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Math. Softw."],"published-print":{"date-parts":[[2009,2]]},"abstract":"<jats:p>\n                    We present the algorithm, error bounds, and numerical results for extra-precise iterative refinement applied to overdetermined linear least squares (LLS) problems. We apply our linear system refinement algorithm to Bj\u00f6rck\u2019s augmented linear system formulation of an LLS problem. Our algorithm reduces the forward normwise and componentwise errors to\n                    <jats:italic toggle=\"yes\">O<\/jats:italic>\n                    (\n                    <jats:italic toggle=\"yes\">\u03b5<\/jats:italic>\n                    <jats:sub>w<\/jats:sub>\n                    ), where\n                    <jats:italic toggle=\"yes\">\u03b5<\/jats:italic>\n                    <jats:sub>w<\/jats:sub>\n                    is the working precision, unless the system is too ill conditioned. In contrast to linear systems, we provide two separate error bounds for the solution\n                    <jats:italic toggle=\"yes\">x<\/jats:italic>\n                    and the residual\n                    <jats:italic toggle=\"yes\">r<\/jats:italic>\n                    . The refinement algorithm requires only limited use of extra precision and adds only\n                    <jats:italic toggle=\"yes\">O<\/jats:italic>\n                    (\n                    <jats:italic toggle=\"yes\">mn<\/jats:italic>\n                    ) work to the\n                    <jats:italic toggle=\"yes\">O<\/jats:italic>\n                    (\n                    <jats:italic toggle=\"yes\">mn<\/jats:italic>\n                    <jats:sup>2<\/jats:sup>\n                    ) cost of QR factorization for problems of size\n                    <jats:italic toggle=\"yes\">m<\/jats:italic>\n                    -by-\n                    <jats:italic toggle=\"yes\">n<\/jats:italic>\n                    . The extra precision calculation is facilitated by the new extended-precision BLAS standard in a portable way, and the refinement algorithm will be included in a future release of LAPACK and can be extended to the other types of least squares problems.\n                  <\/jats:p>","DOI":"10.1145\/1462173.1462177","type":"journal-article","created":{"date-parts":[[2009,2,17]],"date-time":"2009-02-17T08:18:18Z","timestamp":1234858698000},"page":"1-32","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":31,"title":["Extra-Precise Iterative Refinement for Overdetermined Least Squares Problems"],"prefix":"10.1145","volume":"35","author":[{"given":"James","family":"Demmel","sequence":"first","affiliation":[{"name":"University of California, Berkeley"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yozo","family":"Hida","sequence":"additional","affiliation":[{"name":"University of California, Berkeley"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"E. Jason","family":"Riedy","sequence":"additional","affiliation":[{"name":"University of California, Berkeley"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoye S.","family":"Li","sequence":"additional","affiliation":[{"name":"Lawrence Berkeley National Laboratory"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2009,2]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/050643088"},{"key":"e_1_2_1_2_1","unstructured":"Bailey D. 2009. A Fortran-90 double-double precision library. http:\/\/crd.lbl.gov\/~dhbailey\/mpdist."},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01939321"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.5555\/117521.117527"},{"key":"e_1_2_1_5_1","volume-title":"Numerical Methods for Least Squares Problems","author":"Bj\u00f6rck","unstructured":"Bj\u00f6rck, \u00c5. 1996. Numerical Methods for Least Squares Problems. SIAM, Philadelphia, PA."},{"key":"e_1_2_1_6_1","first-page":"322","article-title":"Iterative refinement of linear least squares solution by Householder transformation (Algol programming)","volume":"7","author":"Bj\u00f6rck","year":"1967","unstructured":"Bj\u00f6rck, \u00c5. and Golub, G. 1967. Iterative refinement of linear least squares solution by Householder transformation (Algol programming). BIT 7, 322--337.","journal-title":"BIT"},{"key":"e_1_2_1_7_1","doi-asserted-by":"crossref","unstructured":"Cucker F. Diao H. and Wei Y. 2007. On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems. Math. Comput. 76 258 947--963.","DOI":"10.1090\/S0025-5718-06-01913-2"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/1141885.1141894"},{"key":"e_1_2_1_9_1","unstructured":"Forsythe G. E. and Moler C. B. 1967. Computer Solution of Linear Algebraic Systems. 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