Gaussian Quadrature

A Gaussian Quadrature is a Numerical Analysis that ...

• See: Polynomial, Orthogonal Polynomials, Numerical Analysis, Integral, Function (Mathematics), Weighted Sum, Numerical Integration, Quadrature (Mathematics), Carl Friedrich Gauss, Singularity (Math), Chebyshev–Gauss Quadrature, Gauss–Hermite Quadrature.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Gaussian_quadrature Retrieved:2017-9-16.
  • In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.

    (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points and weights for i 1, ..., n. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as : [math]\displaystyle{ \int_{-1}^1 f(x)\,dx = \sum_{i=1}^n w_i f(x_i). }[/math] Gaussian quadrature as above will only produce good results if the function f(x) is well approximated by a polynomial function within the range [−1, 1]. The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as [math]\displaystyle{ f(x) = \omega(x) g(x) }[/math] , where g(x) is approximately polynomial and ω(x) is known, then alternative weights [math]\displaystyle{ w_i' }[/math] and points [math]\displaystyle{ x_i' }[/math] that depend on the weighting function ω(x) may give better results, where : [math]\displaystyle{ \int_{-1}^1 f(x)\,dx = \int_{-1}^1 \omega(x) g(x)\,dx \approx \sum_{i=1}^n w_i' g(x_i'). }[/math] Common weighting functions include [math]\displaystyle{ \omega(x)=1/\sqrt{1-x^2} }[/math] (Chebyshev–Gauss) and [math]\displaystyle{ \omega(x)=e^{-x^2} }[/math] (Gauss–Hermite).

    It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials.

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