In numerical analysis, GaussâJacobi quadrature (named after Carl Friedrich Gauss and Carl Gustav Jacob Jacobi) is a method of numerical quadrature based on Gaussian quadrature. GaussâJacobi quadrature can be used to approximate integrals of the form
where Æ is a smooth function on [â1, 1] and α, ÎČ > â1. The interval [â1, 1] can be replaced by any other interval by a linear transformation. Thus, GaussâJacobi quadrature can be used to approximate integrals with singularities at the end points. GaussâLegendre quadrature is a special case of GaussâJacobi quadrature with α = ÎČ = 0. Similarly, the ChebyshevâGauss quadrature of the first (second) kind arises when one takes α = ÎČ = â0.5 (+0.5). More generally, the special case α = ÎČ turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called GaussâGegenbauer quadrature.
GaussâJacobi quadrature uses Ï(x) = (1 â x)α (1 + x)ÎČ as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the GaussâJacobi quadrature rule on n points has the form
where x1, âŠ, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, âŠ, λn are given by the formula
where Î denotes the
Gamma function and P(α, ÎČ)
n(x) the Jacobi polynomial of degree n.
The error term (difference between approximate and accurate value) is:
where .
In numerical analysis, GaussâJacobi quadrature (named after Carl Friedrich Gauss and Carl Gustav Jacob Jacobi) is a method of numerical quadrature based on Gaussian quadrature. GaussâJacobi quadrature can be used to approximate integrals of the form
where Æ is a smooth function on [â1, 1] and α, ÎČ > â1. The interval [â1, 1] can be replaced by any other interval by a linear transformation. Thus, GaussâJacobi quadrature can be used to approximate integrals with singularities at the end points. GaussâLegendre quadrature is a special case of GaussâJacobi quadrature with α = ÎČ = 0. Similarly, the ChebyshevâGauss quadrature of the first (second) kind arises when one takes α = ÎČ = â0.5 (+0.5). More generally, the special case α = ÎČ turns Jacobi polynomials into Gegenbauer polynomials, in which case the technique is sometimes called GaussâGegenbauer quadrature.
GaussâJacobi quadrature uses Ï(x) = (1 â x)α (1 + x)ÎČ as the weight function. The corresponding sequence of orthogonal polynomials consist of Jacobi polynomials. Thus, the GaussâJacobi quadrature rule on n points has the form
where x1, âŠ, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, âŠ, λn are given by the formula
where Î denotes the
Gamma function and P(α, ÎČ)
n(x) the Jacobi polynomial of degree n.
The error term (difference between approximate and accurate value) is:
where .