Numerical differentiation formulas

Numerical Method. Numerical differentiation formulas can be used when the data points in the independent variable are equally spaced, such as — — tj At. 

Thus, we plot a dP/dt) versus hi[3 —Pit)/P Qi)] to determine the value of a. We calculate the derivatives numerically, using second-order forward, center, and backward numerical differentiation formulas, and obtain the following ... 

The derivatives and their associated maximum tolerances for the known reactants are calculated by using numerical differentiation formulas. 

In the final step DHPCG calculates the NSIM derivatives for the reactants that are being simulated. Since the derivative forms of T37pe (1), (2) and (3) equations are all linear with respect to the CP (i) s, they are solved simultaneously for the CP(iys by the Gaussian elimination method. The partial derivatives used in the evaluation of the derivative form of the Type (1) equation are calculated with the same numerical differentiation formula that is used in the NONLIN module. 

A second point to consider in constructing Vg(s) is that if numerical differentiation is used to calculate the force constant matrix, then the results may be sensitive to the distribution of points and the step size used in the difference formulas. We have found that frequencies calculated using the GAMESS codes can vary significantly based upon using 2- or 3-point numerical differentiation formulas and a step size ranging from 0.01 to 0.0001 aQ. Of course, for SCF calculations this problem is eliminated with the use of analytic second derivatives as used by Colwell and Handy. 

The Dunham coefficients Yy are related to the spectroscopical parameters as follows 7io = cOe to the fundamental vibrational frequency, Y20 = cOeXe to the anharmonicity constant, Y02 = D to the centrifugal distortion constant, Yn = oie to the vibrational-rotational interaction constant, and Ym = / to the rotational constant. These coefficients can be expressed in terms of different derivatives of U R) at the equilibrium point, r=Re. The derivatives can be either calculated analytically or by using numerical differentiation applied to the PEC points. The numerical differentiation of the total energy of the system, Ecasccsd, point by point is the simplest way to obtain the parameters. In our works we have used the standard five-point numerical differentiation formula. In the comparison of the calculated values with the experimental results we utilize the experimental PECs obtained with the Rydberg-Klein-Rees (RKR) approach [58-60] and with the inverted perturbation approach (IPA) [61,62]. The IPA is method originally intended to improve the RKR potentials. 

For the vibrational frequencies, the CAS(2,2)CCSD method performs noticeably better than the CAS(2,2)CISD[+Q]. This can be seen by examining the standard deviations shown in Table 3.9. In Table 3.11 some selected spectroscopic constants calculated for FH in the ground eiectronic state (X S+) are shown. The results are compared with the experimental values taken from Huber and Herzberg [64], with the exception of the dissociation energy, Dg, which was taken from Lonardo and Douglas [74]. To calculate the spectroscopic constants we used the numerical differentiation formulas. As one can expect, the values of the spectroscopic constants... 

MATLAB has two new stiff integration routines. These are ode 15s and ode23s. The routine odelSs is a variable order (up to order 5) and a variable step size program that is based upon the Klopfenstein modification of classical backward difference formulas called numerical differential formulas (Klopfenstein, 1971). Standard backward difference formulas are also available as an option. In order to determine optimum step size and speed convergence of the implicit corrector formulas, the method depends upon the Jacobian, J, of the derivative function / in... 

Klopfenstein, R. W., Numerical Differentiation Formulas for Stiff Systems of Ordinary Differential Equations, RCA Review 32, 447-462 (1971). 

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

The familiar formulas of numerical differentiation are the derivatives of local interpolating polynomials. All such formulas give bad estimates if there are errors in the data. To illustrate this point consider the case of linear... 

It follows that the formulas of numerical differentiation do not apply to noisy sequence of data. Formulas based on the differentiation of local smoothing polynomials perform somewhat better. These are also of the form... 

The ground state force field, vibrational normal modes and frequencies have been obtained with MCSCF analytic gradient and hessian calculations [176]. Frequencies computed with the DZ basis set are compared with experimental ones in Table 16. The T - So transition moments were obtained using distorted benzene geometries with atomic displacements along the normal modes, and with the derivatives in Eq. 97 obtained by numerical differentiation. The normal modes active for phosphorescence in benzene are depicted in Fig. 12. The final formula for the radiative lifetime of the k spin sublevel produced by radiation in all (i/f) bands is (ZFS representation x,y,z is used [49]) ... 

