Accuracy Considerations

We have already noted that the finite difference approximation to a physical problem improves as smaller and smaller anci smaller increments of Ax and Ay are used. But, we have not said how to estimate the accuracy of this approximation. Two basic approaches are available. 

Compare the numerical solution with an analytical solution for the problem, if available, or an analytical solution for a similar problem. 

Choose progressively smaller values of Ax and observe the behavior of the solution. If the problem has been correctly formulated and solved, the nodal temperatures should converge as Ax becomes smaller. It should be noted that computational round-off errors increase with an increase in the number of nodes because of the increased number of machine calculations. This is why one needs to observe the convergence of the solution. 

It can be shown that the error of the finite-difference approximation to dTIdx is of the order of (Ax/L)2 where L is some characteristic body dimension. 

Analytical solutions are of limited utility in checking the accuracy of a numerical model because most problems which will need to be solved by numerical methods either do not have an analytical solution at all, or if one is available it may be too cumbersome to compute. 

A different study provides some further information coneeming the effects of electron correlation upon calculated intensities. In addition to SCF and MPn type methods, these authors considered also the coupled-cluster model which is an infinite-order generalization of MR Coupled-cluster, limited to single and double excitations (CCSD), was considered, as well as CCSD + T which includes the effects of connected triple excitations. The doubleharmonic approximation was made, consistent with most calculations of H-bonded systems. The [5s3pld/3slp] basis set is of the polarized double- variety, within reach of most workers. 

Results computed tor the HF stretch and for the three internal vibrations of H2O are listed in Table 3.3, along with experimental estimates of the intensities in the final row. Comparison with the first row indicates that SCF intensities seem to be eonsistently too high, sometimes by a factor of less than two, but by an order of magnitude in the case of the symmetric stretch of water. Inclusion of correlation immediately lowers all intensities. MP2 appears satisfactory for all modes, with the exception of v, for H O. Higher levels of eorrelation beyond MP2 introduce smaller reductions, in some cases even lower intensities than experimentally observed. 

Comparison of various modes in additional molecules led to the conclusion that the calculations of stretches of terminal A—H bonds are particularly demanding. With regard to basis set requirements, there seems to be little benefit in enlarging the set beyond the DZP 

Similar sorts of conclusions apply to the frequencies. A systematic study found that a DZP basis set yields vibrational frequencies within about 9% of experimental (harmonic) values. The discrepancy diminishes to 4% when correlation is included via CISD and to 2% with a coupled cluster treatment. Another set of calculations confirmed the eost-effec-tiveness of the MP2 treatment of vibrational frequencies, indicating better agreement with experiment than MP3 on some oceasions. Certain types of modes can be more sensitive to the level of theoretical treatment than others. For example, out-of-plane bending motions for it-bonded systems can require triple- plus two sets of polarization functions, as well as a set of/-functions in the basis set . 

In general, the results just reported, as well as other work, indicate that infrared in- 

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Based on accuracy considerations alone, it seems that time steps of the order At = 16 fs should be possible for the slowest forces for MD [6]. Contrast this with the 0.5-1 fs value needed for the fastest forces. 

PESs for fluorine, hydrogen fluoride, and water using this ansatz showed promise [14]. In all cases the perturbative approach improved the accuracy considerably at a small increase of computational cost. Especially interesting is the possibility of linear scaling. 

D. Computational Effort and Accuracy Considerations Applications of the QMCF MD Methodology... 

In regions where the solution varies slowly, accuracy considerations alone would permit a large time step. However, for a stiff problem where nearby solutions vary rapidly, stability demands a very small time step, even in regions where the solution is changing slowly and the local truncation error (accuracy) can be controlled easily with a large time step. 

If we replace p - pi by its uncertainty u in equation (1.11) then w2 is the minimum value of the mass fraction of the impurity that could be reliably detected. The uncertainty in p - p, includes the uncertainty in the measurement of p but also the uncertainty in pi, the density of the pure sample. The value of pi must be determined for a sample whose purity is known, by some independent means, with an accuracy considerably greater than the one being tested. An example using equation (1.11) and Table 3 is the determination of a small impurity of 1,2-diethylbenzene in 1,4-diethylbenzene. Table 3 contains numerical solutions of equation (1.11) at various densities and uncertainties. 

In every analytical problem a unified approach must be taken to obtain results of satisfactory accuracy. Considerations of interferences due to contamination are of the utmost importance in the analysis of trace elements from air samples. In many cases attention must be given to selection of high-purity acids and other reagents, and particularly to the purity requirements for collection media such as filters. 

