Gauss–Jacobi quadrature

In numerical analysis, Gauss–Jacobi quadrature is a method of numerical quadrature based on Gaussian quadrature. Gauss–Jacobi quadrature can be used to approximate integrals of the form

\int_{-1}^1 f(x) (1-x)^\alpha (1+x)^\beta \,\mathrm{d}x

where ƒ is a smooth function on [−1, 1] and α, β > −1. The interval [−1, 1] can be replaced by any other interval by a linear transformation. Thus, Gauss–Jacobi quadrature can be used to approximate integrals with singularities at the end points. Gauss–Legendre quadrature is a special case of Gauss–Jacobi quadrature with α = β = 0. Similarly, Chebyshev–Gauss quadrature arises when one takes α = β = ±½. More generally, the special case α = β turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called Gauss–Gegenbauer quadrature.

Gauss–Jacobi quadrature uses ω(x) = (1 − x)α (1 + x)β as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the Gauss–Jacobi quadrature rule on n points has the form

\int_{-1}^1 f(x) (1-x)^\alpha (1+x)^\beta \,\mathrm{d}x \approx \lambda_1 f(x_1) + \lambda_2 f(x_2) + \cdots + \lambda_n f(x_n),

where x1, …, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, …, λn are given by the formula

\lambda_i = - \frac{2n+\alpha+\beta+2}{n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)(n+1)!} \frac{2^{\alpha+\beta}}{P'_n(x_i)P_{n+1}(x_i)},

where Γ denotes the Gamma function and Pn the Jacobi polynomial of degree n.

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