Round-off error - Factbites

Round-off error - Factbites
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Topic: Round-off error

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	Round-off error - Wikipedia, the free encyclopedia
		A round-off error is the difference between the calculated approximation of a number and its exact mathematical value.
		Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finite digits to represent infinite digits of real numbers.
		This kind of error is unavoidable for conventional representations of numbers.
	en.wikipedia.org /wiki/Round-off_error (179 words)

	EP208 - Notes for Lecture 5
		The situation is less clear when the round-off error has the opposite sign to the truncation error, the CDA may vary erratically about the true value, though again a relatively stable stationary value in a plot of CDA versus h corresponds to a solution close to a minimum error.
		Another source of error exists when the FDA, CDA or REA are computed; this is the round-off error due to limited precision in numerical arithmetic (numerical values are stored in the computer with a limited number of binary bits).
		CDA = f`(x) + (h^2/6).f```(x) + O(h^4) ----- ------------------------ / \ the derivative the truncation error in the CDA
	www.gantep.edu.tr /~andrew/ep208/notes?lecture=5 (1345 words)

	Numerical errors
		The final error source is the round-off error, due to the fact that real numbers are represented by binary number approximations in a computer.
		If the calculations give results that differ by more than the error tolerance at two different precisions, then round-off is clearly a problem.
		Errors can usually be reduced by reducing step-sizes, but the rate at which this happens--called the order of the algorithm--depends on the algorithm design.
	www.physics.uq.edu.au /people/cochrane/phys3071/phys3071Notes/node30.html (641 words)

	lecture4.txt
		For example, an error of 10cm in the measure of the distance from the Earth to the Moon is not at all the same as an error of 10cm in the measure of the height of the door!
		The errors between these computed values and the exact value can be characterized in two ways: - The "accuracy" of the computed values is a measure of the distance between the computed values and the exact value.
		A problem with using the true error is that it must always be considered in relation with the value.
	www.cs.toronto.edu /~kiringai/courses01/csc270h/lecture4.txt (541 words)

	Error avalanche
		For example, if the output of one function is the input of another, then small round-off errors in the first function can produce much larger errors in the second.
		Error handling systems need to be carefully designed if they are to cope with this and produce meaningful information.
		In computer science, Error avalanche refers to how one small problem can become compounded until it eventually becomes a very large one.
	www.brainyencyclopedia.com /encyclopedia/e/er/error_avalanche.html (212 words)

	Numerical analysis
		The effect of round-off error is partly quantified in the condition number of an operator.
		The study of the generation and propagation of round-off errors in the cause of a computation is an important part of numerical analysis.
		Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic.
	www.sciencedaily.com /encyclopedia/numerical_analysis_1 (1427 words)

	7.2 Round off Errors and the Derivative
		Normally, these very small errors (called round off errors) can be ignored because the "noise" they represent in your evaluation is extremely small compared to the signal, which consists of the value of f itself.
		This usually gives you the power to look for yourself and see where round off error is causing significant error.
		The estimates tend to home in then start to move away again, as the effects of round off error make themselves felt.
	www-math.mit.edu /18.013A/HTML/chapter07/section02.html (518 words)

	CSI801-excercises
		These rounding errors become fatal when two very large numbers are subtracted and result in a catstrophic cancellation.
		This is a single-pass algorithm, but since it takes squares of large numbers, it runs up against the precision limit and rounding errors occur.
	www.science.gmu.edu /~asatsang/csi801/ex1_16b.html (97 words)

	Errors
		In most cases, neither the user nor the programmer has much control over rounding error, since floating-point arithmetic is built into the machine or the programming language.
		The number of digits of accuracy has to do with relative error rather than absolute error (the absolute error is the magnitude of the difference between the true value and the approximation, while the relative error is the absolute error divided by the magnitude of the true value).
		While there is not much programmers can do about the previous types of error, discretization error is entirely under their control, and most of the subject of numerical analysis deals with ways to control it.
	www.math.ubc.ca /~israel/m215/errors/errors.html (750 words)

	A CELL DYNAMICAL SYSTEM MODEL FOR THUNDERCLOUD ELECTRIFICATION
		Z is equal to the scale ratio R / dR, i.e, the numerical value of R, the computed length unit measured in units of round-off error dR.
		The error growth dR on either side of the origin, i.e., clockwise and anticlockwise rotation generates an overall logarithmic spiral shape to W domain with quasiperiodic Penrose tiling pattern for the internal structure as shown in Fig.1.
		Also, the error grows exponentially with number of unit lengths W of the numerical computer output and is responsible for the exponential divergence of two initially close points.
	www.geocities.com /CapeCanaveral/Lab/5833/thunder/THUNDER.html (4598 words)

