Integration Gauss-Legendre numerical

For the radial part the Gauss-Legendre method was used for numerical integration of different functionals using 40 radial points. 

The classical product rules of numerical integration [29] are capable of integrating limited regions around an atom to high precision. They are typically of the form of a Gauss-Legendre polynomial rule in (6, ), multiplied by a radial rule adapted for the asymptotic exponential decay of integrand with distance from the nucleus ... 

Gauss quadratures are numerical integration methods that employ Legendre points. Gauss quadrature cannot integrate a function given in a tabular form with equispaced intervals. It is expressed as ... 

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

These integrals must be evaluated numerically and we do so using the Euler-Maclaurin [26] method for the radial integrations and the Gauss-Legendre [27] method for the angular integration. 

The numerical integration also can be used to calculate the matrix elements of the exchange-correlation potential. For the numerical integration, the atomic partition method proposed by Savin [392] and Becke [393] has been adopted and combined with Gauss-Legendre (radial) and Lebedev (angular) quadratures [394]. The Kohn-Sham LCAO periodic method based on numerical integration at each cycle of the self-consistent-field process is computationally more expensive than the periodic LCAO Hartree-Fock method that is almost fully analytical. 

GAUSSLEGENDRE( FA,B,H,N) numerically evaluates the % integral of the function described by M-file F.M from A to B, % using interval spacing H, by a N-point Gauss-Legendre % quadrature,... 

In [167] the authors obtain a new collocation methods for the numerical solution of second order initial-value problems. This method is based on the approximation of the solutions by the Legendre-Gauss Interpolation. They propose also a multistep version of this method. This multistep version is proved that is very efficient for long time integrations. Numerical results show the efficiency of the new developed methods. 

---