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
- [math]\displaystyle{ \int_{-1}^1 f(x) (1 - x)^\alpha (1 + x)^\beta \,dx }[/math]
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
- [math]\displaystyle{ \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), }[/math]
where x1, …, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, …, λn are given by the formula
- [math]\displaystyle{ \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)}, }[/math]
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:
- [math]\displaystyle{ 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), }[/math]
where [math]\displaystyle{ -1 \lt \xi \lt 1 }[/math].
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
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