For this class of problems, the implicit approach is the basic advantage of the new method. Numerical stability is thus guaranteed, and the size of the integration step is only limited by accuracy considerations. The concomitant decrease in the number of integration steps is the principal gain in computing efficiency. 

Now, even if values of c" i = 1,2,..., ) are known, (25.95) cannot be solved explicitly, as c"+l is also a function of the unknown c, 1 and c" /. However, all k equations of the form (25.95) for i = 1,2,..., k form a system of linear algebraic equations with k unknowns, namely, c"+l, c"+l,..., c"+l. This system can be solved and the solution can be advanced from r to tn+. This is an example of an implicit finite difference method. In general, implicit techniques have better stability properties than explicit methods. They are often unconditionally stable and any choice of Ar and Ax may be used (the choice is ultimately based on accuracy considerations alone). 

Conventional methods to measure the fluorescence quantum yield of solutions attain an accuracy of about 5% at best. It is obvious from Figure 1 that this accuracy is by far not sufficient for the problem at hand. Hence a calorimetric method based on the thermal leasing effect was taken into consideration, because in this way the quantum yield can be determined absolutely (i.e., without recurrence to a luminescence standard) and with high confidence. Such a method appears indicated here in particular, because it measures directly the nonradiative deactivation. In case of very high values of the fluorescence quantum yield, this should enhance the accuracy considerably. 

Besides convergence and accuracy considerations, also stability influences the maximum allowed time interval At. 

The task being auctioned is therefore assigned to the robot that submitted the lowest adjusted bid price, based on (7.3). As such, this closed-loop bid adjustment mechanism can improve bidding accuracy, considerably enhancing the overall team performance. 

Limited accuracy, considerable technique required in exposing and developing emulsions. Counting tracks is time con-sviming and tedious. 

Accuracy considerations. In the previous section we defined fim as the midpoints of the chosen jic-intervals. If generality is maintained, we are clearly free to make modifications here, and finding that accuracy is thereby improved, we do this. In general, the modified /I s are determined so that one or several moment conditions of the form = l/( + 1) are... 

Acoustic transmission measurements were made on a Norton-supplied tensile bar in the molded state. Because this material had not been tested acoustically, there was uncertainty as to the value of its properties, particularly attenuation. PAD found that at 5 MHz, attenuation was sufficiently low that a through transmission signal was easily obtained through the length of the sample. This may allow conducting velocity measurements across the samples at frequencies as high as 20 to 50 MHz, yielding measurement accuracy considerably better than 1%. The velocity and impedance obtained for this sample were 2.3 mm/sec and 5.3 MR, respectively. 

Of course, this section could neither present an exhaustive discussion of dynamic MC algorithms of polymers nor provide a deep insight into the accuracy considerations we only wanted to give the reader a flavor of these problems, and refer to the literature " for more details. 

Linte CA, Lang P, Rettmaim ME, Cho DS, Holmes 3rd DR, Robb RA, et al. Accuracy considerations in image-guided cardiac interventions experience and lessons learned. Int J Comput Assist Radiol Surg 2012 7(1) 13—25. 

If we replace p - pi by its uncertainty in equation (1.11) then W2 is the minimum value of the mass fraction of the impurity that could be reliably detected. The uncertainty in p - pi includes the uncertainty in the measurement of p but also the uncertainty in p, the density of the pure sanqrle. The value of pi must be determined for a sanqrle whose purity is known, by some independent means, with an accuracy considerably greater than the one being tested. An example of the use of equation (1.11) and Table 2 is the determination of a small inqmrity of (Z)-2-pentene in 1-pentene. (Table 2 contains numerical solutions of equation (1.11) at various densities and uncertainties.) The selected densities of 1-pentene and (Z)-2-pentene are (640.61 + 0.47)kg m and (654.74 + 1.17)k gm respectively at 298.15 K. Assuming that the density of the sample with the inqnuity is measmed with an accuracy of 0.4 kg m, then the total uncertainty is (0.4) + (0.47) = 0.62 kg-m l Since pi/p2 = 0.978 then pi -P2I / = 22.8. Interpolation with these values from Table 2 shows that the minimum mass fraction of (Z)-2-pentene that can be detected by the density measurement is 0.042. This corresponds to a purity of 95.8 mass %. The minimum detectable mass fraction of (E)-2-pentene is 0.078 for the same assumptions. Hence density measurements is not a sensitive method for purity determination when the density of the impurity is close to that of the compound under consideration. 

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