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By examining the problem from this perspective, and thoroughly analyzing both the Hutchinson and the Lanczos components of the algorithm, we obtain error bounds that allow us to determine the number of Hutchinson\u2019s sample vectors and Lanczos iterations needed to ensure the detection of all gaps above the target width with high probability. In particular, we conclude that the most efficient strategy is to always use a single random sample vector for Hutchinson\u2019s estimator and concentrate all computational effort in the Lanczos algorithm. Our numerical experiments demonstrate the efficiency and reliability of this approach.\n                  <\/jats:p>","DOI":"10.1007\/s00211-026-01532-8","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T07:26:20Z","timestamp":1772090780000},"page":"847-888","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Estimation of spectral gaps for sparse symmetric matrices"],"prefix":"10.1007","volume":"158","author":[{"given":"Michele","family":"Benzi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michele","family":"Rinelli","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Igor","family":"Simunec","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,26]]},"reference":[{"key":"1532_CR1","doi-asserted-by":"publisher","unstructured":"Lin, L., Lu, J., Ying, L.: Numerical methods for Kohn\u2013Sham density functional theory. 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