Maximum absolute error

Interestingly, if one Taylor series expands Eq. (36) and equates the terms of the same order in kj with Eq. (37) one can derive the standard Lagrangian FD approximations (i.e., require the coefficient of kj to be —1, and require the coefficient of all other orders in kj up to the desired order of approximation to be 0.) A more global approach is to attempt to fit Eq. (36) to Eq. (37) over some range of Kj = kjA values that leads to a maximum absolute error between Eq. (36) and Eq. (37) less than or equal to some prespecrfied value, E. This is the essential idea of the dispersion-fitted finite difference method [25]. 

In light of the previous discussion and contrary to the established practice, we propose the use of the maximum absolute error (6) as the empirical risk to be minimized because it offers the following advantages ... 

Our statistical analysis reveals a large improvement from cc-pCV(DT)Z to cc-pCV(TQ)Z see Fig. 1.4. In fact, the cc-pCV(TQ)Z calculations are clearly more accurate than their much more expensive cc-pcV6Z counterparts and nearly as accurate as the cc-pcV(56)Z extrapolations.The cc-pCV(TQ)Z extrapolations yield mean and maximum absolute errors of 1.7 and 4.0 kJ/mol, respectively, compared with those of 0.8 and 2.3 kJ/mol at the cc-pcV(56)Z level. Chemical accuracy is thus obtained at the cc-pCV(TQ)Z level, greatly expanding the range of molecules for which ab initio electronic-structure calculations will afford thermochemical data of chemical accuracy. 

Table 1.12 Statistical measures of errors for extrapolated CCSD(T) AEs relative to experiment (kJ/mol). A is the mean error, Astd is the standard deviation around the mean error, Aabs is the mean absolute error, and Amax is the maximum absolute error.

If we use B3LYP/VTZ+1 harmonics scaled by 0.985 for the Ezpv rather than the actual anharmonic values, mean absolute error at the W1 level deteriorates from 0.37 to 0.40 kcal/mol, which most users would regard as insignificant. At the W2 level, however, we see a somewhat more noticeable degradation from 0.23 to 0.30 kcal/mol - if kJ/mol accuracy is required, literally every little bit counts . If one is primarily concerned with keeping the maximum absolute error down, rather than getting sub-kJ/mol accuracy for individual molecules, the use of B3LYP/VTZ+1 harmonic values of Ezpv scaled by 0.985 is an acceptable fallback solution . The same would appear to be true for thermochemical properties to which the Ezpv contribution is smaller than for the TAE (e.g. ionization potentials, electron affinities, proton affinities, and the like). 

The best scale factor in the least-squares sense is 0.788 while the mean absolute error of 0.04 kcal/mol is more than acceptable, the maximum absolute error of 0.20 kcal/mol (for SO2) is somewhat disappointing. Representative results (for the W2-1 set) can be found in Table... 

This error can be considerably reduced, at very little cost, by employing B3LYP density functional theory instead of SCF. The scale factor, 0.896, is much closer to unity, and both mean and maximum absolute errors are cut in half compared to the scaled SCF level corrections. (The largest errors in the 120-molecule data set are 0.10 kcal/mol for P2 and 0.09 kcal/mol for BeO.) It could in fact be argued that the remaining discrepancy between the scaled B3LYP/cc-pVTZuc+1 values is on the same order of magnitude as the uncertainty in the ACPF/MTsmall values themselves. 

We present the results for all atoms in Table 2. Three sets of results are given CCSD, CCSD(T), and CASPT2. For the first row atoms we have also performed calculations with two h-type basis functions added to the ANO-RCC basis set. Table 3 gives the root mean square (RMS) errors and also the maximum absolute error. 

Step 2. If the maximum absolute error in the constraints max c,(I +i) is below a chosen threshhold (i.e., 0.25max c,(l ) ), then the Lagrange multipliers are updated by a first-order correction... 

Steps 1-3 are repeated until the maximum absolute error in the constraints falls below a target threshhold. Before the first iteration the Lagrange multipliers may be initialized to zero and the penalty parameter set to 0.1. The constraints are not fully enforced until convergence, and the energy in the primal program approaches the optimal value from below. 

