Gaussian-quadrature integration

Numerical integration methods are widely used to solve these integrals. The Gauss-Miihler method [28] is employed in all of the calculations used here. This method is a Gaussian quadrature [29] which gives exact answers for Coulomb scattering. 

Gaussian Quadrature Gaussian quadrature provides a highly accurate formula based on irregularly spaced points, but the integral needs to be transformed onto the intei val 0 to I. 

Gaussian quadrature can also be used in two dimensions, provided the integration is on a square or can be transformed to one. (Domain transformations might be used to convert the domain to a square.)... 

In the case of 3b, Gaussian quadrature can be used, choosing the weighting function to remove the singularities from the desired integral. A variable step size differential equation integration routine [38, Chapter 15] produces the only practicable solution to 3c. 

The integrals in (28) and (54) are approximated by q Gaussian quadrature points, so that, for each quadrature energy point in (28), there is a set of 9 + 1 first-order differential equations to be solved, since... 

Two features of the trapezoidal method are that we use a uniform spacing between the positions where we evaluate fix) and that every evaluation of fix) (except the end points) is given equal weight. Neither of these conditions is necessary or even desirable. An elegant class of integration methods called Gaussian quadrature defines methods that have the form... 

For an isotropic medium, by substituting the Gaussian quadrature formula for the integral in Eq. (4.78), the integral-differential equation may be reduced into a system of ordinary linear differential equations. Specifically, the integral can be treated as [Chandrasekhar, 1960]... 

These residuals can be calculated by fitting a polynomial to the converged solution and then evaluating the integrals by Simpson s rule or Gaussian quadrature. The number of quadrature points to be used should be of the order of 3N in the region 0 to X. 

The inner-integral of Equation (2) was numerically integrated using a four-point Gaussian quadrature. The mean bubble length was calculated from the first moment of the frequency distribution function given in Equation (2). 

We wish to add a few more comments on the Gaussian quadrature procedure. Consider an integral of the type... 

With a slightly different approach but within the spirit of Gaussian quadrature, one can also find realistic bounds for the integral... 

With the thermodynamic integration method the difference is evaluated with a 12-point Gaussian quadrature. 

The Gaussian quadrature formula was used to perform the integration. The above procedure was repeated for each isomer in the set finally, the mean thermodynamic quantities (5)-(10) were calculated. 

In Gaussian quadrature theory the NDF is called the weight function or measure. The weight function must be nonnegative and non-null in the integration interval and all its moments. 

The fact that the Jacobi matrix is known can be used to evaluate very accurately integrals involving 6a (x,y) by applying Gaussian quadrature. For example, Eq. (3.84) leads to... 

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