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, …, x_n are the roots of the Jacobi polynomial of degree n. The weights λ₁, …, λ_n 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 P_n the Jacobi polynomial of degree n.

References

• Rabinowitz, Philip (2001), "§4.8-1: Gauss–Jacobi quadrature", A First Course in Numerical Analysis (2nd ed.), New York: Dover Publications, ISBN 978-0-486-41454-6.

External links

• Jacobi rule - free software (Matlab, C++, and Fortran) to evaluate integrals by Gauss–Jacobi quadrature rules.
• Gegenbauer rule - free software (Matlab, C++, and Fortran) for Gauss–Gegenbauer quadrature