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	Kahan summation algorithm - Wikipedia, the free encyclopedia
		As the name suggests, this algorithm is attributed to William Kahan.
		So the summation is performed with two accumulators: sum holds the sum, and c accumulates the parts not assimilated into sum, to nudge the low-order part of sum the next time around.
		Thus the summation proceeds with "guard digits" in c which is better than not having any but is not as good as performing the calculations with double the precision of the input.
	en.wikipedia.org /wiki/Kahan_summation_algorithm (863 words)

	William Kahan - Wikipedia, the free encyclopedia
		William Morton Kahan (born June 5, 1933, in Toronto, Ontario, Canada) is a mathematician and computer scientist whose main area of contribution has been numerical analysis.
		Among his many contributions, Kahan was the primary architect behind the IEEE 754 standard for floating-point computation (and its radix-independent follow-on, IEEE 854) and developed the Kahan summation algorithm, an important algorithm for minimizing error introduced when adding a sequence of finite precision floating point numbers.
		Kahan is now a professor of mathematics, computer science, and electrical engineering at the University of California, Berkeley, and continues his contributions to the ongoing revision of IEEE 754.
	en.wikipedia.org /wiki/William_Kahan (431 words)

	[No title]
		a global optimization algorithm in which interval arithmetic is used mathematical institute preprints university of st. andrews, scotland (1987) to appear in: 'proc.
		an algorithm for conclusive investigation of the trajectories of differential systems by means of an interval transformation program (in russian) 'studies in integro-differential equations, no.
		algorithm for simultaneous determination of all roots of algebraic polynomial equations mat.
	www.mat.univie.ac.at /~neum/intlib/autm-q.txt (9837 words)

	William Kahan (Site not responding. Last check: 2007-11-01)
		William Kahan (born June 5, 1933, in Toronto, Alberta, Canada)is an eminent mathematician and computer scientist whose main area of contribution was numerical analysis, the study of accurate and efficient methods of solving numerical problems on a computer with finite precision—a field vitally important in physics and engineering.
		He attended the University of Toronto, where he received his Bachelor's degree in 1954, his Master's degree in 1956, and his Ph.D in 1959, all in the field of mathematics.
		Among his many contributions, Kahan was the primary architect behind the ANSI/IEEE standard for floating-point computation and contributed an important algorithm for minimizing error introduced when adding a sequence of finite precision floating point numbers (see the Kahan Summation Algorithm).
	bopedia.com /en/wikipedia/w/wi/william_kahan.html (183 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Newsgroups: comp.lang.fortran Subject: Re: Kahan's trick (was: Algorithm suggestion?) Date: Wed, 03 Apr 2002 01:55:34 GMT I notice that in most of the discussion of Kahan's summation algorithm people have only been experimenting and not looking at the problem analytically.
		The result from Kahan's algorithm is still exact (though some the partial sums on the way were not even exactly representable and Kahan's algorithm adapted to that).
		The result is: double precision sum is: 67108864.0 single precision sum is: 16777216.0 in error by 75.0% Kahan sum is: 67108864.0 in error by 0.0% The reason for this is that 1(one) is so small compared to 16777216 that you can add it as many times as you want and it doesn't change the answer.
	ftp.cac.psu.edu /pub/ger/fortran/hdk/KahnSum.txt (759 words)

	Eric Fleegal's WebLog : Method for retaining intermediate results in extended precision
		The summation would then be retained in long double precision and narrowed to double precision only when the function returns.
		Keep in mind that it will not work for every algorithm on x86 because any “spilled” registers will still be truncated to 53bit double precision (more info).
		Moreover, many datasets and algorithms may not be sensitive to operand reordering so the speed advantage of the scalar optimized code may be compelling.
	blogs.msdn.com /ericflee/archive/2004/07/21/190741.aspx (950 words)

