Legendre quadrature

These algorithms are frequently stated in terms of integrals over [-1,1], termed Gauss-Legendre quadrature, and the general formula then is... 

TABLE 3.1 Approximations to the Integral J j sinf-n-x) dx = 1 Using the Trapezoidal and Legendre Quadrature Methods... 

These integrals are evaluated numerically with a Gauss-Legendre quadrature. 

In Fig. 2 we report the spherical average potential Vq(R) for the Ne-N2/ O2 and NO systems compared with the corresponding isotropic Ne-Ar case. The Vq(R) term, i.e., thefirst term in the Legendre expansion or the potential surface, is easily obtained from the full PES by a Gauss-Legendre quadrature, according to the formula ... 

The body-frame matrix is then evaluated using Gauss-Legendre quadrature. In particular, we define the quantities... 

Gauss-Legendre quadrature, there are typically only about a third as many quadrature points as finite difference grid points. In addition, these quadratures... 

Finally, (5.86) is approximated by numerical Gauss-Legendre quadrature after a logarithmic transform of the t variable... 

As our angular grid points we use the Gauss-Legendre quadrature points. The grid representation of the wavefunction, is... 

Nquad — 40 for all subsequent scattering calculations to ensure that exact integration is obtained (provided that tg = 50). We note that Gauss-Legendre quadrature with 40 points exactly integrates a polynomial of degree 80, which is precisely the Newton polynomial used. 

Subroutine GAULEG(XX,WW,NDVR) gives NDVR Gauss-Legendre quadrature points XX(1 NDVR) and the corresponding weights WW(1 NDVR). 

Gauss-Legendre Quadrature Gauss-Legendre quadrature, often simply... 

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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