论文标题
具有经典重量功能的高斯正交规则的任意订购渐近扩展
Arbitrary-order asymptotic expansions of Gaussian quadrature rules with classical and generalised weight functions
论文作者
论文摘要
高斯正交规则是具有光滑积分和正权重函数积分数的数值近似的经典工具。 We derive and expicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions在无穷大。结果包括经典正交多项式（Legendre，Jacobi，Laguerre和Hermite）的高斯规则。我们提出了指示这些表达式达到高精度的点数的范围。我们提供了一种算法，该算法可以在经典案例的这些扩展中按任意计算的许多术语计算，而许多算法并非全部对于广义案例。
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.
