I am wondering if anyone can point me to a Gauss quadrature rule on $[0,\infty)$ with $w(x)=x^2\ \mathrm{exp}(-x^2)$. The most similar thing that I can find is the one that is based on the generalized Laguerre polynomial with a weighting function $w(x)=x^a\ \mathrm{exp}(-x)$ here. Thanks!
Gauss quadrature rule with a specific weighting function
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integration
numerical-methods
orthogonal-polynomials
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0Should be this Shizgal, B. (1981). A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. Journal of Computational Physics 41, 309–328, http://dx.doi.org/10.1016/0021-9991(81)90099-1 – 2017-02-06
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0Thanks! That is exactly what I am looking for. – 2017-02-06
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0Would you be interested in an algorithm for creating such quadrature rules? There is not just a single Gauss quadrature rule with given weight and interval $[0,\infty)$, but rather a family of rules of varying orders of accuracy. – 2017-02-09
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0@hardmath I think the paper that slitvinov does provide a set of formula to calculate rules for any order of accuracy, for p = 0, 1, 2. However if you know of a more general algorithm please share with us : ) – 2017-02-12
1 Answers
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Shizgal, B. (1981). A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. Journal of Computational Physics 41, 309–328 doi:10.1016/0021-9991(81)90099-1
A new Gaussian quadrature procedure is developed for integrals of the form $\int_0^\infty \, e^{-y^2} y^p F(y) \, dy $ for $p$ = $0$, $1$ and $2$. Recursion relations are derived for the coefficients in the general three term recurrence relation for the polynomials whose roots are the quadrature abscissae. A comparison with the Gauss-Laguerre quadrature procedure is presented. Solutions of the chemical kinetic Boltzmann equation are obtained with a discrete ordinate method based on this Gaussian quadrature procedure. The results are compared with previous solutions obtained with a polynomial expansion method.