{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T19:59:58Z","timestamp":1793217598383,"version":"4.5.0"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>\n                    We present and analyze several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator. Let f and F be the number of significant bits in the summands and the accumulator, respectively. Then assuming gradual underflow, no overflow, and round-to-nearest arithmetic, up to approximately 2\n                    <jats:sup>F-f<\/jats:sup>\n                    numbers can be added accurately by simply summing the terms in decreasing order of exponents, yielding a sum correct to within about 1.5 units in the last place (ulps). We apply this result to the floating point formats in the IEEE floating point standard. For example, a dot product of single precision vectors of length at most 33 computed using double precision and sorting is guaranteed correct to nearly 1.5 ulps. If double-extended precision is used, the vector length can be as large as 65,537. We also investigate how the cost of sorting can be reduced or eliminated while retaining accuracy.\n                  <\/jats:p>","DOI":"10.1137\/s1064827502407627","type":"journal-article","created":{"date-parts":[[2003,12,5]],"date-time":"2003-12-05T21:00:47Z","timestamp":1070658047000},"page":"1214-1248","source":"Crossref","is-referenced-by-count":62,"title":["Accurate and Efficient Floating Point Summation"],"prefix":"10.1137","volume":"25","author":[{"given":"James","family":"Demmel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yozo","family":"Hida","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,5,16]]},"reference":[{"key":"R1","unstructured":"ANSI\/IEEE,\n                      IEEE Standard for Binary Floating Point Arithmetic\n                      , Std 754\u20101985 edition, IEEE, New York, 1985."},{"key":"R2","unstructured":"ANSI\/IEEE,\n                      IEEE Standard for Radix Independent Floating Point Arithmetic\n                      , Std 854\u20101987 edition, IEEE, New York, 1987."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1109\/TC.1977.1674894"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF01397083"},{"key":"R5","unstructured":"J. Demmel and Y. Hida,\n                      Fast and accurate floating point summation with application to computational geometry\n                      , in Proceedings of the 10th GAMM\u2010IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2002), submitted."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/0914050"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718027"},{"key":"R8","unstructured":"IEEE,\n                      Proposed Revision of the IEEE Floating Point Arithmetic Standard754\n                      , IEEE, New York, 2001."},{"key":"R8","unstructured":"http:\/\/grouper.ieee.org\/groups\/754."},{"key":"R9","unstructured":"Intel Corporation,\n                      Intel Itanium Architecture Software Developer\u2019s Manual, Volume1\n                      , Intel Corporation, Santa Clara, CA, 2002."},{"key":"R9","unstructured":"http:\/\/developer.intel.com\/design\/itanium\/manuals. htm."},{"key":"R10","unstructured":"Intel Corporation,\n                      Intel Architecture Software Developer\u2019s Manual, Volume1, Basic Architecture\n                      , Intel Corporation, Santa Clara, CA, 2002."},{"key":"R10","unstructured":"http:\/\/developer.intel.com\/design\/ pentium\/manuals.htm."},{"key":"R11","unstructured":"W. Kahan,\n                      Doubled\u2010precision IEEE Standard754 Floating Point Arithmetic\n                      , manuscript, 1987."},{"key":"R12","volume-title":"The art of computer programming. Vol. 2: Seminumerical algorithms","author":"Knuth Donald","year":"1969"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02280883"},{"key":"R14","volume-title":"Computer arithmetic in theory and practice","author":"Kulisch Ulrich","year":"1981"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02241816"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1145\/355958.355960"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1145\/362854.362889"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01975722"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01404922"},{"key":"R20","unstructured":"D. Priest,\n                      Algorithms for arbitrary precision floating point arithmetic\n                      , in Proceedings of the 10th Symposium on Computer Arithmetic, Grenoble, France, June 26\u201328, 1991, p. Kornerup and D. Matula, eds., IEEE, New York, pp. 132\u2013145."},{"key":"R21","unstructured":"D. Priest,\n                      On Properties of Floating Point Arithmetics: Numerical Stability and the Cost of Accurate Computations\n                      , Ph.D. thesis, University of California at Berkeley, 1992."},{"key":"R21","unstructured":"ftp.icsi. berkeley.edu\/pub\/theory\/priest\u2010thesis.ps. Z."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1145\/363707.363711"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/BF01581144"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009321"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1145\/512274.512287"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827502407627","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:54:16Z","timestamp":1787334856000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827502407627"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":29,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1137\/S1064827502407627"],"URL":"https:\/\/doi.org\/10.1137\/s1064827502407627","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,1]]}}}