Gaussian numerical integration

Numerical integration methods are widely used to solve these integrals. The Gauss-Miihler method [28] is employed in all of the calculations used here. This method is a Gaussian quadrature [29] which gives exact answers for Coulomb scattering. 

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

If P0(AU) is Gaussian, there is, of course, no reason to carry out a numerical integration, since the integral in (2.15) can be readily evaluated analytically. This yields... 

The oscillator strength for absorption is a very important quantity signifying the nature of the transition. If the absorption spectrum is known, the oscillator strength can be calculated using Eq. (4.20). Instead of numerical integration, one often assumes that the spectrum is approximately gaussian with the same half-width Av (cm-1) as experimentally observed. One then obtains/, the oscillator strength, as... 

Measurements for both state variables, A and T, and both input variables, Aq and To, were simulated at time steps of 2.5 s by adding Gaussian noise to the true values obtained through numerical integration of the dynamic equations. A measurement error with a standard deviation of 5% of the correspoding reference value was considered and the reconciliation of all measured variables (two states and two inputs) was carried out. 

The results of the numerical integration of (5.77) for a Gaussian contour [38] are presented in Fig. 5.12. We may thus conclude that a situation where cos and are of the same order of magnitude is perfectly feasible. 

The inner-integral of Equation (2) was numerically integrated using a four-point Gaussian quadrature. The mean bubble length was calculated from the first moment of the frequency distribution function given in Equation (2). 

The most complicated part of density functional researches is the selection of simple, accurate and fast method for numerical integration over dilferent functionals. For numerical integration it is best to use a spherical polar coordinate system (r,0,volume integral for a given function is a familiar expression ... 

The performance of the robust estimators has been tested on the same CSTR used by Liebman et al (1992) where the four variables in the system were assumed to be measured. The two input variables are the feed concentration and temperature while the two state variables are the output concentration and temperature. Measurements for both state and input variables were simulated at time steps of 1 (scaled time value corresponding to 2.5 s) adding Gaussian noise with a standard deviation of 5% of the reference values (see Liebman et al, 1992) to the true values obtained from the numerical integration of the reactor dynamic model. Same outliers and a bias were added to the simulated measurements. The simulation was initialized at a scaled steady state operation point (feed concentration = 6.5, feed temperature = 3.5, output concentration = 0.1531 and output temperature = 4.6091). At time step 30 the feed concentration was stepped to 7.5. 

The DV-Xa calculations were made with C, symmetry for models A, C, and D, and without symmetry for model B. Numerical atomic orbitals of lx to 5p for Cu, and lx to 2p for C, N and O, and lx for H were used as a basis set for the DV-Xa calculations in the ground and transition states. The sample points used in the numerical integration were taken up to 30000 for each calculation. Self-consistency within 0.001 electrons was obtained for the final orbital populations. Transition probabilities calculated for each model were convoluted by a Gaussian function with a half-width at half-height (HWHH) of 1.0 eV to make transition peak shapes comparable with experimental XANES spectra. 

Analytic and numerical integration is performed using unnormalized Cartesian Gaussian functions. We define such a function as... 

The form of the basis set expansion is chosen to be convenient to model the system and to perform the integrals needed to solve the coupled Eqs. (13). Examples of convenient forms are Gaussians, numerical grids, and plane waves. Plane waves have often (but not exclusively) been chosen as the basis set expansion used in Car-Parrinello simulations for two reasons (i) It is generally desired to simulate extended systems, such as bulk materials, surfaces, and hquids, and plane waves provide a convenient way to model these systems using periodic boundary conditions (ii) forces on atomic nuclei can be calculated very efficiently if the electrons are described by plane waves by making use of the Heilman-Feynman theorem (yide infra). 

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