{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:20:42Z","timestamp":1787329242136,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>An important change of variables for a linear time varying system $\\dot x=A(t)x, t\\ge 0$, is that induced by the QR-factorization of the underlying fundamental matrix solution: $X=QR$, with Q orthogonal and R upper triangular (with positive diagonal). To find this change of variable, one needs to solve a nonlinear matrix differential equation for Q. Practically, this means finding a numerical approximation to Q by using some appropriate discretization scheme, whereby one attempts to control the local error during the integration. Our contribution in this work is to obtain global error bounds for the numerically computed Q. These bounds depend on the local error tolerance used to integrate for Q, and on structural properties of the problem itself, but not on the length of the interval over which we integrate. This is particularly important, since\u2014in principle\u2014Q may need to be found on the half-line $t\\ge 0$.<\/jats:p>","DOI":"10.1137\/06067818x","type":"journal-article","created":{"date-parts":[[2008,3,7]],"date-time":"2008-03-07T18:06:03Z","timestamp":1204913163000},"page":"1166-1189","source":"Crossref","is-referenced-by-count":17,"title":["On the Error in QR Integration"],"prefix":"10.1137","volume":"46","author":[{"given":"Luca","family":"Dieci","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Erik S.","family":"Van Vleck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,3,7]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"L. Ya. Adrianova,\n                      Introduction to Linear Systems of Differential Equations\n                      , Transl. Math. Monogr. 146, AMS, Providence, RI, 1995.","DOI":"10.1090\/mmono\/146"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-97-00902-2"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901385143"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005447"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479896297720"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(83)90098-0"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s002119900083"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901392304"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"L. Dieci and E. S. Van Vleck,\n                      Lyapunov and Sacker-Sell spectral intervals\n                      , J. Dynam. 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Van Vleck,\n                      Exponential Dichotomy for Asymptotically Hyperbolic Two-Dimensional Linear Systems\n                      , 2007, submitted."},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(86)90109-9"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(79)90076-7"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(82)90090-0"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(82)90098-5"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/0714030"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/06067818X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:14Z","timestamp":1787325194000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/06067818X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/06067818X"],"URL":"https:\/\/doi.org\/10.1137\/06067818x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}