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Topic: Rounding error

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	[No title]
		It may arise in several ways: from errors in measuring physical quantities, from errors in storing the data on the computer (rounding errors), or, if the data is itself the solution to another problem, it may be the result of errors in an earlier computation.
		The effects of errors in the data are generally easier to understand than the effects of rounding errors committed during a computation, because data errors can be analyzed using perturbation theory for the problem at hand, while intermediate rounding errors require an analysis specific to the given method.
		Note that the numbers in Table \ref{Army} are consistent with a relative error of $2^{-20}$ in the computer's representation of $0.1$, this constant being used to convert from the system's clock tenths of a second to a second ($2^{-20}$ is the relative error introduced by chopping $0.1$ to $23$ bits after the binary point).
	www.cs.unc.edu /~smp/COMP205/LECTURES/ERROR/lec23/TMP_lec23 (2183 words)

	Article - "Measurement Error Due to Rounding"
		Rounding error is the difference between the true value and the nearest reportable value.
		Rounding is not the only source of measurement error or variation.
		Since the measurement variation and rounding variation are almost equal, rounding is the largest source of measurement variation.
	www.variation.com /techlib/vr-1.html (467 words)

	Rounding Error (Site not responding. Last check: 2007-10-10)
		Rounding (roundoff) error is a phenomenon of digital computing resulting from the computer's inability to represent some numbers exactly.
		Rounding error is a natural consequence of the representation scheme used for integers and floating-point numbers in digital computers.
		Proper handling of rounding error may involve a combination of approaches such as use of high-precision data types and revised calculations and algorithms.
	cricket.cs.drexel.edu /~introcs/Fa03/extras/Rounding (2364 words)

	Method and apparatus for reducing rounding error when evaluating binary floating point polynomials - Patent 5646876
		The final rounding cannot in general give a worst case error smaller than 1/2 ulp of the result, where an ulp (unit in last place) of any quantity is the value of the least significant bit of the floating point representation of that quantity.
		The sum of the remaining terms would already have a rounding error which in general was no less than 1/2 ulp of the sum of the remaining terms, so the total error could be as large or larger than 1/2 ulp of the result plus 1/2 ulp of the sum of the remaining terms.
		Even without this source of error, the extra rounding error, defined as the maximum rounding error introduced prior to the last floating operation, could be as large as 1/2 ulp of the sum of the remaining terms for this example.
	www.freepatentsonline.com /5646876.html (12779 words)

	Rounding Error Test
		Still, a solution must be found, so the browsers sometimes 'round off' the internal calculations, in order to arrive at a simple pixel value for box dimensions.
		However, for complex reasons, the rounding off might occur in the 'wrong' direction, and can result in a visible 1 pixel gap (or sometimes a 1 pixel overlap), next to those boxes.
		If rounding errors occur, they will appear as thin gaps running horizontally and vertically thru the big box.
	www.positioniseverything.net /round-error.html (424 words)

	What Every Computer Scientist Should Know About Floating-Point Arithmetic
		The first section, Rounding Error, discusses the implications of using different rounding strategies for the basic operations of addition, subtraction, multiplication and division.
		The error is 0.5 ulps, the relative error is 0.
		is 2 because a rounding error of.5 ulp wobbles by a factor of
	docs.sun.com /source/806-3568/ncg_goldberg.html (8200 words)

	QOF: Numeric: Rational Number Handling w/ Rounding Error Control
		This combination of output denominator and rounding policy allows the results of financial and other rational computations to be properly rounded to the appropriate units.
		Round to the nearest integer, rounding toward zero when there are two equidistant nearest integers.
		Round to the nearest integer, rounding away from zero when there are two equidistant nearest integers.
	code.neil.williamsleesmill.me.uk /doxygen/group__Numeric.html (2565 words)

	ULPs, Relative Error, and Machine Epsilon (Site not responding. Last check: 2007-10-10)
		The abbreviation ULP represents the measure "unit in the last place." Another measure of the rounding error uses the relative error, which is the difference between the exact number and its approximation divided by the exact number.
		The relative error that corresponds to 1/2 ULP is bounded by:
		, the machine epsilon, is commonly used in discussions of rounding errors because it expresses the smallest floating-point number that you can add to 1.0 with a result that does not round to 1.0.
	h18009.www1.hp.com /fortran/docs/vf-html/pg/pg1ulp.htm (158 words)

	NumerAnalysis: Embarrassments due to rounding error
		Robert Skeel, "Roundoff Error Cripples Patriot Missile," SIAM News, July 1992.
		Rounding error changes Parliament makeup, Debora Weber-Wulff, The Risks Digest, Volume 13, Issue 37, 1992.
		The printout was to two figures, and the actual percentage was rounded to 5.0%.
	www.eng.mu.edu /corlissg/146.2001/noteOn_roundoff.html (478 words)