	BibTeX bibliography kahan-william-m.bib (Site not responding. Last check: 2007-11-01)
		Later, with the introduction of higher-level languages, the computing environment was modified by the language designers who attempted to resolve a three-cornered tug-of-war among implementation efficiency, program portability, and usefulness to the programmer.
		This algorithm is an extension of the well-known recursive doubling technique which computes the sum of n floating-point number in log/sub 2/n parallel steps.
		This algorithm enables a highly accurate result to be obtained with guarantee.
	www.math.utah.edu:8080 /ftp/pub/bibnet/authors/k/kahan-william-m.html (1022 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Kahan, W., Further Remarks on Reducing Truncation Errors, {\it Comm.\ ACM\/} {\bf 8} (1965), 40.
		Extended the analysis of Pichat's algorithm to compute a multi-word representation of the exact sum of n working precision numbers.
		Kahan, W., Paradoxes in Concepts of Accuracy, lecture notes from Joint Seminar on Issues and Directions in Scientific Computation, Berkeley, 1989.
	www.math.psu.edu /local_doc/NTL/quad_float.txt (1370 words)

	[No title]
		The Hong-Sarkar paper Rediscovery of time memory tradeoffs advertises an algorithm by Hellman that, after a huge precomputation, inverts a function with a 128-bit input using about 2^86 function evaluations on a serial computer with 2^93 bits of fast memory.
		Parallel versions of the complicated algorithms are better than serial versions, and there are some situations where they save time, but they are still worse than brute-force search in both of these situations.
		Algorithm 3.40: Computing x-coordinates of multiples of a point on the curve y^2+xy=x^3+Ax^2+b over a field of 2^m elements.
	cr.yp.to.mirror.dogmap.org /2005-590.html (4021 words)

	Floating Point
		Kahan: the mirror for the Hubble space telescope was ground with great precision, but to the wrong specification.
		An algorithm is numerically stable if the output of the algorithm changes by only a small amount when the input data changes by a small amount.
		The obvious algorithm is to add up all the stock prices after Instead a "clever" analyst decided it would be more efficient to recompute the index by adding the net change of a stock after each trade.
	www.cs.princeton.edu /introcs/91float (6370 words)

	Textbook on Computer Arithmetic
		This viewpoint is reflected, e.g., in the detailed coverage of redundant number representations and associated arithmetic algorithms (Chapter 3) that later lead to a better understanding of various multiplier designs and on-line arithmetic.
		Part IV covers division algorithms and their hardware implementations, beginning with the basic shift-subtract algorithms and moving on to high-radix, pre-scaled, modular, array, and convergence dividers.
		Computer Arithmetic: Algorithms and Hardware Designs, is an outgrowth of lecture notes that the author has used for the graduate course “ECE 252B: Computer Arithmetic” at the University of California, Santa Barbara, and, in rudimentary forms, at several other institutions prior to 1988.
	www.ece.ucsb.edu /Faculty/Parhami/text_comp_arit.htm (5832 words)

	CmpE 360 - Spring 2004
		Sum the numbers in the order in which they were generated, this time using a single-precision accumulator.
		compensated summation algorithm (due to Kahan), again using only single precision, to sum the numbers in the order in which they were generated:
		Sum the numbers in order of increasing magnitude (this will require that the numbers be sorted before summing, for which you may use a library sorting routine).
	www.cmpe.boun.edu.tr /courses/cmpe360/spring2004/HW1.php (231 words)

	Java Number Cruncher: The Java Programmer's Guide to Numerical Computing - $43.99 (Site not responding. Last check: 2007-11-01)
		It is not a textbook on numerical methods or numerical analysis, although it certainly shows how to program many key numerical algorithms in Java.
		We'll examine these algorithms, enough to get a feel for how they work and why they're useful, without formally proving why they work.
		Because Java is ideal for doing so, we'll also demonstrate many of the algorithms with interactive, graphical programs.
	www.informit.com /bookstore/product.asp?isbn=0130460419 (1139 words)

