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

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

where ƒ is a smooth function on Template:Math and Template:Math. The interval Template:Math 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 Template:Math. Similarly, the Chebyshev–Gauss quadrature of the first (second) kind arises when one takes Template:Math. More generally, the special case Template:Math turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called Gauss–Gegenbauer quadrature.

Gauss–Jacobi quadrature uses Template:Math as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the Gauss–Jacobi quadrature rule on Template:Math points has the form

$\underset{-1}{\overset{1}{\int }}f(x)(1-x{)}^{\alpha }(1+x{)}^{\beta }dx\approx {\lambda }_{1}f(x_1)+{\lambda }_{2}f(x_2)+\dots +{\lambda }_{n}f(x_n),$

where Template:Math are the roots of the Jacobi polynomial of degree Template:Math. The weights Template:Math are given by the formula

${\lambda }_i=-\frac{2n+\alpha +\beta +2}{n+\alpha +\beta +1}\frac{\mathrm{\Gamma }(n+\alpha +1)\mathrm{\Gamma }(n+\beta +1)}{\mathrm{\Gamma }(n+\alpha +\beta +1)(n+1)!}\frac{2^{\alpha +\beta }}{P_n^{(\alpha ,\beta )\prime }(x_i)P_{n+1}^{(\alpha ,\beta )}(x_i)},$

where Γ denotes the Gamma function and Template:Math the Jacobi polynomial of degree n.

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

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

where $-1<\xi <1$.

References

• Template:Citation.

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