Gaussian Quadrature - Other Forms

Other Forms

The integration problem can be expressed in a slightly more general way by introducing a positive weight function ω into the integrand, and allowing an interval other than . That is, the problem is to calculate

for some choices of a, b, and ω. For a = −1, b = 1, and ω(x) = 1, the problem is the same as that considered above. Other choices lead to other integration rules. Some of these are tabulated below. Equation numbers are given for Abramowitz and Stegun (A & S).

Interval ω(x) Orthogonal polynomials A & S For more information, see ...
Legendre polynomials 25.4.29 Section Gauss–Legendre quadrature, above
(−1, 1) Jacobi polynomials 25.4.33 Gauss–Jacobi quadrature
(−1, 1) Chebyshev polynomials (first kind) 25.4.38 Chebyshev–Gauss quadrature
Chebyshev polynomials (second kind) 25.4.40 Chebyshev–Gauss quadrature
[0, ∞) Laguerre polynomials 25.4.45 Gauss–Laguerre quadrature
[0, ∞) Generalized Laguerre polynomials Gauss–Laguerre quadrature
(−∞, ∞) Hermite polynomials 25.4.46 Gauss–Hermite quadrature

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