{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T10:31:27Z","timestamp":1774261887802,"version":"3.50.1"},"reference-count":30,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2005,7,8]],"date-time":"2005-07-08T00:00:00Z","timestamp":1120780800000},"content-version":"vor","delay-in-days":3782,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Overdetermined linear systems often arise in applications such as signal processing and modern communication. When the overdetermined system of linear equations <jats:italic>AX \u2248\ufe01 B<\/jats:italic> has no solution, compatibility may be restored by an orthogonal projection method. The idea is to determine an orthogonal projection matrix <jats:italic>P<\/jats:italic> by some method M such that [<jats:italic>\u00c3 B\u0303<\/jats:italic>] = <jats:italic>P<\/jats:italic>[<jats:italic>A B<\/jats:italic>], and <jats:italic>\u00c3X = B\u0303<\/jats:italic> is compatible. Denote by <jats:italic>X<\/jats:italic><jats:sub><jats:italic>M<\/jats:italic><\/jats:sub> the minimum norm solution to <jats:italic>\u00c3X = B\u0303<\/jats:italic> using method M. In this paper conditions for compatibility of the lower rank approximation and subspace properties of \u00c3 in relation to the nearest rank\u2010<jats:italic>k<\/jats:italic> matrix to <jats:italic>A<\/jats:italic> are discussed. We find upper and lower bounds for the difference between the solution <jats:italic>X<\/jats:italic><jats:sub><jats:italic>M<\/jats:italic><\/jats:sub> and the SVD\u2010based total least squares (TLS) solution <jats:italic>X<\/jats:italic><jats:sub><jats:italic>SVD<\/jats:italic><\/jats:sub> and also provide a perturbation result for the ordinary TLS method. These results suggest a new algorithm for computing a total least squares solution based on a rank revealing QR factorization and subspace refinement. Numerical simulations are included to illustrate the conclusions.<\/jats:p>","DOI":"10.1002\/nla.1680020206","type":"journal-article","created":{"date-parts":[[2005,11,1]],"date-time":"2005-11-01T17:52:14Z","timestamp":1130867534000},"page":"135-153","source":"Crossref","is-referenced-by-count":9,"title":["Orthogonal projection and total least squares"],"prefix":"10.1002","volume":"2","author":[{"given":"Ricardo D.","family":"Fierro","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James R.","family":"Bunch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2005,7,8]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1684(89)90039-X"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01397471"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01436075"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90103-0"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1137\/0913043"},{"key":"e_1_2_1_7_2","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1137\/0911029","article-title":"Computing truncated singular value decomposition least squares solutions by rank revealing QR factorizations","volume":"11","author":"Chan T. 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