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

$\lambda _{i}=-{\frac {2n+\alpha +\beta +2}{n+\alpha +\beta +1}}\,{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)(n+1)!}}\,{\frac {2^{\alpha +\beta }}{P_{n}^{(\alpha ,\beta )\,\prime }(x_{i})P_{n+1}^{(\alpha ,\beta )}(x_{i})}},$ where Γ denotes the Gamma function and P(α, β)
n
(x)
the Jacobi polynomial of degree n.