Polynomial Quadrature Method

Computing Free-Energy Profiles Using Multidimensional Potentials of Mean Force and Polynomial Quadrature Methods... 

Up to this point, the development of this method is equivalent to that of the trapezoidal rule. The Gauss quadrature method goes a step beyond this in order to make the error term in Eq. (4.95) vanish. To do so, the integral of the error term is expanded in terms of 2nd-degree Legendre polynomial (see Sec. 3.10) ... 

The criterion to calculate the value of these variables is a variant of the orthogonal collocation method. In a generic iteration, knowing the values of these variables, it is possible to build N piecewise Hermite polynomials. Using these polynomials, it is possible to calculate the value of the N variables and their first and second derivatives in the internal points for each element These internal points are the Gauss quadrature points and they are 2 for a third-order, 3 for the fourth-order, 4 for the fifth-order, and 5 for the sixth-order polynomials. 

It is noticed that a quadrature rule is applied in (12.477) and (12.478). In order to reduce time consuming operations, the quadrature points are normally chosen the same as the collocation points in the approximation of the norm integrals of the least-squares method. The quadrature points and the collocation points are defined at the same locations when both type of points are determined as the roots of the same type of orthogonal polynomial of the same order. In this case [f] = fj coincides with [f] =flQ. 

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