Accuracy approximate error

Of course, the words arbitrary domain cannot be understood in a literal sense. Before giving further motivations, it is preassumed that the boundary F is smooth enough to ensure the existence of a smooth solution u = u x,t) of the original problem (l)-(2). In the accurate account of the approximation error and accuracy we always take for granted that the solution of the original problem associated w ith the governing differential equation exists and possesses all necessary derivatives which do arise in the further development. 

The accuracy of the error equations (Eqs. (22) and (23)] also depends on the selected wavelet. A short and compactly supported wavelet such as the Haar wavelet provides the most accurate satisfaction of the error estimate. For longer wavelets, numerical inaccuracies are introduced in the error equations due to end effects. For wavelets that are not compactly supported, such as the Battle-Lemarie family of wavelets, the truncation of the filters contributes to the error of approximation in the reconstructed signal, resulting in a lower compression ratio for the same approximation error. 

In modem approaches to drug discovery there are numerous approximations, assumptions and accuracy or error limitations. Inevitably, this affects the reliability of the data, information and knowledge with the obvious implication that some of the knowledge will be incorrect. Appreciating the differences between the types of parameters can help to ensure that when possible the appropriate statistical techniques are used and that the reliability of knowledge is considered when making important decisions. 

Error thus arises from two sources. Lack of precision (random errors) can be estimated by a statistical analysis of a series of measurements. Lack of accuracy (systematic errors) is much more problematic. If a systematic error is known to be present, we should do our best to correct for it before reporting the result. (For example, if our apparatus has not been calibrated correctly, it should be recalibrated.) The problem is that systematic errors of which we have no knowledge may be present. In this case the experiment should be repeated with different apparatus to eliminate the systematic error caused by a particular piece of equipment better still, a different and independent way to measure the property might be devised. Only after enough independent experimental data are available can we be convinced of the accuracy of a result— that is, how closely it approximates the true result. 

Naman s spectral steadiness criterion was used for analysis of the accuracy and steadiness of the finite difference scheme (2.44) and for estimation of the approximation error. According to this criterion, the scheme must be steady for any elementary conditions, including for the first harmonic Yq = where p is the... 

The accuracy of the nonpolar solvation model performance is crucial to the success of other expanded versions of the differential geometry formalism. In particular, as the electrostatic effect and its associated approximation error are excluded, the major factor impacting the nonpolar solvation model is the solvent-solute boundary, which is governed by the DG-based formalism. Therefore, the nonpolar model provides the most direct and essential validation of the DG-based models. In our recent work [1], the DG-based nonpolar solvation (DG-NP) model was tested using a... 

In the present work Gallerkin s method of weighted residuals is used to derive the weak form of the equilibrium equations. Hence, the first step towards finite element discretisation of the governing equations is the definition of shape functions for the domain variables, i.e. displacement, pore water pressure and pore air pressure. Introducing these shape functions into equations 13, 14 and 15 the governing equations are approximated with a certain accuracy. The approximation errors, termed... 

As shown in numerical analysis textbooks, the accuracy of Eq. 7-32 is influenced by the integration interval. However, discrete-time models involving no approximation errors can be derived for any linear differential equation under the assumption of a piecewise constant input signal, that is, the input variable u is held constant over At, Next, we develop discrete-time modeling methods that introduce no integration error for piecewise constant inputs, regardless of the size of At,... 

Since both the FDA and FEA have local approximation errors of both are capable of arbitrary accuracy pro-... 

Cases 9 and 10 illustrate the type of convergence study which is possible with methods in which the local approximation error has a known parameter dependence. In case 9 the error should be at least a factor of (1.5) = 5 times larger than case 10. Thus one can confidently and conservatively ascribe RHF-limit accuracy to the digits that are stable in the two calculations. That is, in case 10 the uncertainty should be no greater than the fifth decimal place for the total and orbital energies and the third-fourth for the moments. 

In [13], an efficient residual error estimation scheme has been introduced for controlling the quality of the approximation. This gives us a stopping criterion for the iteration guaranteeing that the quality of the approximation fits to the accuracy requirements of the stepsize control. 

This discussion may well leave one wondering what role reality plays in computation chemistry. Only some things are known exactly. For example, the quantum mechanical description of the hydrogen atom matches the observed spectrum as accurately as any experiment ever done. If an approximation is used, one must ask how accurate an answer should be. Computations of the energetics of molecules and reactions often attempt to attain what is called chemical accuracy, meaning an error of less than about 1 kcal/mol. This is suf-hcient to describe van der Waals interactions, the weakest interaction considered to affect most chemistry. Most chemists have no use for answers more accurate than this. 

For very-high-accuracy ah initio calculations, the harmonic oscillator approximation may be the largest source of error. The harmonic oscillator frequencies... 

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