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Topic: Approximation error

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	Approximation - Wikipedia, the free encyclopedia
		Approximations may be used because incomplete information prevents use of exact representations.
		The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.
		Approximation theory is a branch of mathematics, a quantitative part of functional analysis.
	en.wikipedia.org /wiki/Approximation (478 words)

	ApproximationAlgorithms - PineWiki
		An approximation algorithm returns a solution to a combinatorial optimization problem that is provably close to optimal (as opposed to a heuristic that may or may not find a good solution).
		Approximation algorithms are typically used when finding an optimal solution is intractable, but can also be used in some situations where a near-optimal solution can be found quickly and an exact solution is not needed.
		Fully polynomial-time approximation schemes are the holy grail of approximation algorithms; they do not appear to exist for many problems, but when they are available, they are often almost as useful as an optimizing algorithm would be.
	pine.cs.yale.edu /pinewiki/ApproximationAlgorithms (1275 words)

	Preprints
		Both upper and lower bounds for the approximation error are derived in terms of the components of the a posteriori error estimator.
		The numerical approximation is stabilized by using a SUPG approximation for the constitutive equation.
		The approximation is stabilized by using a SUPG approximation for the constitutive equation.
	www.math.clemson.edu /~vjervin/papers/index.html (1749 words)

	Springer Online Reference Works
		in whose metric it is expedient to estimate the error of approximation.
		The investigation and estimation of the error of approximation is important from the practical point of view and at the same time is the branch of the approximation of functions that is richest in ideas.
		Here the Hausdorff distance is preferable when choosing the measure of the error of approximation; this is also the case in some cases of the approximation of functions, for instance when a discontinuous function has to be approximated by a smooth function (see [16]).
	eom.springer.de /a/a012950.htm (2622 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The second fundamental type of error, which we call {\em approximation error}, persists even for noise-free input units, and is due to error in the ``fit'' of the approximating function $F$ to the target function $f$ (fig.\ \ref{fig-error}).
		Estimation error is visible across the entire domain, though it is concentrated in the small region just to the right of center where the input probability is peaked.
		Approximation error is concentrated in the central region which contains high spatial frequencies, with minor secondary peaks in other regions, including the region of high input probability.
	www-lnc.usc.edu /~brannon/sb90.html (2208 words)

	Approximation Summary
		The largest restriction on these approximations is that the function and all its derivatives must vanish at zero in order for the polynomial to be manageable.
		Approximation is not a skill which is emphasized in some formal portions of mathematics, but it is essential to any branch of applied mathematics.
		Computer science uses approximate calculations for almost any command, and physics and engineering are almost as dependent upon approximation and interpolation in their experimental phases.
	www.bookrags.com /Approximation (1314 words)

	Gamasutra - Features - "A Non-Integer Power Function on the Pixel Shader" (08.01.02]
		The normalized error function is also plotted in order to show how the error is distributed to the left and right of the point where both curve crosses.
		In order to optimize the approximation error, one should seek values of A and B for which the maximal error on the left and right side are equal.
		When the left error is greater than the right error, the crossing point is moved to the left, otherwise it is moved to the right.
	www.gamasutra.com /features/20020801/beaudoin_01.htm (2056 words)

	Optimality
		This demonstrates that the mean-squared error is hence the normalised variance of the function values over the matched states.
		Therefore, the approximated mean-squared error is a suitable approximation to the mean-squared error.
		Therefore, it fulfils the requirement of the Principle of Orthogonality (Theorem 3.2), that the approximation is orthogonal to the error.
	www.cs.bath.ac.uk /~jd224/lcsfa/node3.html (952 words)

	Ellipse Perimeter Approximations
		This approximation, proposed by Lindner, possibly in 1904, is conceptually similar to Peano's in its second form in the sense that it uses the arithmetic mean and corrects it by means of a factor which is a function of h.
		This corresponds to a zero in the relative error ε(y) and a sharp negative peak on the log-log error-function plot of ε(y).
		The blue error curves (1 and 5) correspond to negative deviations with S'(a,b)<=S(a,b); in all the remaining cases the deviations are non-negative.
	www.ebyte.it /library/docs/math05a/EllipsePerimeterApprox05.html (5424 words)

	3 Polygonal approximation
		The general problem of approximation a given two-dimensional piecewise linear curve by another coarser one is of fundamental importance in computer graphics, vectorization tasks, vector map processing (see Figs.
		In practice, the most of practical error measures in use are based on distance between vertices of the input curve and the approximation linear segments.
		approximation of multiple-object vector data: vector map of "Europe"; 365 objects with 160,000 vertices (a); fragment of the vector map after 20:1 data reduction (b).
	cs.joensuu.fi /~koles/approximation/Ch3_1.html (903 words)

