Polynomial Basis Functions and Quadratures

Abstract

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