Numerical differentiator

In fig. 2 an ideal profile across a pipe is simulated. The unsharpness of the exposure rounds the edges. To detect these edges normally a differentiation is used. Edges are extrema in the second derivative. But a twofold numerical differentiation reduces the signal to noise ratio (SNR) of experimental data considerably. To avoid this a special filter procedure is used as known from Computerised Tomography (CT) /4/. This filter based on Fast Fourier transforms (1 dimensional FFT s) calculates a function like a second derivative based on the first derivative of the profile P (r) ... 

One therefore needs a smooth density estimation techniques that is more reliable than the histogram estimates. The automatic estimation poses additional problems in that the traditional statistical techniques for estimating densities usually require the interactive selection of some smoothing parameter (such as the bin size). Some publicly available density estimators are available, but these tended to oversmooth the densities. So we tried a number of ideas based on numerical differentiation of the empirical cdf to devise a better density estimator. 

The well-known inaccuracy of numerical differentiation precludes the direct calculation of pressure by the insertion of the computed velocity field into Equation (3.6). This problem is, however, very effectively resolved using the following variational recovery method Consider the discretized form of Equation (3.6) given as... 

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

Numerical differentiation should be avoided whenever possible, particularly when data are empirical and subject to appreciable observation errors. Errors in data can affect numeric derivatives quite strongly i.e., differentiation is a roughening process. When such a calculation must be made, it is usually desirable first to smooth the data to a certain extent. 

To avoid numerical differentiation (which is inherently unstable) one uses the fact that an eigenvalue can be expressed as Ai = v Tvf where are the corresponding normalized left and right eigenvectors. Differentiation of the eigenvalue with respect to any parameter is then equivalent to the differentiation of the transfer matrix, and one finds... 

The simple-minded approach for minimizing a function is to step one variable at a time until the function has reached a minimum, and then switch to another variable. This requires only the ability to calculate the function value for a given set of variables. However, as tlie variables are not independent, several cycles through tlie whole set are necessary for finding a minimum. This is impractical for more than 5-10 variables, and may not work anyway. Essentially all optimization metliods used in computational chemistry tlius assume that at least the first derivative of the function with respect to all variables, the gradient g, can be calculated analytically (i.e. directly, and not as a numerical differentiation by stepping the variables). Some metliods also assume that tlie second derivative matrix, the Hessian H, can be calculated. 

The state mixing term, the first in the r.h.s., usually dominates, at least in the presence of avoided crossings. Its determination reduces to a simple problem of interpolation of the Hu matrix elements, according to eq.(16). The second term corresponds, for large R, to the electron translation factor (see for instance [38]). This term depends on the choice of the reference frame that is, for baricentric frames, it depends on the isotopic masses. It contains the Gn matrix, which may be determined by numerical differentiation of the quasi-diabatic wavefunctions [16] this calculation is more demanding, especially in the case of many internal coordinates. It is therefore interesting to adopt the approximation ... 

Since most of the numerical differential equation solvers require the equations to be integrated to be of the form... 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

The tensor of the static first hyperpolarizabilities P is defined as the third derivative of the energy with respect to the electric field components and hence involves one additional field differentiation compared to polarizabilities. Implementations employing analytic derivatives in the Kohn-Sham framework have been described by Colwell et al., 1993, and Lee and Colwell, 1994, for LDA and GGA functionals, respectively. If no analytic derivatives are available, some finite field approximation is used. In these cases the P tensor is preferably computed by numerically differentiating the analytically obtained polarizabilities. In this way only one non-analytical step, susceptible to numerical noise, is involved. Just as for polarizabilities, the individual tensor components are not regularly reported, but rather... 

To evaluate the concentration dependence of D, a series of sorption experiments are performed at successive concentration intervals. A numerical differentiation is then performed on the plot of Da(c ) versus r to obtain a first approximation for the relationship between Da and cx. Similarly, the cx dependence of Dd, where d denotes desorption, can be determined from desorption data. This estimation method works quite satisfactorily for cases where D has a mild dependence on Ci and both Da(cx) and Dd(cx) give good estimates of D(cx). It is to be... 

