Gauss–Jacobi quadrature (original) (raw)
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 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 where x1, …, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, …, λn are given by the formula where .
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| dbo:abstract | 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 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 where x1, …, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, …, λn are given by the formula 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: where . (en) |
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| rdfs:comment | 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 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 where x1, …, xn are the roots of the Jacobi polynomial of degree n. The weights λ1, …, λn are given by the formula where . (en) |
| rdfs:label | Gauss–Jacobi quadrature (en) |
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