Gauss–Jacobi quadrature
In numerical analysis, Gauss–Jacobi quadrature is a method of numerical quadrature based on Gaussian quadrature. Gauss–Jacobi quadrature can be used to approximate integrals of the form
where ƒ is a smooth function on and. The interval 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. Similarly, the Chebyshev–Gauss quadrature of the first kind arises when one takes. More generally, the special case turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called Gauss–Gegenbauer quadrature.
Gauss–Jacobi quadrature uses as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the Gauss–Jacobi quadrature rule on points has the form
where are the roots of the Jacobi polynomial of degree. The weights are given by the formula
where Γ denotes the Gamma function and the Jacobi polynomial of degree n.
The error term is:
where.
