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*To*: stds-754@xxxxxxxx*Subject*: [stds-754] Re: GAMM-IMACS Proposal for Accurate Floating-Point Vector Arithmetic*From*: "Nelson H. F. Beebe" <beebe@xxxxxxxxxxxxx>*Date*: Tue, 22 Nov 2005 17:45:08 -0700 (MST)*Cc*: beebe@xxxxxxxxxxxxx, Ulrich Kulisch <ae35@xxxxxxxxxxxxxxxxxxxx>*Sender*: stds-754@xxxxxxxx

In mail to this list on Thu, 20 Oct 2005 10:30:12 +0200, Ulrich Kulisch <ae35@xxxxxxxxxxxxxxxxxxxx> kindly forwarded the GAMM-IMACS Proposal for Accurate Floating-Point Vector Arithmetic. One implementation of this proposal would use internal accumulators wide enough to hold numbers over the entire exponent range, requiring widths of hundreds to tens of thousands of bits (for the 32-bit, 64-bit, 80-bit, and 128-bit formats). For several decades, researchers in floating-point arithmetic have investigated ways to improve the accuracy of floating-point summation and dot product in purely-software implementations, and there are two new articles cited below that add to this knowledge. The fparith.bib file at http://www.math.utah.edu/pub/tex/bib/index-table-f.html#fparith includes them, and now marks all such articles with the special string "accurate floating-point summation" in keywords values, making it easy to find them. The first article below in particular avoids sorting, avoids branches, and avoids higher intermediate precision, producing several different summation algorithms that, it appears to me, largely solve the problem that the GAMM-IMACS proposal addresses. The article's authors also note the great utility of hardware fused-multiply-add instructions for use in the core of their algorithms, providing further evidence of their desirability. If this article does indeed solve the problem, then further investigations are still merited to compare the efficacy of a wide-accumulator solution (in software or hardware) with the techniques of this article, possibly augmented with hardware assists. That in turn can guide the committee in deciding whether to support the proposal in the emerging revised IEEE 754 standard. @String{j-SIAM-J-SCI-COMP = "SIAM Journal on Scientific Computing"} @Article{Ogita:2005:ASD, author = "Takeshi Ogita and Siegfried M. Rump and Shin'ichi Oishi", title = "Accurate Sum and Dot Product", journal = j-SIAM-J-SCI-COMP, volume = "26", number = "6", pages = "1955--1988", month = nov, year = "2005", CODEN = "SJOCE3", DOI = "10.1137/030601818", ISSN = "1064-8275 (print), 1095-7197 (electronic)", MRclass = "5-04, 65G99, 65-04", bibdate = "Mon Nov 21 14:52:48 MST 2005", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SISC/26/6";, URL = "http://epubs.siam.org/sam-bin/dbq/article/60181";, abstract = "Algorithms for summation and dot product of floating-point numbers are presented which are fast in terms of measured computing time. We show that the computed results are as accurate as if computed in twice or K-fold working precision, $K\ge 3$. For twice the working precision our algorithms for summation and dot product are some 40\% faster than the corresponding XBLAS routines while sharing similar error estimates. Our algorithms are widely applicable because they require only addition, subtraction, and multiplication of floating-point numbers in the same working precision as the given data. Higher precision is unnecessary, algorithms are straight loops without branch, and no access to mantissa or exponent is necessary.", acknowledgement = ack-nhfb, keywords = "accurate dot product; accurate floating-point summation; fast algorithms; high precision; verified error bounds", } @Article{Zhu:2005:NDA, author = "Yong-Kang Zhu and Jun-Hai Yong and Guo-Qin Zheng", title = "A New Distillation Algorithm for Floating-Point Summation", journal = j-SIAM-J-SCI-COMP, volume = "26", number = "6", pages = "2066--2078", month = nov, year = "2005", CODEN = "SJOCE3", DOI = "10.1137/030602009", ISSN = "1064-8275 (print), 1095-7197 (electronic)", MRclass = "65G05, 65B10", bibdate = "Mon Nov 21 14:52:48 MST 2005", bibsource = "http://epubs.siam.org/sam-bin/dbq/toc/SISC/26/6";, URL = "http://epubs.siam.org/sam-bin/dbq/article/60200";, abstract = "The summation of $n$ floating-point numbers is ubiquitous in numerical computations. We present a new distillation algorithm for floating-point summation which is stable, efficient, and accurate. The algorithm iteratively ``distills'' the summands without discarding any significant digit until the partial sums cannot change the whole sum. It uses standard floating-point arithmetic and does not rely on the choice of radix or any other specific assumption. Furthermore, the error bound of our algorithm is independent of $n$ and less than 1 ulp.", acknowledgement = ack-nhfb, keywords = "accurate floating-point summation; distillation; rounding error", } ------------------------------------------------------------------------------- - Nelson H. F. Beebe Tel: +1 801 581 5254 - - University of Utah FAX: +1 801 581 4148 - - Department of Mathematics, 110 LCB Internet e-mail: beebe@xxxxxxxxxxxxx - - 155 S 1400 E RM 233 beebe@xxxxxxx beebe@xxxxxxxxxxxx - - Salt Lake City, UT 84112-0090, USA URL: http://www.math.utah.edu/~beebe - -------------------------------------------------------------------------------

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