Table 8.3 Mean and maximum absolute errors (kcal mol ) in enthalpies of activation and forward reaction for different methods...

The error in Ky t) generated in the first scheme from the experimental autocorrelation function was also examined by taking the generated Ky(t) and using it as input to the 2nd scheme to try to recover the original autocorrelation function. The original autocorrelation functions were all recovered in this manner within a maximum absolute error < 0.002 for all times, t < 10"12 s. 

The computed eigenenergies are compared with exact ones. In Fig. 40-43 we present the maximum absolute error logio(Err) where... 

The numerical results obtained for the thirty-three methods were compared with the analytic solution of the Woods-Saxon potential. Figure 1 shows the maximum absolute error Err — — log10 ivcurate — Ecomputed in the computation of all resonances En, n = 1(1)4, for step length equal to A = The nonexistence of a value indicates that the corresponding maximum absolute error is larger than 1. 

Modified Woods-Saxon Potential Coulombian Potential. - In Figure 2 the maximum absolute error, defined as Err = -log10 Eaccurate - Ecomputed, in the computation of all resonances E ,n — 1(1)4 obtained with another potential in (121), for step length equal to = and for the methods mentioned above, is shown. This potential is... 

The true solutions to the Woods-Saxon bound-states problem were obtained correct to fourteen decimal places using the analytic solution and the numerical results obtained using the above mentioned methods were compared with this true solution. In Figure 3 we present the maximum absolute error, defined as Err = — og o Eaccurate — ECOmputed, in the computation of the eigevnalues E ,n = 0(4) 12, for step length equal to h =... 

The numerical results obtained for the twelve methods were compared with the analytic solution of the Woods-Saxon potential. Figure 4 shows the maximum absolute error... 

We find the solution to (35) using the secant method with a trust radius of a/4 at each iteration. The algorithm was terminated once the integral on the right-hand side of (35) was less than 10 in absolute magnitude. The results are presented in Table 1, along with the maximum absolute error as defined by... 

NFE counts the computational cost for each method. So, the comparison is based on the maximum absolute error whieh is obtained with the specific NFE i.e. with the specific computational cost for each method. 

The design and operation of a highly accurate large flow cryogenic calibration stand has proven quite successful. The test system has been used to calibrate flowmeters using liquid oxygen, liquid nitrogen and distilled water. The maximum absolute error of the test system as obtained from the error analysis is 0.28 for the calibration of volumetric-type flowmeters, and 0.17 for the calibration of mass type flowmeters. 

Model MSE values for training pattern Maximum absolute % error ... 

The ANN topology was adopted as a predictive tool. The cutting speed, feed rate, drill diameter and drill point geometry were nsed as the input parameters. The drilling-induced damage was the output. The maximum absolute error for training patterns was found to be 12.7 per cent and the minimum was 0.1 per cent, and for most cases the error was less than 5 per cent. The maximum absolute error for testing patterns was found to be 13.19 per cent and the minimum was 0.35 per cent (Mishra et al., 2010). 

An ANN predictive approach was developed with inputs as the drill point geometry, the feed rate and the cutting speed with residual tensile strength of composite with drilled holes as the output. The maximum absolute error for testing... 

The maximum absolute error Incurred In the use of the above formulae to predict non-dimensional minimum film thickness In comparison with the numerically determined value was 4.9Z for the Reynolds boundary condition case and 2.4% for the half-Sommerfeld boundary condition (with respect to the value set In the numerical solution). The corresponding mean errors over the range of conditions examined were 1.2% and -0.3% respectively. The excellence of the fits represented by the equations (18) and (19) are shown In Figures 3 and 4 respectively. 

Not surprisingly, the TAE exhibits the slowest basis set convergence behavior of all the properties considered here. For a set of 13 molecules with very well-established atomization energies, the mean and maximum absolute errors on the valence part of the total atomization energy are given here, as a function of the basis set ... 

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