{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:43:20Z","timestamp":1753886600593,"version":"3.41.2"},"reference-count":19,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2009,7,22]],"date-time":"2009-07-22T00:00:00Z","timestamp":1248220800000},"content-version":"vor","delay-in-days":202,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>The minimization of a quadratic function within an ellipsoidal trust region is an important\nsubproblem for many nonlinear programming algorithms. When the number of variables is large,\none of the most widely used strategies is to project the original problem into a small dimensional\nsubspace. In this paper, we introduce an algorithm for solving nonlinear least squares problems. \nThis algorithm is based on constructing a basis for the Krylov subspace in conjunction with a\nmodel trust region technique to choose the step. The computational step on the small dimensional\nsubspace lies inside the trust region. The Krylov subspace is terminated such that the termination\ncondition allows the gradient to be decreased on it. A convergence theory of this algorithm is\npresented. It is shown that this algorithm is globally convergent.<\/jats:p>","DOI":"10.1155\/2009\/435851","type":"journal-article","created":{"date-parts":[[2009,7,22]],"date-time":"2009-07-22T15:00:07Z","timestamp":1248274807000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Newton\u2010Krylov Type Algorithm for Solving Nonlinear Least Squares Problems"],"prefix":"10.1155","volume":"2009","author":[{"given":"Mohammedi R.","family":"Abdel-Aziz","sequence":"first","affiliation":[]},{"given":"Mahmoud M.","family":"El-Alem","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2009,7,22]]},"reference":[{"key":"e_1_2_8_1_2","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/19\/2\/201"},{"volume-title":"Iterative Solution of Nonlinear Equations in Several Variables","year":"1970","author":"Ortega J. 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Minimization of a large scale quadratic function subject to an ellipsoidal constraint 1994 no. TR94-27 Department of Computational and Applied Mathematics Rice University Houston Tex USA."},{"key":"e_1_2_8_16_2","doi-asserted-by":"publisher","DOI":"10.1137\/0719026"},{"key":"e_1_2_8_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00933149"},{"key":"e_1_2_8_18_2","doi-asserted-by":"publisher","DOI":"10.2307\/2007504"},{"key":"e_1_2_8_19_2","doi-asserted-by":"publisher","DOI":"10.1137\/0906042"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2009\/435851.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2009\/435851.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2009\/435851","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,22]],"date-time":"2024-06-22T14:54:14Z","timestamp":1719068054000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2009\/435851"}},"subtitle":[],"editor":[{"given":"Irena","family":"Lasiecka","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1155\/2009\/435851"],"URL":"https:\/\/doi.org\/10.1155\/2009\/435851","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2009,1]]},"assertion":[{"value":"2008-12-15","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-02-02","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-07-22","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"435851"}}