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The rich variety of numerical techniques in the literature for tackling the GPEVP has ingredients from linear algebra, partial differential equations, and numerical optimization as well as gradient flows on Riemannian manifolds. We review this heterogeneous body of literature with a focus on a unified treatment of seemingly different approaches, algorithms, and method properties, and we point to open problems and future challenges in the field.<\/jats:p>","DOI":"10.1137\/22m1516324","type":"journal-article","created":{"date-parts":[[2025,5,8]],"date-time":"2025-05-08T03:36:49Z","timestamp":1746675409000},"page":"256-317","source":"Crossref","is-referenced-by-count":12,"title":["The Gross\u2013Pitaevskii Equation and Eigenvector Nonlinearities: Numerical Methods and Algorithms"],"prefix":"10.1137","volume":"67","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6432-5504","authenticated-orcid":false,"given":"Patrick","family":"Henning","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, Ruhr-University Bochum, DE-44801 Bochum, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9443-8772","authenticated-orcid":false,"given":"Elias","family":"Jarlebring","sequence":"additional","affiliation":[{"name":"KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,5,8]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1126\/science.1060182"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/0-8176-4492-X"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.64.063603"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/15M1008622"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1137\/16M1108200"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-021-01216-5"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2022036"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-024-02612-3"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1145\/321296.321305"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1126\/science.269.5221.198"},{"key":"ref11","doi-asserted-by":"crossref","first-page":"729","DOI":"10.4208\/cicp.2008.v4.p729","volume":"4","author":"Antoine X.","year":"2008","journal-title":"Commun. 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