An Error Estimate

Following the procedure outlined for a muffin-tin potential in Sect.5.7, we 

Provided that the errors of the LMTO method have been taken into account 

For closely packed structures, where is small, one may therefore neglect 

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Unfortunately, this local error Cr cannot be calculated, since we do not know the exact solution to the QCMD equations. The clue to this problem is given by the introduction of an approximation to Let us consider another discrete evolution with an order q > p and define an error estimation via er t + z i) - z t). 

Neville s algorithm constructs the same unique interpolating polynomial and improves the straightforward Lagrange implementation by the addition of an error estimate. 

To overcome these limitations, Vasantharajan and Biegler (1990) propose a new formulation based on the residual function (see Russell and Christiansen, 1978), Since the ODE residual equations are satisfied only at the collocation points, a straightforward way to compute an error estimate... 

To make an error estimation, we evaluate the second-order transition probability by taking into account the second term in Eq. (2.29),... 

Once more, for the non-negligible local Peclet and Damkohler numbers, taking the simple mean over the section does not lead to a good approximation and our numerical simulations, presented in the last section, will confirm these theoretical results. For an error estimate analogous to Theorem 1, we refer to the articles Mikelic and Rosier (2007) and Choquet and Mikelic (2008). 

The electron-diffraction measurements reproduce the twist angle to 42° with an error estimate of about 5 °. This value corresponds to the angle of maximum probability. An exact location of the minimum of the potential function is not easy to derive with present knowledge, but the discrepancy between the electron-diffraction value and the one obtained by ab initio calculation is too large to be accepted. 

The critical temperature Tc of each pseudocomponent is calculated using the API Technical Data Book, Eq. 4D1.1 [16]. This equation is shown here in Table 1.9, code lines 3260 through 3360. Normally, this equation is good for most all types of hydrocarbons, having an error estimation of 6°F. This equation has been noted to have a maximum... 

Some simple relationships giving an estimate of the discretization error in the case of multistep formulae have been proposed. Very often, the difference between the values deduced from predictor and corrector equations is used as an error estimate. 

Si NMR chemical shifts were calculated for each molecule relative to the theoretical shielding for tetramethylsilane (TMS), at the HF/6-311+G(2d,p)86 level using the GIAO method,94 as implemented in Gaussian 94 and Gaussian 98. Shifts for gas-phase molecules are reported because the inclusion of solvation via the SCRF method was found to have little effect on the predicted shifts.83 Comparison of calculated shifts with experimental values for compounds with well-known structures yielded an error estimate of about 1 to 8% for quadra-coordinated silicon and 2 to 9% for penta-coordinated silicon. 

Since A is known and D is either given or measured in the experiment, it is possible to extract Aal and 5 from the parameters Q and Q, provided /% and values are also known. Typical values of / , for the uCB liquid crystals are 1.55-1.65 aud, luckily, these are slowly varying fnnctions of T and wavelength in the isotropic phase. For 5CB an r valne of 1.587 can be nsed with an error estimated at less than 0.5%. 

For R = mim ux,Uy), one can obtain an error estimate also for the Yeh and Berkowitz method, which allows to tune this method. So why should one actually implement the ELC term First of all because it leads to a faster algorithm, normally by a factor of two or more, since the box size can be made smaller due to the smaller gap size, and therefore one can use a smaller mesh size at constant accuracy. In addition, many implementations of P M (such as the ESPResSo one) allow only for cubic simulation boxes. But (18) shows that in fact the ratios A /Aj, rsp. Xz/ dominate the error behaviour, so that one cannot reduce the error in the Yeh and Berkowitz approach with a cubic 3d-method. For such an implementation, ELC is a must. 

Suppose that there exists a sequence of rules 7o, /i,..., with increasing accuracy, namely / is the most accurate rule. If an integrand has no singularity in the interval, we apply h, k = 0,1, consecutively until a satisfactory approximation is obtained. Let > 0 be an error estimate for h. Let hint be defined by... 

Selection of Optimal Tree. The optimal tree (most accurate tree) is the one having the highest predictive ability. Therefore, one has to evaluate the predictive error of the subtrees and choose the optimal one among them. The most common technique for estimating the predictive error is the cross-validation method, especially when the data set is small. The procedure of performing a cross validation is described earlier (see section 14.2.2.1). In practice, the optimal tree is chosen as the simplest tree with a predictive error estimate within one standard error of minimum. It means that the chosen tree is the simplest with an error estimate comparable to that of the most accurate one. 

In the discussion of Schwendeman s paper laurie re-emphasized the point that the Costain uncertainties should always be added to the Kraitchman coordinate in propagating the uncertainty to an error estimate for a derived parameter, schwen-deman agreed that the Costain uncertainty refers to the absolute value of the Kraitchman coordinate, which is known to err by being too small, nelson observed that as a result these uncertainties should be classified as systematic errors. schwendeman agreed and pointed out that in his own method of analysis the Costain uncertainties were propagated in such a manner as to produce asymmetric uncertainties that, in turn, reflect systematic errors. 

AhydH was measured in acetic acid solution. An error estimate was not given. 

In contrast to the clear tendency of the measured condensed fraction to decrease upon dilution, the behavior of the condensation radius Rm appears to be more complicated. There does not seem to exist a simple monotonic convergence of R towards Ru- Rather, for high densities the measured condensation distance is larger than the Manning radius, while for the investigated low densities it is smaller. Unfortunately, a clear-cut statement is difficult, since the localization of the point of inflection in P as a function of ln(r) is only possible with an error estimated to be of the order of 1%. 

This result lends support to our conclusion about the five effects that do not fit the straight line in the normal plot. The values of two others, the 12 and 123 interaction effects, lie practically at the confidence limit. The conclusions do not change much in relation to the analysis of the full design, but it is important to recognize that to obtain an error estimate we are combining variances that differ by up to four orders of magnitude. This is a violation of the normal error hypothesis that is the basis for a... 

The largest standard error in this case was found to be SE = 0.0312, with an error estimate for the fitting of = <3goc SE. 

In this context, the value of d calculated using (1.52) assumes the meaning of an error estimation of the solution also for methods other than the secant method. The test ti+i — h < e, is therefore replaced with di < independent of the method adopted. 

Fig. 6 illustrates the cross-entropy measured for the avaUahility metric by both the 2A— and A —approaches. It depicts that the findings of Moura Droguett (2009) on the accuracy of the 2A-method are conservative. For the availability measure, the 2A-method presents an error estimate smaller than A -method over the number of steps and that tends... 

The CV or RSD, the units of which are obviously per cent, is an example of a relative error, i.e. an error estimate divided by an estimate of the absolute value of the measured quantity. Relative errors are frequently used in the comparison of the precision of results which have different units or magnitudes, and are again important in calculations of error propagation. 

Most of these technologies are based on neural networks models, and also provide an error estimate for die predictions, allowing the user to validate the... 

Table 42 shows that the saturation vapor pressure and orthobaric density of vapor and liquid have been measured repeatedly. We adopted experimental data of MEI [4.3] as reference values of p. These data were obtained statistically for a high-purity substance and with an error estimated by the authors at 0.1-0.2%. The results of comparison with the data of other researchers are shown in Fig. 34. 

A four-point finite-difference subroutines was used to estimate the error remaining after the numerical integration. This subroutine gives the numerical value of an integral / and an error estimate E by using a quartic interpolation. Hence... 

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