Laurent-Hermite-Gauss Quadrature

This paper exends the results presented in Gustafson and Hagler (in press) by explicating the (2n)-point Laurent-Hermite-Gauss quadrature formula of parameters γ, λ > 0: ʃ−∞∞ f(x)e−[1λ(x−yx)]2 dx = ∑j=±1 ∑k=1nf(hγ,λn,k,j)Hγ,λn,k,j + Eγ,λ2n[f(x)], where the abscissas hn, k, j(γ, λ) and weights Hn, k, j(γ, λ) are given in terms of the abscissas and weights associated with the classical Hermite-Gauss Quadrature, as prescribed in Gustafson and Hagler (J. Comput. Appl. Math. 105 (1999) to appear). By standard numerical methods, it is shown in the present work that, for fixed γ, λ > 0, Eγ,λ2n[f(x)] = g(4n)(v) n!(4n)! 2nπλ2n+1 for some v in (−∞, ∞), provided g(x):=x2n ƒ(x) has a continuous (4n)-th derivative. The resolution as γ → 0+, with λ=1, of the transformed quadratures introduced in Gustafson and Hagler (in press) to the corresponding classical quadratures is presented here for the first time, with the (2n)-point Laurent-Hermite-Gauss quadrature providing an example, displayed graphically in a figure. Error comparisons displayed in another figure indicate the advantage in speed of convergence, as the number of nodes tends to infinity, of the Laurent-Hermite-Gauss quadrature over the corresponding classical quadrature for certain integrands.