In experimental kinetics studies one measures (directly, or indirectly) the concentration of one or more of the reactant and/or product species as a function of time. If these concentrations are plotted against time, smooth curves should be obtained. For constant volume systems the reaction rate may be obtained by graphical or numerical differentiation of the data. For variable volume systems, additional numerical manipulations are necessary, but the process of determining the reaction rate still involves differentiation of some form of the data. For example,... 

The term in parentheses on the left side of equation B may be determined from the data in several ways. The bromine concentration may be plotted as a function of time and the slope of the curve at various times determined graphically. Alternatively, any of several methods of numerical differentiation may be employed. The simplest of these is used in Table 3.1.1 where dC/dt is approximated by AC/At. Mean bromine concentrations corresponding to each derivative are also tabulated. 

In order to ameliorate the sharply sloping background obtained in an STS spectrum, the data are often presented as di,/dFh vs. Vb, i.e. the data are either numerically differentiated after collection or Vb has a small modulation applied on top of the ramp, and the differential di,/d Vb is measured directly as a function of Vb. The ripples due to the presence of LDOS are now manifest as clear peaks in the differential plot. dt,/dFb vs. Vb curves are often referred to as conductance plots and directly reflect the spatial distribution of the surface electronic states they may be used to identify the energy of a state and its associated width. If V is the bias potential at which the onset of a ripple in the ijV plot occurs, or the onset of the corresponding peak in the dt/dF plot, then the energy of the localised surface state is e0 x F. Some caution must be exercised in interpreting the differential plots, however, since... 

The modeling of the internal pore diffusion of a wax-filled cylindrical single catalyst pore was accomplished by the software Comsol Multiphysics (from Comsol AB, Stockholm, Sweden) as well as by Presto Kinetics (from CiT, Rastede, Germany). Both are numerical differential equation solvers and are based on a three-dimensional finite element method. Presto Kinetics displays the results in the form of diagrams. Comsol Multiphysics, instead, provides a three-dimensional solution of the problem. 

Differentiation. The main application of numerical differentiation in this book is to find a rate, dC/dt, from tabular data (C, t). In view of the... 

Numerical differentiation may be quite sensitive to the correlating equation. In problem PI.03.01, the results with four different curvefits do not agree well although the curvefits themselves are statistically satisfactory. In problem PI.0302, however, the agreement between the higher polynomial fits is more nearly acceptable. 

In addition, by numerically differentiating the analytical gradients, the harmonic vibrational frequencies can be obtained. 

For reaction mechanisms that have explicit solutions to the set of differential equations, it is always also possible to define the derivatives dC /dp explicitly. In such cases the Jacobian can be calculated in explicit equations and time consuming numerical differentiations are not required. The equations are rather complex, although implementation in Matlab is straightforward. The calculation of numerical derivatives is always possible and for mechanisms that require numerical integration, it is the only option. 

The Newton-Raphson procedure was used to find e satisfying F(e) = 0. Iterations began at high conversion and the derivative dF/de was found by numerical differentiation. Convergence was obtained in 5 iterations, with 10 critical point evaluations, in about 10 seconds. The computer used was the University of Calgary Honeywell HIS-Multics system. 

Since the operations to get frequency response from step-test data involve numerical differentiation of the data, the results are less reliable than pulse test data as frequency is increased. 

By calculating A.U (R) and Al/ (i ) separately, we can straightforwardly calculate the total adiabatic correction V (R) for any isotopes of A and B. The adiabatic corrections are calculated by numerical differentiation of the multi-configurational self-consistent field (MCSCF) wave functions calculated with Dalton [23]. The nurnerical differentiation was performed with the Westa program developed 1986 by Agren, Flores-Riveros and Jensen [22],... 

Thus, the slope of a plot of In P against 1/Tis AH /ZR, and numerical differentiation (Appendix A) of experimental vapor-pressure data will provide values of AHm/Z as a function of temperamre and pressure. If Z is known, AH can be calculated. 

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