	The Accuracy Of Floating Point Summation - Higham (ResearchIndex)
		The usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers.
		Five summation methods and their variations are analysed here.
		2 Summation algorithm with corrections and some of its applica..
	citeseer.ist.psu.edu /higham93accuracy.html (836 words)

	Eric Fleegal's WebLog
		With carefully coded algorithms, single precision can yield very accurate results; however, most users (even most computer scientists) are not trained to devise such algorithms for all but the simplest cases.
		True, William Kahan and other numerical VIPs have long criticized Microsoft for the lack of 80bit long doubles.
		However influential these people may be in their industry, it’s not they but our customers who largely determine which features get incorporated and which features get postponed.
	blogs.msdn.com /ericflee (5154 words)

	docs.sun.com: What Every Scientist Should Know About Floating-Point Arithmetic (Site not responding. Last check: 2007-11-01)
		The third part discusses the Kahan summation formula, which was used as an example in Section, "Systems Aspects," on page 37.
		Proof The algorithm for addition with k guard digits is similar to that for subtraction.
		error bound for the Kahan summation formula (Theorem 8) is not as good as using double precision, even though it is much better than single precision.
	docs.sun.com /app/docs/doc/800-7895/6hos0aoub?l=ja&a=view (3164 words)

	Citations: Parallel Algorithms for the Rounding-Exact Summation of Floating-Point Numbers - Leuprecht (ResearchIndex) (Site not responding. Last check: 2007-11-01)
		Kahan states that these algorithms appear to have average run times of order at least n log n.
		If the floating point arithmetic is faithful, the following algorithm computes an expansion y = P m i=1 y i with m n such that y = P x i.
		Then the following algorithm carried out with faithful arithmetic computes a t digit expansion y = P m i=1 y i with m n such that y = P x i.
	citeseer.ist.psu.edu /context/367184/0 (702 words)

	Java Number Cruncher : The Java Programmer's Guide to Numerical Computing by Ronald Mak - 0130460419
		An authority on mapping pure math to computer math, he explains how to use the often-overlooked computational features of Java, and does so in a clear, non-theoretical style.
		Without getting lost in mathematical detail, you'll learn practical numerical algorithms for safely summing numbers, finding roots of equations, interpolation and approximation, numerical integration, solving differential equations, matrix operations, and solving sets of simultaneous equations.
		You'll also enjoy intriguing topics such as searching for patterns in prime numbers, generating random numbers, computing thousands of digits of pi, and creating intricately beautiful fractal images.
	www.allbookstores.com /book/0130460419 (301 words)

	Citations: On accurate floating-point summation - Malcolm (ResearchIndex) (Site not responding. Last check: 2007-11-01)
		Classes of Integrands The experiments carried out are intended to demonstrate that summation algorithms other than recursive summation can provide more accurate integral approximations, and to examine the additional e ort needed to....
		presents an algorithm analogous to our Algorithm 3, where he breaks up each f bit number s i into q words each with f q nonzero bits, sums the resulting q n words into f bit accumulators exactly, and then sums the accumulators in (roughly) decreasing order.
		Modifications of Wolf s algorithm were presented in 1965 by Ross [2] and Kahan [8] Kahan also presented the compensated summation method, which manipulates only one accumulator.
	citeseer.ist.psu.edu /context/512964/0 (723 words)

	Stuff: January 2004 Archives (Site not responding. Last check: 2007-11-01)
		Enter the clever people at Tegic and their T9 algorithm, now a part of AOL.
		With the T9 algorithm, all you do is press the key with the letter you want once.
		I think pretty much all cellphones available now have this algorithm; you just might have to turn it on if it's not on already.
	weblog.nabeelazar.com /archive/2004_01.php (942 words)

