Laguerre quadrature

The Gauss-Laguerre quadrature is defined such that Wma- o ap = a- Furthermore, the second summation in Eq. (8.74) corresponds to the moments of the gamma PDF,... 

As in Eq. (8.102), the weights and abscissas in this expression come from (i) the ECQMOM, (ii) Gauss-Laguerre quadrature, and (iii) Gauss-Hermite quadrature. Likewise, Mi and M2 can be chosen as large as needed in order to minimize the quadrature error in the spatial fluxes and drag terms. 

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. Dover, New York, 1965 the Laguerre quadrature is given on p. 890. 

Table EJ, Quadrature Points and Weights for Laguerre Quadrature...

Rewrite the appropriate parts of the UTILITY routines to incorporate the much more efficient Laguerre quadrature [64.H 67.H], rather than the strip integration which is now used to calculate the rate constant... 

Choice of the numerical quadrature and its optimization have received a great deal of attention (see for example Refs. [24, 25] and references therein for older literature). However, our literature search did not result in an unequivocal decision what could be the best choice for our purpose. After some experimentation, we found that a scaled adaptive Laguerre quadrature with 15 grid points can be used as an universal radial quadrature for low-energy electron-molecule scattering calculations. The radial points were obtained as scaled Laguerre grid points... 

Applying Gauss-Laguerre quadrature (for reference see e.g. [5]), we obtain 1... 

Monteagudo et al characterized the asphaltenes as a continuous ensemble for which the distribution function was taken from the fractal aggregation theory. The asphaltene family was discretized in pseudo-components by the Gauss-Laguerre quadrature. Only the asphaltene polydispersity was taken into account. All other components were represented by as solvent whose properties (molar volume and solubility parameter) were calculated form a cubic equations of state. Aggregation of asphaltenes was considered to be a reversible process. And it was assumed the phase equilibrium was between a liquid phase and a pseudo-liquid phase containing only asphaltenes. 

The use of a finite-basis expansion to represent the continuum is reminiscent of the use of quadratures to represent an integration. Heller, Reinhardt and Yamani (1973) showed that use of the Laguerre basis (5.56) is equivalent to a Gaussian-type quadrature rule. The underlying orthogonal polynomials were shown by Yamani and Reinhardt (1975) to be of the Pollaczek (1950) class. 

Note that a different set of Laguerre polynomials, parameterized by k > 0, is used for each value of a. Thus, the weights and abscissas must be computed separately for each value of a. As with any Gaussian quadrature, Eq. (3.86) is exact when g(f) is a polynomial of order less than 2N (Gautschi, 2004). 

Three related approaches to the calculation of the recombination rate constant are described subsequently, the second two being approximations to the first. In each case the integral over E and the sum over J in the numerator of Eq. (3.3) are approximated by /V-point Laguerre and 2M-point extended Simpson s rule quadratures,30 respectively, yielding... 

The Gauss quadrature formula developed in this section is known as the Gauss-Legendre quadrature because of the use of the Legendre polynomials. Other orthogonal polynomials, such as Chebychev, Laguerre, or Hermite, may be used in a similar manner to develop a variety of Gauss quadrature formulas. 

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