algorithm in finite precision arithmetic
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.
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BibTeX
@misc{chen2026simple,
title = {{A simple stability analysis of the Lanczos algorithm in finite precision arithmetic}},
author = {Tyler Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.21268},
eprint = {2608.21268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21268},
}
RIS
TY - UNPB
TI - A simple stability analysis of the Lanczos algorithm in finite precision arithmetic
AU - Tyler Chen
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2608.21268
UR - https://arxiv.org/abs/2608.21268
AB - 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.
ER -