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 $x_{1}, …, x_{n}$ are the roots of the Jacobi polynomial of degree $n$. The weights $λ_{1}, …, λ_{n}$ 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$$.