	[No title]
		This algorithm first calculates the vector __ \ n_star_est = (1/n) > (p[i] - y) * length(p[i] - y))**(d + 1) /_ i=1...n where, as usual, n is the number of points and d the number of dimensions.
		Bentley's single-fragment algorithm is faster than any other algorithm in the average case, but in the worst case it degrades to worse than Prim's using a Fibonacci heap.
		Prim's algorithm was chosen for implementation because it has the best worst-case behavior of any of the algorithms.
	www.stanford.edu /~blp/uniformity/uniformity-2001.04.27.text (9609 words)

	Design, Implementation and Testing of Extended and Mixed Precision BLAS (ResearchIndex) (Site not responding. Last check: 2007-11-01)
		Abstract: This paper describes the design rationale, a C implementation, and conformance testing of a subset of the new Standard for the BLAS (Basic Linear Algebra Subroutines): Extended and Mixed Precision BLAS.
		Permitting higher internal precision and mixed input/output types and precisions permits us to implement some algorithms that are simpler, more accurate, and sometimes faster than possible without these features.
		12 Application of a new algorithm for the symmetric eigenproble..
	citeseer.ifi.unizh.ch /376177.html (825 words)

	European Conference on Computer Algebra (EUROCAL)
		Algorithmic determination of the Jacobson radical of monomial algebras
		Algorithms for the Character Theory of the Symmetric Group
		An algorithm for constructing detaching bases in the ring of polynominals over a field
	wotan.liu.edu /docis/dbl/eurcal/index.html (1647 words)

	Higher precision maths - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-11-01)
		As an alternative to sorting, you may want to look at Kahan's summation algorithm (formula).
		Tellingly, all 6 implementations of the dotproduct yield every so slightly different results (depending on sort-order, Kahan summation, etc) but the variation is of a neglegible order now.
		I will rewrite several instances of code to comply with the improved summations given by the Kahan algorithm.
	www.gamedev.net /community/forums/topic.asp?topic_id=330625 (1447 words)

	[No title]
		In particular, global summation and dot products of distributed arrays are very susceptible to rounding errors.
		We analyzed several accurate summation methods and found that two methods are particularly effective to improve (ensure) reproducibility: Kahan's self-compensated summation and Bailey's double-double precision summation.
		We provide an MPI operator MPI\_SUMDD to work with MPI collective operations to ensure a scalable implementation on large number of processors.
	www.netlib.org /bibnet/authors/k/kahan-william-m.bib (1086 words)

	Science Chat - Math Num Analysis (Site not responding. Last check: 2007-11-01)
		I'm looking for an algorithm for a pentadiagonal solver which I can implement as a preconditioner to a large non-symmetric dense conjugate gradient squared routine, I have noticed that the solution time decreases as I go from vanilla...
		I need FFT delphi algorithm for signals which size is not power of 2.
		i have an estimation algorithm now: beta_n=g(alpha_n,data) alpha_n+1=h(alpha_n,beta_n,data) (g,h are known functions) this scheme...
	www.science-chat.org /articles-4347-301-cat1.html (3438 words)

	Math Forum Discussions
		>> Actually this is not a correct algorithm, just an approximation.
		The errors come in due to the general problem of summation of
		>> method of summation, this reduces the error to about 7 digits.
	mathforum.org /kb/thread.jspa?threadID=1343636&messageID=4537763 (331 words)

	Learn more about Kahan summation algorithm in the online encyclopedia. (Site not responding. Last check: 2007-11-01)
		Learn more about Kahan summation algorithm in the online encyclopedia.
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	www.onlineencyclopedia.org /k/ka/kahan_summation_algorithm.html (140 words)

	Stuff: Sum more stuff (Site not responding. Last check: 2007-11-01)
		This has always been one of my favorite algorithms.
		It makes you realize that even the simplest computations require more planning and work than you might expect.
		This algorithm isn't foolproof (you can find counter-examples where this code fails), but that's true for pretty much any mathematical algorithm.
	weblog.nabeelazar.com /archive/000002.php (507 words)

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