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This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.<\/jats:p>","DOI":"10.1007\/s00454-023-00620-y","type":"journal-article","created":{"date-parts":[[2024,1,24]],"date-time":"2024-01-24T15:07:01Z","timestamp":1706108821000},"page":"442-466","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Cone of $$5\\times 5$$ Completely Positive Matrices"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7739-4031","authenticated-orcid":false,"given":"Max","family":"Pfeffer","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Jos\u00e9 Alejandro","family":"Samper","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,1,24]]},"reference":[{"key":"620_CR1","doi-asserted-by":"crossref","unstructured":"Afonin, A., Hildebrand, R., Dickinson, P.J.C.: The extreme rays of the $$6\\times 6$$ copositive cone. 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