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Author Malcolm, Michael A.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Language English
Subject Keyword Error analysis ♦ Floating-point summation
Abstract cumulation of floating-point sums is considered on a computer which performs t-digit base β floating-point addition with exponents in the range —m to M. An algorithm is given for accurately summing n t-digit floating-point numbers. Each of these n numbers is split into q parts, forming q·n t-digit floating-point numbers. Each of these is then added to the appropriate one of &eegr; auxiliary t-digit accumulators. Finally, the accumulators are added together to yield the computed sum. In all, q·n + &eegr; - 1 t-digit floating-point additions are performed. Let &ngr; = ⌈(M + m + 1)/(&eegr; + 1)⌉. If n ≤ (1/q)β⌈((q-1)/q)t⌈-&ngr;+1 (*), then the relative error in the computed sum is at most ⌈(t + 1)/&ngr;⌉β1-t. Further, with an additional q + &eegr; - 1 t-digit additions, the computed sum can be corrected to full t-digit accuracy.For example, for the IBM/360 (β = 16, t = 14, M = 63, m = 64), typical values for q and &eegr; are q = 2 and &eegr; = 32. In this case, (*) becomes n ≤ 1/2 × 164 = 32,768, and we have ⌈(t + 1)/&ngr;⌉β1-t = 4 × 16-13.
Description Affiliation: Stanford Univ., CA (Malcolm, Michael A.)
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2005-08-01
Publisher Place New York
Journal Communications of the ACM (CACM)
Volume Number 14
Issue Number 11
Page Count 6
Starting Page 731
Ending Page 736


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Source: ACM Digital Library