True derivative

The absorbance spectrum in Figure 54-1 is made from synthetic data, but mimics the behavior of real data in that both are represented by data points collected at discrete and (usually) uniform intervals. Therefore the calculation of a derivative from actual data is really the computation of finite differences, usually between adjacent data points. We will now remove the quotation marks from around the term, and simply call all the finite-difference approximations a derivative. As we shall see, however, often data points that are more widely spread are used. If the data points are sufficiently close together, then the approximation to the true derivative can be quite good. Nevertheless, a true derivative can never be measured when real data is involved. 

On the other hand, computing only the numerator term is not recommended when results are to be compared between different instruments or laboratories. It is also not recommended when performing theoretical studies are of interest, or when the results of experiments are to be compared to theoretical expectations, since it does not, in general, reflect the actual value of the true derivative. Given the minor computational burden, however, the proper computation of including the division should always be done. Here we start with the examination of the numerator term alone for its pedagogical value. 

We continue in our next chapter by examining the behavior of the derivative calculation when the division of the Ay term is divided by the AX term, to form an approximation to the true derivative. 

Figure 55-7 First derivatives calculated using different spacings for finite difference approximation to the true derivative. The underlying curve is the 20 run bandwidth absorbance band in Figure 54-1, with data points every nm. Figure 55-7a Difference spacings = 1-5 nm Figure 55-7b Spacings = 5 10 run Figure 55-7c Spacings = 40-90 nm. (see Color Plate 21)...

Can we explain all these effects Of course we can, and in fact the explanation is almost obvious. When spacings are small, the computed derivative is a good approximation to the true derivative. As long as this is the case, the exact value of AX used to compute the derivative is unimportant, because as we saw in Figure 55-5, the first difference AT increases almost linearly with AX, therefore all values of AX give the same result for the computation, because AT/AX is constant regardless of spacing. 

As we observed from Figure 55-5, however, as AX continues to increase, Ay no longer increases proportionately. Strictly speaking, this happens immediately when AX becomes finite, and the question of whether the amount is noticeable is a matter of degree, how much difference it makes in a particular application. Nevertheless, whatever point that is, the initial increase in AX carries a corresponding increase in Ay, and beyond that point it is no longer proportional. At that point, the computed value of the estimate of the true derivative starts to decrease. 

Most of our discussion so far has centered on the use of the two-point-difference method of computing an approximation to the true derivative, but since we have already brought up the Savitzky-Golay method, it is appropriate here to consider both ways of computing derivatives, when considering how they behave when used for quantitative calibration purposes. 

Figure 57-12 This diagram shows how, as the spacing at which the derivative is computed increases, the error in the approximation to the true derivative decreases, even for the same error in the data.

Hyaluronic Add Sulfate.—Meyer and Chaffee108 showed that this mucopolysaccharide of the cornea was a true derivative of hyaluronic acid, since it could be hydrolyzed enzymically by what is now known as hyaluronidase. ... 

Figure 4-25. The top panel displays the true derivatives and those computed as the quotient of differences the middle and bottom panels show the result of a 2nd and 4th degree polynomial fitted through 11 data points.

In most practical cases the original relations (Eq. 16) are nonlinear and the linear least-squares treatment must be iterated to obtain convergence. The elements of the Jacobian X must be recalculated with each new iteration step. Although the least-squares procedure is said to be rather tolerant with respect to the precision of the Jacobian X, true derivatives should be used if ever possible, because finite difference schemes will most often require detailed considerations with respect to the allowed step width. Even then the results may show a tendency to oscillate long before a convergence limit due to the algorithms used or the number of digits carried is reached. With true derivatives, however, this limit is attainable. 

Typke has introduced the rs-fit method [7] where Kraitchman s basic principles are retained. A system of equations is set up for all available isotopomers of a parent (not necessarily singly substituted) and is solved by least-squares methods for the Cartesian coordinates (referred to the PAS of the parent) of all atomic positions that have been substituted on at least one of the isotopomers The positions of unsubstituted atoms need not be known and cannot be determined. The method is presented here with two recent improvements true derivatives are used for the Jacobian matrix X, and the problem of the observations and theircovariances, which is rather elaborate, is fully worked out. The equations are always given for the general asymmetric rotor, noting that simplifications occur in more symmetric situations, e.g. for linear molecules, which could nonetheless be treated within the framework presented. 

The principal considerations in% choosing a finite-difference method for (7) are accuracy, stability, computation time, and computer storage requirements. Accuracy of a method refers to the degree to which the numerically computed temporal and spatial derivatives approximate the true derivatives. Stability considerations place restrictions on the maxi-... 

In Standard Numerov-Cooley, integration or step-by-step use of Eqn. (101) begins both at close-in and at far-out extremes where the values of P and K are near zero, assuming a bound state. These are guessed to be very small values. Then, the inward and outward functions that are obtained are matched in slope and value at some midway point by iterative adjustment of the energy. In Eqn. (101), the zero-order energy E is known already and so the process requires no iteration. This also means the integration needs to be done in only one direction. The F that is found will be a mixture of the true derivative wavefunction and the zero-order wavefunction, and so the last step is a projection step to ensure orthonormality. 

---