Gauss–Jacobi quadrature

In numerical analysis, Gauss–Jacobi quadrature (named after Carl Friedrich Gauss and Carl Gustav Jacob Jacobi) is a method of numerical quadrature based on Gaussian quadrature. Gauss–Jacobi quadrature can be used to approximate integrals of the form

$\int _{-1}^{1}f(x)(1-x)^{\alpha }(1+x)^{\beta }\,dx$

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, the Chebyshev–Gauss quadrature of the first (second) kind arises when one takes α = β = −0.5 (+0.5). 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

$\int _{-1}^{1}f(x)(1-x)^{\alpha }(1+x)^{\beta }\,dx\approx \lambda _{1}f(x_{1})+\lambda _{2}f(x_{2})+\ldots +\lambda _{n}f(x_{n}),$

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

$\lambda _{i}=-{\frac {2n+\alpha +\beta +2}{n+\alpha +\beta +1}}\,{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)(n+1)!}}\,{\frac {2^{\alpha +\beta }}{P_{n}^{(\alpha ,\beta )\,\prime }(x_{i})P_{n+1}^{(\alpha ,\beta )}(x_{i})}},$

where Γ denotes the Gamma function and P^{(α, β)}
_n(x) the Jacobi polynomial of degree n.

The error term (difference between approximate and accurate value) is:

$E_{n}={\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)\Gamma (n+\alpha +\beta +1)}{(2n+\alpha +\beta +1)[\Gamma (2n+\alpha +\beta +1)]^{2}}}{\frac {2^{2+\alpha +\beta +1}}{(2n)!}}f^{(2n)}(\xi ),$

where $-1<\xi <1$.

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