{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T19:18:00Z","timestamp":1790709480965,"version":"4.1.0"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2003,1]]},"abstract":"<jats:p>\n                    We consider the least squares problem with a quadratic equality constraint (LSQE), i.e., minimizing |Ax-b|\n                    <jats:sub>2<\/jats:sub>\n                    subject to $\\|x\\|_2=\\alpha$, without the assumption $\\|A^\\dagger b\\|_2&gt;\\alpha$ which is commonly imposed in the literature. Structure and perturbation analysis are given to demonstrate the sensitivity of the LSQE problem. We present a projection method combined with correction techniques (PMCT) for solving numerically the LSQE problem when the LSQE problem is ill-conditioned. We also give a detailed convergence analysis of our algorithms to illustrate the convergence behavior. Our algorithms have some obvious advantages over Newton's method and variants. Numerical experiments indicate that PMCT is much more efficient than Newton's methods when the LSQE problem is ill-conditioned; PMCT has a 90% success rate in terms of convergence, while commonly used Newton-type iterations almost always fail.\n                  <\/jats:p>","DOI":"10.1137\/s0895479801398712","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"188-212","source":"Crossref","is-referenced-by-count":27,"title":["A Projection Method for Least Squares Problems with a Quadratic Equality Constraint"],"prefix":"10.1137","volume":"25","author":[{"given":"Zhenyue","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yushan","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)00092-1"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0909061"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01403888"},{"key":"R4","first-page":"238","volume":"31","author":"Bj\u00f6rck \u00c5.","year":"1991","journal-title":"BIT"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971484"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0142040"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177731317"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01932285"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0717028"},{"key":"R10","unstructured":"L. Eld\u00e9n and H. Park,\n                      A Procrustes Problem on the Stiefel Manifold\n                      , Technical Report LiTH\u2010MAT\u2010R\u201097\u20106, Department of Mathematics, Link\u00f6ping University, Sweden, 1995."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396656"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90494-1"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01436075"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/1015032"},{"key":"R15","volume-title":"Matrix computations","author":"Golub Gene","year":"1996"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385796"},{"key":"R17","unstructured":"M. Gulliksson,\n                      Algorithms for the Partial Procrustes Problems\n                      , Report UMINF\u201095, ISSN\u20100348\u20100542, Department of Computing Science, Ume\u00e5 University, Sweden, 1995."},{"key":"R18","first-page":"253","volume":"3","author":"Hanke Martin","year":"1993","journal-title":"Surveys Math. Indust."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF02149761"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/0905037"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971217"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144597321909"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/0716012"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1093\/comjnl\/13.3.309"},{"key":"R25","unstructured":"C. R. Rao and S. K. Mitra,\n                      Generalized Inverse of Matrices and Its Applications\n                      , John Wiley, New York, 1971."},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1007\/BF02162161"},{"key":"R27","unstructured":"M. Rojas and Sorensen,\n                      A Trust\u2010Region Approach to the Regularization of Large\u2010Scale Discrete Ill\u2010Posed Problems\n                      , Technical Report 99\u201026, Department of Computational and Applied Mathematics, Rice University, Houston, 1999."},{"key":"R28","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear Systems\n                      , 2nd ed., SIAM, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"R29","volume-title":"Matrix perturbation theory","author":"Stewart G.","year":"1990"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479801398712","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:41:04Z","timestamp":1787334064000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479801398712"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["10.1137\/S0895479801398712"],"URL":"https:\/\/doi.org\/10.1137\/s0895479801398712","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,1]]}}}