Gauss-Jacobi Quadrature

We have discussed the differentiation of the interpolation polynomial. Now, we turn to the process of quadrature. This is often needed in solutions, such as chemical reaction rates obtained as an integration of a concentration profile. 

The interpolation polynomial, yyv i(x) of Eq. 8.104, is continuous, and therefore we can integrate it with respect to x, using a weighting function W(x), as 

When we interchange the summation sign and the integral sign, the so-called quadrature relation arises 

For a specific choice of weighting function, say the Jacobi weight 

If the N interpolation points are chosen as N zeroes of the Jacobian polynomial of degree N, the quadrature is called the Gauss-Jacobi quadrature, and Wj are called the Gauss-Jacobi quadrature weights. 

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For computational purposes, the following formula for the Gauss-Jacobi quadrature weights can be obtained using the properties of the Lagrangian interpolation polynomials lj x)... 

In many applications it is desirable to generate Gauss-type quadrature rules with preassigned abscissas, e.g. for solving boundary value problems. For the Gauss-Lob atto-Jacobi quadratures, we prescribe go = -1 and gp = 1. A matrix J must thus be constructed such that Amin (7) = -1 and Amax(T) = 1. This implies that a polynomial pp+i(g) must be determined so that ... 

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