	Numbers, Expressions, and Rounding Error
		One annoying aspect of your error checking is the number of times the alert box is displayed for the user when the form has multiple errors.
		After finishing the Rounding Error analysis, you should return to your Forms Error Checking activity from earlier in the lab.
		Rounding error proved to be the bane of early computer users, precisely because of the inexactness of arithmetic with floating-point numbers.
	www.mcs.drexel.edu /~introcs/Fa04/labs/L7-conds_FP/GroupLab.html (2113 words)

	kbAlertz: (107462) - Spreadsheet formulas and functions in Microsoft Works for Windows may return incorrect results in ... (Site not responding. Last check: 2007-10-10)
		This error can be corrected up to 13 significant digits (14 would not correct the error in all cases) using the ROUND function.
		For each formula or function that is returning an incorrect result, place the ROUND function in the same cell with the original formula or function as the "x" parameter and a number from 0 to 13 as the "NumberOfPlaces" parameter to indicate the number of significant digits after the decimal desired.
		These errors occur when all of the bits in a binary number cannot be used in a calculation (computers use a binary number format to store numbers internally).
	kbalertz.com /kb_107462.aspx (529 words)

	[Prime] Re: rounding error (Site not responding. Last check: 2007-10-10)
		However, prime95 can miss detecting a hardware error and then your final result will be bad.
		If you've had more than a 5 errors there is a good chance that one was missed.
		It could be overheating, in which case cleaning the fans and heatsink and checking the air flow may solve your problem.
	www.hogranch.com /pipermail/prime/2004-June/000061.html (248 words)

	Embarrassments due to rounding error
		I would appreciate any interesting anecdotes on real-life embarrassments due to rounding error.
		The rounded calculation gave a value of 1098.892.
		The Greens appeared to have a cliff-hanging 5.0%, until it was discovered (after the results had been announced) that they really had only 4.97%.
	www.ece.utah.edu /~ece6340/LECTURE/links/Embarrassments%20due%20to%20rounding%20error.htm (430 words)

	PRB: Rounding Error Casting Double to Long (12297) (Site not responding. Last check: 2007-10-10)
		The error is caused by differences in arithmetic precision.
		When rounding is the preferred behavior, the application can add or subtract 0.5 as appropriate so truncation provides the correct result.
		Because the error is present only in the least significant part of the number, errors occur only when a calculation loses precision in an intermediate value.
	www.sunsite.org.uk /sites/ftp.microsoft.com/MISC/KB/en-us/12/297.HTM (375 words)

	A P P E N D I X D - What Every Computer Scientist Should Know About Floating-Point Arithmetic
		Since rounding error is inherent in floating-point computation, it is important to have a way to measure this error.
		In statements like Theorem 3 that discuss the relative error of an expression, it is understood that the expression is computed using floating-point arithmetic.
		All the operations mentioned in the standard are required to be exactly rounded except conversion between decimal and binary.
	docs.sun.com /source/819-0499/ncg_goldberg.html (7967 words)

	Mathematics -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-10)
		attempts to get round the usual criteria for publication of (additional info and facts about mathematical paper) mathematical papers in a (additional info and facts about learned journal) learned journal after (additional info and facts about peer review) peer review, with assumptions of bias;
		The case of (additional info and facts about Kurt Heegner) Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work.
		And like (The branch of physics that studies celestial bodies and the universe as a whole) astronomy, mathematics owes much to amateur contributors such as (French mathematician who founded number theory; contributed (with Pascal) to the theory of probability (1601-1665)) Fermat and (additional info and facts about Mersenne) Mersenne.
	www.absoluteastronomy.com /encyclopedia/m/ma/mathematics.htm (3164 words)

	Rounding Error vs. Truncation Error (Site not responding. Last check: 2007-10-10)
		This module demonstrates the tradeoff between rounding error and truncation error when solving a mathematical problem using finite-precision, floating-point arithmetic.
		This module demonstrates the tradeoff between rounding error and truncation error in computing the change in the surface area of the Earth if its radius changes by a given amount.
		For a small change, the exact formula suffers substantial rounding error, whereas the approximate formula is very accurate.
	www.cse.uiuc.edu /eot/modules/floating_point/rounding_truncation (257 words)

	Sangrae Lee Abstract (Site not responding. Last check: 2007-10-10)
		Customary survey quantile point estimators and inference methods have been based on the implicit assumption that recorded measurements have rounding errors that are negligible.
		The first diagnostic evaluates the approximate mean squared error of a customary unadjusted survey quantile estimator, averaged over the rounding interval.
		The second diagnostic compares this averaged mean squared error with the approximate mean squared error of a quantile estimator that would have been obtained in an idealized setting that has no rounding error.
	web.utk.edu /~wparr/leeabstract.html (210 words)

	Opera JavaScript Flaw--Rounding Error (Site not responding. Last check: 2007-10-10)
		According to ECMA-262, if the fractional portion of a number is less than 0.5, the number is to be rounded down to the next lower integer.
		If the fractional portion is 0.5 or greater, the number is rounded up to the next higher integer.
		In Opera, however, any number with a fractional portion greater than zero is rounded up to the next higher integer.
	home.hiwaay.net /~taylorc/www/user-agents/opera/javascript-bugs/rounding.html (232 words)

