Gaussian quadrature orthogonal polynomials

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. 

On each element, i, Np interior collocation points are chosen as the roots of an Npth degree orthogonal polynomial [65]. The first- and second-order differentials with respect to space are then approximated using two matrices, A and B, obtained by solving the Gaussian-Jacobi quadratures [63,65]. For boimdary problems, the endpoints can be included in the calculation of the spatial derivatives. Thus, at the fcth collocation point on the zth element, we have... 

Why are we interested in orthogonal polynomials Because, as will be explained in the next section, their roots are the nodes of Gaussian quadrature approximations. 

Fm basis sets corresponding to classical orthogonal pol3momials (harmonic oscillator functions, Qiebyshev polynomi, Legendre and associated L endre lynomials, etc.) the approximation 2.5 is equivalent to the approximatimi of V by Gaussian quadrature. For... 

In the following, we shall first show how Gaussian quadrature can be simplified for the special case of even polynomials and weight functions such as those in (9.11.3). The orthogonal polynomials needed for the calculation of Coulomb integrals (9.11.3) are then introduced, and finally we show how a Gaussian-quadrature scheme for the evaluation of Coulomb integrals can be developed based on the McMurchie-Davidson and Obara-Saika schemes. 

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