Quadratic terms in die property expansions are considered to be first-order in electrical anharmonicity, cubic terms are taken to be second-order, etc. Similarly, cubic terms in the vibrational potential are considered to be first-order in mechanical anharmonicity, quartic terms are second-order, and so forth. The notation (n, m) is used hereafter for the order of electrical (n) and mechanical (m) anharmonicity whereas the total order (n -I- m) is denoted by I, II,. Although our definition of orders is reasonable other choices are possible. Two key questions are (1) How important are anharmonicity contributions to vibrational NLO properties and (2) What is the convergence behavior of the double perturbation series in electrical and mechanical anharmonicity Both questions will be addressed later. Here we note that compact expressions, complete through order II in electrical plus mechanical anharmonicity, have been presented [19]. The formulas of order I contain either cubic force constants or second derivatives of the electrical properties with respect to the normal coordinates. Depending upon the level of calculation at least one order of numerical differentiation is ordinarily required to determine these anharmonicity parameters. For electrical properties, the additional normal coordinate derivative may be replaced by an electric field derivative using relations such as d p./dQidQj = —d E/dldAj.ACd, = —dk,/rjF where F is the field and k j is... 

The NONLIN module is responsible for intializing the concentration vector, C(t), for l i NRCT. Here NRCT is the number of reactants. If there are no equilibrium reactions, then C i) is set to IC i), the initial concentration vector, for 1 < f < NRCT. If equilibrium reactions do exist, then the type (2) equations (with derivatives set to zero) and the Type (1) and Type (3) equations are all solved simultaneously for the equilibrium concentrations of all reactants. Because the equilibrium equations are generally nonlinear, the Newton-Raphson iteration method is used to solve these equations. Also, since there is no symbol manipulation capability in the current version of CRAMS, numerical differentiation is used to calculate the required partial derivatives. That is, the rate expressions cannot at this time be automatically differentiated by analytical methods. A three point differentiation formula is used 27) ... 

Step 2. Numerically differentiate ptotai vs. t to generate dptou /dt via an nth-order-correct finite difference formula at each discrete data point. 

From a fundamental point of view, integration is less demanding than differentiation, as far as the conditions imposed on the class of functions. As a consequence, numerical integration is a lot easier to carry out than numerical differentiation. If we seek explicit functional forms (sometimes referred to as closed forms) for the two operations of calculus, the situation is reversed. You can find a closed form for the derivative of almost any function. But even some simple functional forms cannot be integrated expliciUy, at least not in terms of elementary functions. For example, there are no simple formulas for the indefinite integrals J e dx or J dx. These can, however, be used for definite new functions, namely, the error function and the exponential integral, respectively. 

The formulae were incorporated in a computer program [82], The input is an experimental array of the 1st derivative peak-to-peak ESR amplitude against the microwave power that is read from a previously prepared file and initial trial parameter values for the Lorentzian and Gaussian line-widths and the microwave power Po) at saturation that are provided interactively. The corresponding theoretical amplitudes were obtained by numerical differentiation of the Voigt function (1). A non-linear least squares fit of the calculated saturation curve to the experimental data is performed. Output data consist of a graph of the experimental data and the... 

It is useful at this point to note that the Newton forward difference formula is utilized here for the development of the numerical integration formula, while the Newton backward difference formula was previously used (in Chapter 7) for the integration of ordinary differential equations of the initial value type. 

A characteristic of DAEs besides their form is their differentiation index [32]. For a definition and an example see Appendix C. It is an indicator for the problems to be encountered with the numerical solution of a set of DAEs. Systems of index > 1 are usually called higher index DAEs and the higher the index the more severe numerical difficulties can be. As the mathematical description of problems in various disciplines often leads to DAE system, they have been a research subject for more than two decades. A large body of publications and a number software programs for their numerical solution have emerged. DAE systems of index 1 can be safely numerically computed by means of the backward differentiation formula (BDF) [33, 34] implemented in solvers such as the well known and widely used DASSL code [35]. 

The derivatives of energy in the above equations can be obtained by numerical differentiation (Finite Field approach, FF). For instance, to calculate the second- and fourth-order derivatives which correspond to polarizability and 2nd hyperpolarizability the following seven point formulaes can be used ... 

Several algorithms have been developed to obtain the set of equations. The result is a set of differential-algebraic equations (DAEs) solved using a backward differential formulae (BDF) numerical method. 

Differential Method In order to use the differential method of data analysis, it is necessary to differentiate the reactant concentration versus space-time data obtained in a plug-flow PBR. There are three methods of differentiation that are commonly used (i) graphical equal-area differentiation, (ii) numerical differentiation or finite difference formulas, and (iii) polynomial fit to the data followed by analytical differentiation. The aim of differentiation is to obtain point values of the reaction rate ( Ra)p at each reactant concentration Q4 or conversion xa or space time (.W/Fao), as required. All three differentiation methods can introduce some error to the evaluation of -Ra)p- Information on and illustration of the various differentiation techniques are available in the literature [23, 26]. 

Other modes of numerical differentiations are based on Lagrangian differentiation formulas, Thylor exptmsions, and Spline and Splaus functions. Also, differentiation is used after discrete Fourier transformation. In this case, noise can be eliminated by neglecting the higher terms of the polynomial, but often it is not etisy to rind the limit of noise frequencies and the higher frequencies of the effective signals. 

---