	EP208 - Notes for Lecture 1
		As round-off errors can be a significant source of error in a numerical method (in addition to the truncation error) we will look more closely at the nature of the round-off error and how it can be reduced.
		Remember: although double- and quad-precision can reduce round-off errors they have no effect on the size of truncation errors; truncation errors are inherent to the numerical method and not to the internal representation of numbers in a computer.
		In this lecture, we will look at the nature of truncation errors and round-off errors.
	www.gantep.edu.tr /~andrew/ep208/notes?lecture=1 (1236 words)

	Excel Statistical Functions: INTERCEPT
		Round off errors have become so severe that division by 0 occurs when D3 = 8.
		Earlier versions of Excel give wrong answers in these cases because the effects of round-off errors are greater with the computational formula that these versions use.
		However, cells D7:D13 show the round off errors that you obtain with the earlier versions.
	support.microsoft.com /default.aspx?kbid=828234&product=xl2003 (995 words)

	scan2002[Abstracts]
		CESTAC is a method to estimate and control round-off errors in floating-point computation.
		At the end of the computation, several results are available, each with a different round-off error propagation.
		The idea behind this method is to execute the same computation several times with different round-off error propagation.
	scan2002.lip6.fr /abstracts.php?fname=Roselyne&lname=Chotin (219 words)

	Pseudo-Randomness of Round-Off Errors in Discretized Linear Maps on the Plane (ResearchIndex)
		Abstract: We analyze the sequences of round-off errors of the orbits of a discretized planar rotation, from a probabilistic angle.
		3 global stability and scaling in a model of Hamiltonian round..
		4 errors and p-adic numbers (context) - Bosio, Vivaldi - 2000
	citeseer.ist.psu.edu /545680.html (565 words)

	INRIA - INédit 44
		A second step in the reliability study concerns round-off errors, a field in which the competence of the Arenaire team is recognized worldwide.
		In effect, even if an algorithm is reliable, it nonetheless accumulates round-off errors since the speed, angle and all other parameters that describe the flight of an aircraft are expressed by real numbers.
		The point is to find a compromise between a very precise algorithm with a very large number of round-off errors and a simpler algorithm that can manage these errors better.
	www.inria.fr /actualites/inedit/inedit44_interna.en.html (511 words)

	Math Digest
		Floating-point arithmetic can't avoid round-off errors either: if a number cannot be exactly represented using this notation, it must be approximated by its closest floating-point neighbor.
		As accurate as digital computers are, round-off errors are hard to avoid when performing arithmetic computations with non-integer numbers.
		And while perhaps not the best solution to practical problems requiring a single result, Hayes suggests that interval arithmetic could be at least part of the means to more accurately solve real-world problems.
	80-www.ams.org.library.uor.edu /mathmedia/mathdigest/200312-interval.html (205 words)

	mp_arc 99-479
		We explore some connections between round-off errors in linear planar rotations and algebraic number theory.
		Round-off errors and p-adic numbers (254K, Postscript) Dec 15, 99
		These constructs are based on symbolic dynamics and on the representation of the discrete phase space as a ring of integers in a quadratic number field.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=99-479 (126 words)

	(Jezequel F.) Round-off Error Propagation in the Solution of the Heat Equation by Finite Differences
		Abstract: The effect of round-off errors on the numerical solution of the heat equation by finite differences can be theoretically determined by computing the mean error at each time step.
		(Jezequel F.) Round-off Error Propagation in the Solution of the Heat Equation by Finite Differences
		The theoretical formulas provide an approximation of the experimental round-off errors.
	www.jucs.org /jucs_1_7/round_off_error_propagation (148 words)

	RBD, Reliability Block Diagrams, Monte Carlo Simulation- Relex Feature Article
		In general, computing results using explicit closed-form expressions is inefficient for complex systems because of long computation times and results that are prone to round-off errors.
		However, if analytical results are not possible or prone to round-off errors, then simulation should be used.
		If a system is complex, the resulting formulas may become complex, making the computational process lengthy and the results prone to round-off errors.
	www.relexsoftware.com /resources/art/art_rbd2.asp (1132 words)

	Practical Automation Inc.
		For the raster printers, where the barcodes approach the minimum feature size of the barcode, the application engineer must be aware of an additional error source that can come from round off errors related to barcode generation program.
		It works well in most cases, however, when the target barcode size, that will be printed on the printer, starts to approach the minimum element size for the barcode, mathematical round off errors can cause barcode inaccuracies when forming the line and space patterns.
		For the FGL printers the minimum feature size must be considered when establishing the minimum size for the barcode.
	www.practicalautomation.com /cgi-bin/faqs.pl?display=66 (417 words)

