Gauss–Jacobi quadrature - Alchetron, the free social encyclopedia

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

∫ − 1 1 f ( x ) ( 1 − x ) α ( 1 + x ) β 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

∫ − 1 1 f ( x ) ( 1 − x ) α ( 1 + x ) β d x ≈ λ 1 f ( x 1 ) + λ 2 f ( x 2 ) + ⋯ + λ n f ( x n ) ,

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

λ i = − 2 n + α + β + 2 n + α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) Γ ( n + α + β + 1 ) ( n + 1 ) ! 2 α + β P n ′ ( x i ) P n + 1 ( x i ) ,

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

References

Gauss–Jacobi quadrature Wikipedia

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