Computational numerical quadrature

Maurits, N.M., Altevogt, P., Evers, O.A., Fraaije, J.G.E.M. Simple numerical quadrature rules for Gaussian Chain polymer density functional calculations in 3D and implementation on parallel platforms. Comput. Theor. Polymer Sci. 6 (1996) 1-8. 

A typical CFD model for acid base and equilibrium chemistry solves Eqs. (25)-(29), and then uses Eq. (55) to approximate ft. Once ft is known, the expected values of the reactant concentrations are computed by numerical quadrature from the formula... 

For example, (A) and (B) can be computed using Eqs. (49) and (50), respectively. Note that instead of Eq. (55), we could use the simpler expression for given by Eq. (33), which avoids the need for numerical quadrature. In both cases, the mean and variance of the mixture fraction are identical (and thus both models account for finite-rate mixing effects.) In practical applications, the differences in the predicted values of () can often be small (Wang and Fox, 2004). 

This expression looks complicated but can be computed with minimum effort by using numerical quadrature software. Allowing a size distribution for the outgassing of argon from minerals (Turner, 1968) makes it easier to understand why more argon is lost at low temperature from natural crystals than predicted by the uniform size distribution, regardless of mineral geometry, o... 

This expression was computed by the numerical inverse Laplace transform embedded in the numerical quadrature. As previously, we used p = 1,4, 6, E [A] = P2/A2 = 1, and A2 = 4h 1. Figure 9.23 illustrates the influence of the p parameter on the shape of the state probability profile the larger p, the most... 

The computational effort for the formation of the KS (or Fock) matrix is dominated by the evaluation of the two-electron repulsion integrals (ERIs) which are required for the Coulomb and exact-exchange contributions and, in the case of DFT, also the numerical quadrature of the exchange-correlation (XC) contribution. Efforts to accelerate these steps are summarized in Table 1 and reviewed in the remainder of this section. 

In the generalized gradient approximation (GG A) to DFT, the XC potential depends on the electron density p and its gradient Vp and is a complicated function in three-dimensional space. This makes an analytical solution of the XC integrals impossible and numerical quadrature is used to compute the XC matrix elements,... 

Abstract An improvement is made to an automatic quadrature due to Ninomiya (J. Inf. Process. 3 162-170, 1980) of adaptive type based on the Newton-Cotes rule by incorporating a doubly-adaptive algorithm due to Favati, Lotti and Romani (ACM Trans. Math. Softw. 17 207-217,1991 ACM Trans. Math. Softw. 17 218-232, 1991). We compare the present method in performance with some others by using various test problems including Kahaner s ones (Computation of numerical quadrature formulas. In Rice, J.R. (ed.) Mathematical Software, 229-259. Academic, Orlando, FL, 1971). 

Kahaner, D.K. Computation of numerical quadrature formulas. In Rice, J.R. (ed.) Mathematical Software, 229-259. Academic, Orlando, FL (1971)... 

The problem of computing dispersion energies is reduced to the computation of polarizabilities for a sufficient number of frequencies, so that the Casimir-Polder integral can be obtained by numerical quadrature [93]. An alternative to this quadrature is the substitution of the product of the polarizabilities by a sum over Hartree-Fock orbitals... 

This method has proved extremely effective for the computation of AOs for atoms, it is certainly competitive in accuracy with the much more atom-specific method of solution of the radial equation by numerical quadrature. In particular the energy and overlap integrals associated with these basis functions for atoms are quite trivial to evaluate. 

When required, Equation (6.14) should be divided by a symmetry factor to account for the interchange of identical particles in symmetric molecules. While this expression may at first sight seem difficult to evaluate, it can actually be computed quite efficiently. The quantity N E, x) should be precalculated on a large grid of E, x) values which is used to construct an interpolant. The numerical integrations can then be carried out extremely rapidly using numerical quadrature. 

Since the integrations over atomic basins entail numerical quadratures, the populations of AIMs are expensive to compute. Despite the recent improvements in their cost/accuracy ratios,"" it is not uncommon for such calculations to be more time-consuming than the generation of the electronic wavefunction under analysis, particularly in the cases of relatively small systems computed at low levels of theory. 

In this section, the accuracy of the sparse grid collocation technique for computing numerically multidimensional integrals is examined. A test function is examined whose exact numerical integral is available. For the one-dimensional quadrature rules used in the Smolyak s algorithm, only the nested quadrature rules, namely, the Clenshaw-Curtis(CC) rule and the Gauss-Patterson(GP) rule, have been used. 

To compute the 2N + 1 coefficients of this expansion, we might consider using numerical quadrature for the necessary integrals however, as TV increases, so does the required number of quadrature points, as the sine and cosine basis functions vary more rapidly witii increasing m. Thus, the amount of work necessary to obtain an approximate Fourier representation in this manner scales as TV. Below, we consider an alternative method that reqnires only NXogiN TV operations. 

If we were to apply numerical quadrature directly to the formula above, the required work scales with the number of time values NstsN. One may use FFT methods to compute the convolution in a much smaller number of operations that scales only as N o%2N using the convolution theorem. ... 

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