Polynomial Basis Functions and Quadratures

摘要: Spectral and pseudospectral methods in chemistry physics are based on classical nonclassical orthogonal polynomials defined terms of a three term recurrence relation. The coefficients the relations for can be calculated with Gautschi-Stieltjes procedure. round-off errors that occur use Gram-Schmidt orthogonalization procedure is demonstrated both polynomials. trapezoidal, Simpson’s Newton-Cotes integration rules derived as Fejer, Clenshaw-Curtiss, Gauss-Lobatto Gauss-Radau algorithms. Sinc interpolation Fourier sine basis functions compared Lagrange interpolation. Nonclassical Maxwell Bimodal infinite domain respect to weight \(w(x) = x^2\exp (-x^2)\) \(x^2\exp [-(x^4/4\epsilon -x^2/2\epsilon )]\), respectively, introduced kinetic theory problems. Gaussian quadrature rule Rys function e^{-cx^2},\; x \in [-1,\ 1]\), used evaluate integrals molecular quantum mechanics presented. For \(c \rightarrow 0\) \infty \), Legendre scaled Hermite polynomials, respectively. Two dimensional quadratures, such Lebedev cubature, two density functional electronic structure calculations well nonlinear Boltzmann equation theory. Stieltjes moment problem related inversion data chemical reconstruct photoelectron cross sections.