	3 Polygonal approximation
		Pikaz and Dinstein [197] found min-# approximation of the polygonal curve where the city-block metric is used to measure distance between the approximation and the input curve.
		Proximity of the approximation curve to the input one is measured by the sum of squared errors, but the deviation (error value) for every individual vertex is beyond the control of the algorithm.
		The obtained polygonal approximation is minimal with respect to the number of vertices under a given maximal error.
	cs.joensuu.fi /~koles/approximation/Ch3_5.html (1838 words)

	Approximation error - Wikipedia, the free encyclopedia
		In the mathematical sub-field of numerical analysis the approximation error in some data is the discrepancy between an exact value and some approximation to it.
		The numerical stability of an algorithm in numerical analysis indicates how the error is propagated by the algorithm.
		When calculating using approximate values it is important to be able to calculate the errors involved.
	en.wikipedia.org /wiki/Approximation_error (184 words)

	Levels of Detail & Polygonal Simplification
		Approximation error is measured using the distance to the original mesh.
		The tolerance volume is computed as a sphere around each vertex, and represents the accumulation of the error introduced by simplification which lead to the creation of the vertex (by applying the edge collapse operator).
		This geometry removal algorithm is original in that it does not measure the approximation error, but uses a clustering process to ensure that simplification is restricted to certain areas.
	www.acm.org /crossroads/xrds3-4/levdet.html (5289 words)

	1_2_3a_update
		Approximation theory is concerned with using one function to approximate another, usually to save effort in using the function.
		One characteristic of least squares approximations is that the error tends to be larger near the ends of the intervals.
		Notice that the errors in each of these two examples varies from zero at the breakpoints to a absolute value that is close to the target relative error.
	il.water.usgs.gov /proj/feq/fequtl98.i2h/1_2_3a_update.html (5508 words)

	On the Relationship between Generalization Error and Sample Complexity for Radial Basis Functions and Incremental ... (Site not responding. Last check: 2007-11-03)
		Approximation error is defined by the difference between the best possible network parameter set and the target function to approximate.
		The generalization error bound can be viewed as a benchmark that at least a well-chosen RBF should yield.
		Furthermore, the error bound does not assume any specific form of input distribution and we want to reduce the number of neurons by knowing the statistics of the training data.
	www.engr.uconn.edu /~chenhm/home/329pre/pre_RBF.htm (1421 words)

	The Complex Variable Boundary Element Method (More)
		Additionally, the error analysis of the CVBEM approximation functions is workable by the easy-to-understand concept of relative error.
		A sophistication of the relative error analysis is the generation of an approximative boundary upon which the CVBEM approximation function exactly solves the boundary conditions of the boundary value problem (of the Laplace equation), and the goodness of approximation is easily seen as a closeness-of-fit between the approximative and true problem boundaries.
		Due to the convenient approximation error evaluation afforded by the CVBEM, the modeling approach is extremely useful in developing highly accurate approximations for two-dimensional potential problems.
	www.hromadka.net /COMPLE1_More.html (869 words)

	Approximations and Errors (Site not responding. Last check: 2007-11-03)
		The two major forms of numerical error are round-off error and truncation error.
		Truncation error is the discrepancy introduced by the fact that the numerical method employs an approximation to represent exact mathematical operations and quantities.
		Accuracy refers to the proximity of an approximate number or measurement to the true value it is supposed to represent.
	www.site.uottawa.ca /~basu/notes/errors.html (391 words)

	Farfield: Approximating & filtering noisy data with RBFs
		One way to approximate noisy data is to specify a fitting tolerance as described in the RBF FAQ.
		The error bar fitter finds the smoothest function within value ranges specified at the data points, while spline smoothing balances the smoothness of the fitted RBF against fidelity to the raw data.
		This is approximately equivalent to specifying that the width of the Gaussian kernel used by FastRBF corresponds to the mesh/grid spacing.
	www.farfieldtechnology.com /products/toolbox/theory/smoothrbf.html (1467 words)

	In Situ Adaptive Tabulation Overview
		The error tolerance for z is steadily ramped from 0.01 to 0.99.
		The error tolerance of the current frame is indicated on the figure.
		Explicit error control of the approximation of y is maintained by defining a region where the linear approximation is considered accurate.
	www.che.utexas.edu /~john/research/isat.htm (1084 words)

