d-fold Hermite–Gauss quadrature

摘要

We extend results presented in Gustafon and Hagler (J. Comput. Appl. Math. 105 (1999) 317–326); Hagler (Ph.D. Thesis, University of Colorado, 1997; J. Comput. Appl. Math. 104 (1999) 163–171; Hagler et al. (Lecture Notes in Pure and Applied Mathematics Series, Vol. 1999, Marcel Dekker, New York, 1998, pp. 187–208) by giving a construction of systems of orthogonal rational functions from systems of orthogonal polynomials and explicating the (2dn)-point d-fold Hermite–Gauss quadrature formula of parameters γ,λ>0:∫−∞∞f(x)e−[v[d](γ,λ)(x)]2dx=∑k=12dnf(hd,n,k(γ,λ))Hd,n,k(γ,λ)+Ed,n(γ,λ)[f(x)],where v[d](γ,λ)(x) is the d-fold composition of v(γ,λ)(x)=(1/λ)(x−γ/x) and where the abscissas hd,n,k(γ,λ) and weights Hd,n,k(γ,λ) are given recursively in terms of the abscissas and weights associated with the classical Hermite–Gauss quadrature. Error analysis, tables of numerical values for nodes, and examples and comparisons are included.