Approximate derivative calculation

Eq. (1) can be applied to again, to determine the approximate derivative at the next higher order. The approximate derivative calculation at a higher order can be generalized as ... 

Approximate Derivative Calculated by Using Continnous Wavelet Transform. 

Using the data obtained from the silver nitrate experiments, we have derived a simple approximation to calculate the cavity polarisability as a function of diameter [22]. If we apply this model to cobalt nitrate, the derived threshold for filling is 0.8 nm [32] this result qualitatively agrees with our observations that cobalt nitrate-filled cavities are much narrower ( 2 nm) than obtained with silver nitrate (= 4 nm). 

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)...

If the function were so complex that an analytical derivative could not be obtained explicitly, an approximate derivative would have to be calculated numerically make a small change in temperature AT, evaluate fat T + AT and... 

Approximate derivation o Tobwg)- Given values of and the basic BWG treatment also leads to explicit equations for the ordering temperature, T ", but the omission of sro inevitably leads to calculated values that are appreciably higher than shown by experiment. If the simplicity of the BWG method is to be retained, an empirical correction factor (x) has to be included, where X = Typical equations for various structural transitions are given... 

Equations (14-37) and (14-42) represent two different ways of obtaining an effective factor, and a value of Ae obtained by taking the reciprocal of S, from Eq. (14-42) will not check exactly with a value of A, derived by substituting At = 1/Si and A2 = 1/S2 into Eq. (14-37). Regardless of this fact, the equations generally give reasonable results for approximate design calculations. 

The benefit of QSAR is a more efficient lead optimization process. If a good QSAR formula can be derived, the activity of leads can be approximated by calculation without... 

Graaff R, Aamoudse JG, Zijp JR, Sloot PM A, Demul FFM, Greve J, Koelink MH. Reduced light-scattering properties for mixtures of spherical-particles a simple approximation derived from Mie calculations. Applied Optics 1992, 31, 1370-1376. 

The knowledge of the surface potential for the dispersed systems, such as metal oxide-aqueous electrolyte solution, is based on the model calculations or approximations derived from zeta potential measurements. The direct measurement of this potential with application of field-effect transistor (MOSFET) was proposed by Schenk [108]. These measurements showed that potential is changing far less, then the potential calculated from the Nernst equation with changes of the pH by unit. On the other hand, the pHpzc value obtained for this system, happened to be unexpectedly high for Si02. These experiments ought to be treated cautiously, as the flat structure of the transistor surface differs much from the structure of the surface of dispersed particle. The next problem may be caused by possible contaminants and the surface property changes made by their presence. 

A possible way to solve the convergence problems consists in using high order energy derivatives in the Taylor energy expansion. The drawback is that a higher derivative calculation is really expensive. Consequently some authors had included these terms in some approximate way / 12,1 4, 28/. This, combined, for instance, with the Direct Cl method of Knowles and Handy, using Slater determinants instead of CSF s, overcomes some problems. 

Model-based approaches allow fast derivative computation by relying on a process model, yet only approximate derivatives are obtained. In self-optimizing control [12,21], the idea is to use a plant model to select linear combinations of outputs, the tracking of which results in optimal performance, also in the presence of uncertainty in other words, these linear combinations of outputs approximate the process derivatives. Also, a way of calculating the gradient based on the theory of neighbouring extremals has been presented in [13] however, an important limitation of this approach is that it provides only a first-order approximation and that the accuracy of the derivatives depends strongly on the reliability of the plant model. 

The crudest approximation to y(jc + h) would be h + A f a b The quadratic approximation involves finding the partial derivatives of f and evaluating them at a, b). Since this is inconvenient, the method known as the Runge-Kutta method uses derivatives calculated at certain other points to give an accuracy equivalent to terms of the Taylor series up to and including A y (o)/4 . Denote by the first approximation to y a + /t) — y(a) then... 

---