Gaussian quadrature formulas

Using the Gaussian quadrature formulas gives the following values for various ... 

Some other typical Gaussian quadrature formulas are (a,b) W(x) Gauss-... 

Weights and zeros for the above formulas (and for other Gaussian formulas) may he found in references such as Stroud Gaussian Quadrature Formulas, Prentice-Hall, 1966). 

Using the Gaussian quadrature formulas gives the following values for various values ofm. Clearly, three internal points, requiring evaluation of the integrand at only three points, gives excellent results. 

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]... 

Stroud, A. J., and Secrest, D., Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, NJ, 1966. 

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. 

Gaussian quadrature is particularly useful in the case where the integrand y(x) is known that is, we have the flexibility of choosing the discrete points, Xj (also called the quadrature points). The Gaussian quadrature formula for N quadrature points is of the form... 

The definition of the Gaussian quadrature formula in Eq. E.30 implies that the determination of this formula is determined by the selection of N quadrature points and N quadrature weights that is, we have 2N parameters to be found. With these degrees of freedom (2N parameters), it is possible to fit a polynomial of degree 2N - 1. This means that if Xj and are properly chosen, the Gausssian quadrature formula can exactly integrate a polynomial of degree up to 2N — 1. 

This set of four parameters for the Gaussian quadrature formula can integrate any polynomial up to degree 3 exactly, for only two quadrature points. 

Thus, we see that the Gaussian quadrature formula using N quadrature points... 

We can also apply the Gaussian quadrature formula to the case when we have a weighting function in the integrand as... 

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