{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T23:11:47Z","timestamp":1769555507200,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,22]],"date-time":"2024-05-22T00:00:00Z","timestamp":1716336000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004706","name":"Zhejiang Normal University Postdoctoral Research Fund","doi-asserted-by":"publisher","award":["ZC304022938"],"award-info":[{"award-number":["ZC304022938"]}],"id":[{"id":"10.13039\/501100004706","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004706","name":"Zhejiang Normal University Postdoctoral Research Fund","doi-asserted-by":"publisher","award":["61976196"],"award-info":[{"award-number":["61976196"]}],"id":[{"id":"10.13039\/501100004706","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004706","name":"Zhejiang Normal University Postdoctoral Research Fund","doi-asserted-by":"publisher","award":["LZ22F030003"],"award-info":[{"award-number":["LZ22F030003"]}],"id":[{"id":"10.13039\/501100004706","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of China","award":["ZC304022938"],"award-info":[{"award-number":["ZC304022938"]}]},{"name":"Natural Science Foundation of China","award":["61976196"],"award-info":[{"award-number":["61976196"]}]},{"name":"Natural Science Foundation of China","award":["LZ22F030003"],"award-info":[{"award-number":["LZ22F030003"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["ZC304022938"],"award-info":[{"award-number":["ZC304022938"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["61976196"],"award-info":[{"award-number":["61976196"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LZ22F030003"],"award-info":[{"award-number":["LZ22F030003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The conditioning theory of the ML-weighted least squares and ML-weighted pseudoinverse problems is explored in this article. We begin by introducing three types of condition numbers for the ML-weighted pseudoinverse problem: normwise, mixed, and componentwise, along with their explicit expressions. Utilizing the derivative of the ML-weighted pseudoinverse problem, we then provide explicit condition number expressions for the solution of the ML-weighted least squares problem. To ensure reliable estimation of these condition numbers, we employ the small-sample statistical condition estimation method for all three algorithms. The article concludes with numerical examples that highlight the results obtained.<\/jats:p>","DOI":"10.3390\/axioms13060345","type":"journal-article","created":{"date-parts":[[2024,5,22]],"date-time":"2024-05-22T11:03:47Z","timestamp":1716375827000},"page":"345","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Conditioning Theory for ML-Weighted Pseudoinverse and ML-Weighted Least Squares Problem"],"prefix":"10.3390","volume":"13","author":[{"given":"Mahvish","family":"Samar","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinzhong","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6704-0301","authenticated-orcid":false,"given":"Huiying","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,22]]},"reference":[{"key":"ref_1","unstructured":"Nashed, M.Z. 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