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

:$\int_^1 f(x) (1 - x)^\alpha (1 + x)^\beta \,dx$

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 (second) 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

:$\int_^1 f(x) (1 - x)^\alpha (1 + x)^\beta \,dx \approx \lambda_1 f(x_1) + \lambda_2 f(x_2) + \ldots + \lambda_n f(x_n),$

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

:$\lambda_i = -\frac \, \frac \, \frac ,$

where Γ denotes the Gamma function and the Jacobi polynomial of degree ''n''.

The error term (difference between approximate and accurate value) is:
:$E_n = \frac \frac f^(\xi),$
where $-1 < \xi < 1$.

References

* .

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

{{DEFAULTSORT:Gauss-Jacobi quadrature
Numerical integration