{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:32Z","timestamp":1787333732955,"version":"build-2736575974"},"reference-count":11,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p>Assuming standard floating-point arithmetic (in base $\\beta$, precision $p$) and barring underflow and overflow, classical rounding error analysis of the LU or Cholesky factorization of an $n\\times n$ matrix $A$ provides backward error bounds of the form $|\\Delta A| \\le \\gamma_n |\\widehat L| |\\widehat U|$ or $|\\Delta A| \\le \\gamma_{n+1} |\\widehat R^T| |\\widehat R|$. Here, $\\widehat L$, $\\widehat U$, and $\\widehat R$ denote the computed factors, and $\\gamma_n$ is the usual fraction $nu\/(1-nu) = nu + {\\mathcal O}(u^2)$ with $u$ the unit roundoff. Similarly, when solving an $n\\times n$ triangular system $Tx = b$ by substitution, the computed solution $\\widehat x$ satisfies $(T+\\Delta T)\\widehat x = b$ with $|\\Delta T| \\le \\gamma_n |T|$. All these error bounds contain quadratic terms in $u$ and limit $n$ to satisfy either $nu&lt;1$ or $(n+1)u &lt; 1$. We show in this paper that the constants $\\gamma_n$ and $\\gamma_{n+1}$ can be replaced by $nu$ and $(n+1)u$, respectively, and that the restrictions on $n$ can be removed. To get these new bounds the main ingredient is a general framework for bounding expressions of the form $|\\rho-s|$, where $s$ is the exact sum of a floating-point number and $n-1$ real numbers and where $\\rho$ is a real number approximating the computed sum $\\widehat s$. By instantiating this framework with suitable values of $\\rho$, we obtain improved versions of the well-known Lemma 8.4 from [N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002] (used for analyzing triangular system solving and LU factorization) and of its Cholesky variant. All our results hold for rounding to nearest with any tie-breaking strategy and whatever the order of summation.<\/jats:p>","DOI":"10.1137\/130927231","type":"journal-article","created":{"date-parts":[[2014,5,27]],"date-time":"2014-05-27T12:07:46Z","timestamp":1401192466000},"page":"684-698","source":"Crossref","is-referenced-by-count":20,"title":["Improved Backward Error Bounds for LU and Cholesky Factorizations"],"prefix":"10.1137","volume":"35","author":[{"given":"Siegfried M.","family":"Rump","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Claude-Pierre","family":"Jeannerod","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,5,27]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/1037130"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1145\/227699.227701"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"N. J. Higham,\n                      Accuracy and Stability of Numerical Algorithms\n                      , 2nd ed., SIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898718027"},{"key":"atypb4","unstructured":"IEEE Standard 754-2008:\n                      IEEE Standard for Floating-Point Arithmetic\n                      , http:\/\/ieeexplore. ieee.org\/servlet\/opac?punumber$=$4610933 (2008)."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/120894488"},{"key":"atypb6","unstructured":"W. Kahan,\n                      A brief tutorial on gradual underflow\n                      , http:\/\/www.cs.berkeley.edu\/$\\sim$wkahan (2005)."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-011-0342-4"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/050645671"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009321"},{"key":"atypb10","unstructured":"P. H. Sterbenz,\n                      Floating-Point Computation\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1974."},{"key":"atypb11","first-page":"354","volume":"10","author":"Wilkinson J. H.","year":"1974","journal-title":"IMA Bull."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130927231","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:19:58Z","timestamp":1787332798000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130927231"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,1]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["10.1137\/130927231"],"URL":"https:\/\/doi.org\/10.1137\/130927231","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,1]]}}}