Cubic-spline fit

Figure C2.17.13. A model calculation of the optical absorjDtion of gold nanocrystals. The fonnalism outlined in the text is used to calculate the absorjDtion cross section of bulk gold (solid curve) and of gold nanoparticles of 3 mn (long dashes), 2 mn (short dashes) and 1 mn (dots) radius. The bulk dielectric properties are obtained from a cubic spline fit to the data of [237]. The small blue shift and substantial broadening which result from the mean free path limitation are...

Figure 2 (a) The pair interaction V as a function of distance PdsoRhso alloy, (b) Spinodal curve for Pdj.Rhi j alloy system. The points indicate calculated points while the solid line is the cubic spline fit through the points. 

The first figure compares the plot of this equation with a cubic spline fit of the original data. 

Figure 1 graphically depicts the numerical data relevant to our application listed by Lundberg et al. Different sets of curves off vs. X are provided for individual values of d /d0. Discrete data were provided in the numerical tables of the original work to produce the continuous traces in Figure 1, a cubic spline fitting was used. 

Data array calculations with cubic spline fitting and numerical integration appears to yield data of much greater accuracy. 

In spite of the flexibility introduced into equation (62) by using 2D splines to describe DM(a, v), a comparison of classical trajectories run on a model potential with those run on a rotated Morse-cubic spline fitted to that model does not show good point-by-point agreement (see ref. 54 and references therein). 

By solving these equations using the method proposed by Mauri[10], a and (3 can be easily calculated, then the discrete components compositions and continuous fraction distribution in the outlet can be obtained. In the calculation, the distribution function was calculated by a cubic spline fit method by Ying [9],... 

Figure 6-15 Effect of Temperature on SLR —AG (At) of a 1% High-Methoxyl Pectin/60% Sucrose Gel during Heating from 20 to 90°C, Right after a Cooling Scan from 90 to 20°C. denote AG jAt calculated after cubic spline fit the line is for AG jAt calculated after fitting a polynomial equation to the data denote the smoothed AG jAt values calculated after cubic spline fit.

Figure 6-16 Effect of Temperature on SLR of a Cured Gel (20°C, 48 hr) during Heating from 20 to 90°C. Values of AG At values calculated by derivation of a polynomial equation fitted to the data (continuous line), or after cubic spline fit and smoothed using a moving average procedure ( ) ( ) denote G values.

More recently, quadratic and cubic spline fits of the curves were presented [3], The first required a look-up table of eighteen constants. The second required the selection of one of two sets of equations and a 48-element look-up table. 

Figure 1 H + H2, J = 0. (a) Cumulative reaction probability. The solid curve is a spline fit to the accurate quantal results, and the dashed curve was obtained by integrating the synthetic density in b. (b) Density of reactive states. The solid curve is obtained by analytically differentiating a cubic spline fit to the accurate quantum mechanical CRPs. The heavy dashed curve is the fit of Eqs. (14) and (15). The arrows are positioned at the fitted values of T, and the feature numbers and assignments above the arrows correspond to Table 2. (Reprinted with permission from Ref. 8, copyright 1991, American Chemical Society.)...

Fig. 2. Reaction probabilities for D + H2(v = 1, j = 0) HD(v = 0, j = 0) + H are plotted as a function of total energy for all values of J < 19. The solid curves and data points do not include the geometric phase. The short dashed curves and open squares include the geometric phase. The numbers labeling each set of curves denote the value of J. The curves are shifted to make viewing easier. The flat part of the curves near 0.5 eV corresponds to zero probability and indicates the value of the shift. The data points are calculated values and the curves are a cubic spline fit.

Once the radical concentration as a function of time has been determined, the final step in determining the termination rate coefficient can be taken. This involves the differentiation of this radical concentration versus time, according to equation 3.7 (the analytically determined second derivative of the cubic spline fit of [AT] versus time is used for this purpose). In doing so, figure 3.3 is obtained. As expected, the described calculation method yields values for kt which are identical to the input values used in the calculations. 

The discrepancy between input and output values is, however, somewhat greater than in the previous graphs. For intermediate chain lengths, the differences are smaller than 10 % but increases towards smaller chain lengths, approximately up to 110 to 210 %. It is difficult to point out the exact reason for this increased discrepancy towards smaller chain lengths, but it can most probably be attributed to the use of the cubic spline fit procedure. If the correct fitting function (equation 3.2) would be used to fit the monomer versus time profile over the smallest chain lengths, then the inaccuracy becomes much smaller. Hence, the cause can not be found in inaccurate numbers in the calculated monomer consumption trace. Nevertheless,... 

The termination rate coefficient as a function of chain length, calculated from the above graph is shown in figure 3.16, as well as the obtained results had the MWD not been rescaled according to [i ]o. It can be seen that the minor deviation of 2.5% in the number MWD integral due to the use of a cubic spline fit, has indeed a dramatic effect on the determined values of kt if the MWD and [i ]o are not fine tuned to each other. In the... 

The resulting plot is shown in figure 4.10 from which it is clear that the use of a cubic spline fit is a dead end route, not only is the scatter enormous, even negative values are found for the radical concentration versus time. 

Figure 4.9 Dimensionless monomer concentration versus time experimental data (o, identical to figure 4.8) and cubic spline fit (— ).

Calculation was done according to equation 4.25 using a cubic spline fit procedure. 

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