	Franco Vivaldi: algebraic dynamics
		In work with D Bosio and I Vladimirov, I have shown that the dynamics of round-off errors in discretized irrational rotations admits a natural representation as expanding dynamics over the p-adic integers.
		As a result, one obtains a central limit theorem for the propagation of round-off errors.
		J H Lowenstein and I have developed an arithmetical theory of invertible (hamiltonian) round-off errors for linear rational rotations, building on previous work with S Hatjispyros.
	www.maths.qmw.ac.uk /~fv/research/algdyn.html (590 words)

	mp_arc 95-47
		Mathematical model of these round-off errors is proposed and new phenomena in numerical simulations of dynamical systems (differential equations) and other numerical procedures are discussed.
		It is shown also that these new phenomena could appear not only with Pentium floating point division bugs, but in a more general situation, when one tooks into account round-off errors on the each elementary step of calculations, rather than on the final step only.
		We consider some unusual phenomena that appear in long consecutive calculations due to a new sort of round-off errors of Pentium floating point division bug type.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=95-47 (117 words)

	Periodicity and transport from round-off errors, Franco Vivaldi
		We investigate the effects of round-off errors on quasi-periodic motions in a linear symplectic planar map.
		Periodicity and transport from round-off errors, Franco Vivaldi
		By discretizing coordinates uniformly we transform this map into a permutation of $\Z^2$, and study motions near infinity, which correspond to a fine discretization.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1048515812 (145 words)

	CMPT 101 Lectures
		Round-off errors occur when significant digits are truncated from the least significant end of the value.
		If a parity error occurs (a bit has flipped), then the parity will be wrong and an error may have occured but not with certainty.
		Furthermore, no parity error occurs, there may still be an error in the data.
	www.artsci.gmcc.ab.ca /courses/cmpt101oa/lecture12.htm (321 words)

	Mini Seminar for Second Year Ph.D.s
		But in practical situations, not only we observe the process at discrete times but also each observation is subject to measurement errors, one of these being the round-off effect.
		Theoretical results and simulation results about the asymptotic behavior of this estimator when the observations are subject to round-off errors will be presented.
		Comparisons to the i.i.d error case will be shown, and future work will be reported.
	galton.uchicago.edu /seminars/studentabstracts/yyli_3-9-05.html (169 words)

	OptiVec: VF_setRspEdit
		The maximum round-off error in the construction of Flt accumulates roughly to (size * big * prec), where big is the largest element of Flt and prec the relative floating-point precision.
		Due to accumulated round-off, filter elements that should be zero may be not so, but contain small non-zero numbers.
		During deconvolutions, the editing has also another meaning: for all "lost" frequencies (i.e., those for which the Fourier transform of the response function contains only numbers near zero), the filter is set to 0 instead of the inverse of these small numbers.
	www.optivec.com /vecfuncs/setRspEdit.htm (247 words)

	Calculating the Mean and Standard Deviation
		We want to calculate both the mean and standard deviation in one pass through the data without round-off errors.
		We avoid round-off errors by carrying the maximum possible precision.
		This method also allows you to notice peculiar observations as soon as they appear.
	www.stormloader.com /dickbeldin/Home/amsd.htm (200 words)

	Excel Statistical Functions: RSQ
		Earlier versions of Excel exhibit incorrect answers in these cases because the effects of round-off errors are more profound with the computational formula that is used by these versions.
		Note that CORREL is still well-behaved, but round-off errors in PEARSON have become so severe that division by 0 occurs in RSQ and PEARSON^2 when D2 = 8.
		In versions of Excel that are earlier than Excel 2003, PEARSON may exhibit round-off errors.
	support.microsoft.com /default.aspx?kbid=828131&product=xl2003 (1099 words)

	780.20: 2082 Session 2a and 2b
		Read the discussion about round-off errors and the quadratic equation in the background notes for Session 2 (section 2b).
		Look at a round-off error demonstration with quadratic equations
		Identify and fix the errors, until it compiles and runs correctly (verify the results against the comments in the file).
	www.physics.ohio-state.edu /~ntg/780/2082_sessions_2005/780_2082_02.php (1397 words)

	April5.html
		Round off errors can be especially visible to the human eye when there is a smooth change in intensity across a surface.
		Roundoff errors become noticable in these images because the transition from one pixel value to the next occurs along a curve.
		Round off errors, and non-linearities of the Human Visual System (HVS) [corresponds (sort of) to 13.1.1 of text]
	www.cim.mcgill.ca /~langer/557B/April5.html (1408 words)

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