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In this paper, we presented a new analysis framework to show how the inexact inner products influence the Krylov subspace methods that are based on the Arnoldi process. A new metric was developed to quantify the inexactness of the Arnoldi process with inexact inner products. In addition, the proposed metric can be used to approximately estimate the loss of orthogonality in the practical use of the Arnoldi process. The discrepancy in residual gaps between Krylov subspace methods employing inexact inner products and their corresponding exact counterparts was discussed. Numerical experiments on several examples were reported to illustrate our theoretical findings and final observations were presented.&lt;\/p&gt;<\/jats:p>","DOI":"10.3934\/nhm.2025002","type":"journal-article","created":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T10:42:27Z","timestamp":1736160147000},"page":"15-34","source":"Crossref","is-referenced-by-count":2,"title":["Theory of Krylov subspace methods based on the Arnoldi process with inexact inner products"],"prefix":"10.3934","volume":"20","author":[{"given":"Meng","family":"Su","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chun","family":"Wen","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhao-Li","family":"Shen","sequence":"additional","affiliation":[{"name":"College of Science, Sichuan Agricultural University, Ya'an, Sichuan 625000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefano","family":"Serra-Capizzano","sequence":"additional","affiliation":[{"name":"Department of Science and High Technology, University of Insubria, Como Campus 22100, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2025002-1","doi-asserted-by":"crossref","unstructured":"Z. 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