	An Error-Controlled Octree Data Structure for Large-Scale Visualization
		Using an octree-based approximation, one generates a data-dependent, spatial decomposition that adapts to the underlying complexity of the input data.
		To calculate the maximum error for an RGB value, we calculate the maximum error for the red, green, and blue color components and then choose the maximum of these three errors.
		When both trees have the same depth (four), the RMS error of the representation of the small subregion is greater than the error of the representation of the main region.
	www.acm.org /crossroads/xrds6-3/lsv_oct.html (3141 words)

	An Efficient, Error-Bounded Approximation Algorithm for Simulating Quasi-Statics of Complex Linkages
		We demonstrate that in some cases our algorithm is able to achieve up to two orders of magnitude of performance improvement, while providing a high-precision, error-bounded approximation of the quasi-statics of the simulated linkage.
		This paper introduces an algorithm for automatic simplification of the quasi-statics of articulated bodies (in the quasi-statics case, the articulated bodies velocities are zero at all times).
		Based on a user-defined maximum error threshold, the algorithm determines a set of joints which contribute most to the articulated body acceleration, and culls away the other joints (implicitly assuming that their acceleration is zero).
	gamma.cs.unc.edu /AQ (484 words)

	IDAV Publication: Using Graphs for Fast Error Term Approximation of Time-varying Datasets
		Error tables are used to look up the error between different time steps of a time-varying dataset, their generation itself can be expensive.
		The error computed on an edge of a tree describes the distance between the two nodes on that edge.
		Evaluating the error between two different time steps requires traversing a path between the two leaf nodes, accumulating the errors on the traversed edges.
	graphics.cs.ucdavis.edu /graphics/publications/print_pub?pub_id=163 (256 words)

	Linear Approximation and Error Estimation
		For this reason, the linear function whose graph is the tangent line to y = f(x) at a specified point (a, f(a)) is called the linear approximation of f(x) near x = a.
		Thus, the the linear approximation to f(x) near x = a is given by
		Once you have an error estimate for the radius, you might wonder how this error might effect the calculation of the volume id the ball bearing.
	www.zweigmedia.com /ThirdEdSite/calctopic1/linearapprox.html (835 words)

	Dr. Dobb's \| Algorithms for High-Precision Finite Differences \| July 22, 2001
		In fact, it is often the case that the actual error in evaluating a function varies widely at adjacent points.
		However, it is inconvenient to do this, and so instead, the relative condition error is required to be at least 0.001, since the truncation error generally rises as the condition error falls.
		Table 1 lists the errors incurred by forward difference approximations to the first partial derivative with respect to x at the point (0,0) (the exact value is-2).
	www.ddj.com /184410554?pgno=1 (2996 words)

	Approximation Error Maps (ResearchIndex) (Site not responding. Last check: 2007-11-03)
		Global error measures are insuficiently informative when the space A is not spatially homogeneous.
		We introduce here the concept of approximation error map, a mathematical description of how the approximation errors are distributed over the domain -- not for a single function f...
		1 Approximation properties of splines on spherical triangulati..
	citeseer.ist.psu.edu /431866.html (356 words)

	On Optimizing Knot Positions for Multi-dimensional
		Using results from univariate polynomial approximation theory, spline approximation theory and multivariate tensor product approximation theory, we derive a local upper bound for the
		The knots are optimized based on minimization of two global approximation errors:
		We demonstrated our knot optimization algorithm with 2D surface fitting simulation experiments using B-spline models defined in both 2D Cartesian and polar coordinates. In all cases, the B-spline model control points were computed using a least-squares fit to noiseless observations of the true function.
	www.eng.auburn.edu /users/dengx/optKnots.htm (161 words)

	Approximation Error
		This yields an extension as well as a simplification of the asymptotic error formulas that have been published previously.
		For a specified approximation error, this allows us to predict the sampling rate reduction that can obtained by using splines instead Daubechies wavelets.
		However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from SPIE.
	bigwww.epfl.ch /publications/unser9804.html (306 words)

	Dr. Dobb's \| Common-Fraction Approximation of Real Numbers \| July 22, 2001
		Assuming that neither 0 or 1 are a satisfactory approximation to
		Intervals on successively higher Farey Sequences are searched until either the numerator of the new interval boundary in the next-higher sequence exceeds the limit N or the desired tolerance is achieved.
		If the absolute best approximation is desired with the magnitude constraint, a tolerance of 0 can be specified.
	www.ddj.com /184409653 (1511 words)

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