	Rounding error? - HERO GAMES Discussion Boards
		I tried setting the "Number of decimal places for rounding" (in file->preferences) to 1, 2 and 5 - but the result was always the same.
		This should presumably round to 2, but HD rounds it to 1.
		This is the way the Hero System rounding rules are setup and this is the way that HD works.
	www.herogames.com /forums/showthread.php?t=12199 (944 words)

	The Patriot Missile Failure (Site not responding. Last check: 2007-10-10)
		Specifically, the time in tenths of second as measured by the system's internal clock was multiplied by 1/10 to produce the time in seconds.
		The small chopping error, when multiplied by the large number giving the time in tenths of a second, led to a significant error.
		Indeed, the Patriot battery had been up around 100 hours, and an easy calculation shows that the resulting time error due to the magnified chopping error was about 0.34 seconds.
	www.ima.umn.edu /~arnold/disasters/patriot.html (549 words)

	possible rounding error in STR END calculations - HERO GAMES Discussion Boards
		I am not sure if this is an error or just a quirk of the rounding rules.
		So your base STR rounds to 1 END cost and the Power rounds to 2 END Cost which are then added for 3 END Cost.
		Every purchase is rounded separately, in accordance with the rules of the system.
	www.herogames.com /forums/showthread.php?t=13470 (611 words)

	Rounding Error (Site not responding. Last check: 2007-10-10)
		For x large, truncation errors due to finite precision result in loss of significant figures in x-(x
		To avoid this round-off error in this case, re-write x-(x
		Rounding errors are a problem if 4ac is small compared to b
	www.physics.udel.edu /~jim/rounderror.htm (67 words)

	Ex Astris Scientia - Starship Size Table
		Depending on the chosen reference, there are overall errors of up to 15% even for the ships whose sizes are well-known.
		The explanation for the Constellation is simple because the nacelles of the depicted schematic are too small, yielding a size of at most 260m instead of the correct length of at least 300m.
		The <370m from the Encyclopedia II could be an error which was later reproduced in the DS9TM, although it is also confirmed by Stipes.
	www.ex-astris-scientia.org /articles/size_table.htm (888 words)

	Nonindependent Randomized Rounding and an Application to Digital Halftoning
		We investigate the problem of rounding a given [0,1]-valued matrix to a 0,1 matrix such that the rounding error with respect to $2 \times 2$ boxes is small.
		We present a randomized algorithm computing roundings with expected error at most 0.5463 per box, improving the 0.75 nonconstructive bound of Asano et al.
		Thus, by equipping the rounding process with some of the problem information, we reduce the rounding error significantly compared to independent randomized rounding, which leads to an expected error of 0.82944 per box.
	epubs.siam.org /sam-bin/dbq/article/43015 (238 words)

	Rounding error in program
		Yes, because 18.5 should be rounded to 18, not 19.
		This was changed by the International Standards Organisation in 1992, to rounding to the nearest even integer.
		Using sprintf like that doesn't result in the your number being rounded; the fractional part is simply truncated.
	www.codecomments.com /message281874.html (967 words)

	ACC: DSum Causes Rounding Error on Large Currency Fields (95/97)
		Using the domain aggregate (totals) function DSum() to total a currency field may cause rounding errors for values exceeding fourteen significant digits.
		If you use the Currency data type, rounding errors typically do not occur because Currency values are scaled integers rather than floating point numbers.
		However, the Sum() and DSum() functions store an accumulated total as a floating point number, which is limited to approximately 14 significant digits.
	support.microsoft.com /default.aspx?scid=kb;en-us;Q177360 (756 words)

	GnuCash: Numeric: Rational Number Handling w/ Rounding Error Control (Site not responding. Last check: 2007-10-10)
		The same as gnc_numeric_mul, but uses error for accumulating conversion roundoff error.
		Equivalence predicate: Convert both a and b to denom using the specified DENOM and method HOW, and compare numerators the results using gnc_numeric_equal.
		The same as gnc_numeric_sub, but uses error for accumulating conversion roundoff error.
	cvs.gnucash.org /docs/HEAD/group__Numeric.html (2378 words)

	DevX.com Forums - Javascript rounding error (Site not responding. Last check: 2007-10-10)
		There is a rounding error problem that I encountered using Java script.
		Note: There is no rounding error when 4.225 times a number that is smaller
		lost, that's why we have a rounding error.
	news.devx.com /printthread.php?t=6496 (217 words)

	Increment formulations for rounding error reduction in the numerical solution of structured differential systems -- ... (Site not responding. Last check: 2007-10-10)
		Increment formulations for rounding error reduction in the numerical solution of structured differential systems
		Strategies for reducing the effect of cumulative rounding errors in geometric numerical integration are outlined.
		The focus is, in particular, on the solution of separable Hamiltonian systems using explicit symplectic integration methods and on solving orthogonal matrix differential systems using projection.
	library.wolfram.com /infocenter/Articles/5406 (141 words)

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