Quadrature points

The accuracy of this method increases nhen increasing M in equation 33, i. e. the dimension of the system of ordinary differential equations 36. Usually, due to the monomiodal sh ie of the PSD considered in this work, M = 3 provides a satisfactory approxinatlon of the solution for the same reason, a low mmher of quadrature points (<5) is required in the evaluation of the integral terms in equations 20, 21 and 36. 

Discretized Path Integral Exact Quantum Result Quadrature Points... 

An advantage of this technique is that it allows for getting as many sample points as needed at each location along the interval of interest. In particular, it is possible to obtain very good statistics even in transition regions which are rarely visited otherwise. This leads in general to an efficient calculation and small statistical errors. Nevertheless, despite its many successes, this method has some difficulties. First, the system needs to be prepared such that has the desired value (at which dA/d needs to be computed) and an equilibration run needs to be performed at this value of . Second, it is not always obvious to determine how many quadrature points are needed to calculate the integral (deff j<9 ) d . 

Calculations are usually performed by considering a set of quadrature points Xk between 0 and 1 and associated weights cvfc, e.g., Gaussian points and weights. At each Xk, an MD simulation is performed and the average of <)MX (x, p., )/dX is computed. Finally the free energy is computed using... 

Figure 4 Distribution of Lanczos eigenvalues in the H02 system (adapted with permission from Ref. 40) and Gauss-Chebyshev quadrature points.

The integrals in (28) and (54) are approximated by q Gaussian quadrature points, so that, for each quadrature energy point in (28), there is a set of 9 + 1 first-order differential equations to be solved, since... 

These residuals can be calculated by fitting a polynomial to the converged solution and then evaluating the integrals by Simpson s rule or Gaussian quadrature. The number of quadrature points to be used should be of the order of 3N in the region 0 to X. 

Formulated in this way, the problem reduces to one originally considered by Gauss (see, e.g., Stroud and Secrest, 1966). Shibata et al. (1987) review and extend methods for doing that. The important point is that theorems are available that can be used to determine the best quadrature points and the corresponding weighting constants, while in the case of a discrete description the choice of... 

In fact only a finite number of bound-state functions are needed for convergence. The role of each bound state in the numerical solution is as another quadrature point for the appropriate value of L. 

The direct amplitudes involving are analogous to the distorted-wave Born approximation and are calculated by (10.31). The T-matrix element in the second amplitude of (10.51), which has the observed resonances, is calculated by solving the problem of electron scattering on He" ". The solution consists of half-on-shell T-matrix elements at the quadrature points for the scattering integral equations (6.87). The same points are used for the k integration of (10.51). 

Index set of Gaussian quadrature (or collocation) points RTD function at yth quadrature point in ith element (point [i,j]) Dimensionless concentration at point [i,j]... 

As in the monovariate case, the one-point quadratures used to approximate the source terms on the right-hand side are usually inaccurate. Thus, only cases with A > 1 will normally be of interest. For A = 2, the NDF is described by two quadrature points. Since each point a in phase space is determined by three parameters w , i,a, 2,a, six mixed moments are required in order to close the problem. There is no unique way to choose the moments. 

Alopaeus, V., Laakkonen, M. Aittamaa, J. 2006 Numerical solution of moment-transformed population balance equation with fixed quadrature points. Chemical Engineering Science 61, 4919-4929. 

For a general set of points these equations are nonsymmetric due to the non-Hermiticity of although in certain cases in which Gauss quadrature points are utilized, Goodisman has shown this quantity to be Hermitian. 

Based on the Christoffel-Darboux formula (20) it can be shown that this procedure leads to a functional expansion which becomes an interpolation formula on the integration points, vK<7 ) = (<7 )- As an example consider an expansion by the Chebychev orthogonal polynomials g,(q) = T (q) with the constant weights W(qi) = 2/tt. The quadrature points q, are the zeros of the Chebychev polynomial of degree N + 1. On inserting Eq. (16) into the functional expansion Eq. (2) becomes... 

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