Gauss–Jacobi quadrature

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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

∫ − 1 1 f ( x ) ( 1 − x ) α ( 1 + x ) β d x {\displaystyle \int _{-1}^{1}f(x)(1-x)^{\alpha }(1+x)^{\beta }\,dx} {\displaystyle \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

∫ − 1 1 f ( x ) ( 1 − x ) α ( 1 + x ) β d x ≈ λ 1 f ( x 1 ) + λ 2 f ( x 2 ) + … + λ n f ( x n ) , {\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}),} {\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}),}

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

λ i = − 2 n + α + β + 2 n + α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) Γ ( n + α + β + 1 ) ( n + 1 ) ! 2 α + β P n ( α , β ) ′ ( x i ) P n + 1 ( α , β ) ( x i ) , {\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})}},} {\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})}},}

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 = Γ ( n + α + 1 ) Γ ( n + β + 1 ) Γ ( n + α + β + 1 ) ( 2 n + α + β + 1 ) [ Γ ( 2 n + α + β + 1 ) ] 2 2 2 + α + β + 1 ( 2 n ) ! f ( 2 n ) ( ξ ) , {\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 ),} {\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 ),}

where − 1 < ξ < 1 {\displaystyle -1<\xi <1} {\displaystyle -1<\xi <1}.

References

  • 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