AI 中文总结
介绍定制高斯求积法，针对无法直接获取相关法则的情况，阐述矩行列式法和施蒂尔杰斯程序法这两种计算方法，并在Julia包中实现，利用特定编程和算法评估误差，还说明了从R访问的方式。
AI 中文摘要
n点高斯求积法则通过函数的n次加权求值的加权平均值来近似函数的加权积分，对于次数至多为2n - 1的多项式是精确的，使用较少求值就能有高精度。与连续变量的经典正交多项式相关的权函数的此类法则容易获得，若不然则需定制。计算这些法则最易理解的两种方法是矩行列式法和施蒂尔杰斯程序法。我们在Julia包CustomGaussQuadrature中实现了它们，该包使用类型泛型数值编程和自适应高精度算法来评估舍入引起的近似误差，并描述了通过JuliaConnectoR从R进行访问的方法。
英文摘要
An $n$-point Gauss quadrature rule approximates the weighted integral of a function by a weighted sum of $n$ evaluations of this function and is exact for polynomials of degree at most $2n-1$. Such rules can be highly accurate with relatively few evaluations of this function. For weight functions associated with classical orthogonal polynomials of a continuous variable (such as Legendre, Hermite and Laguerre), these quadrature rules are readily available. We suppose that this is not the case, so that these rules must be custom-made. We present CustomGaussQuadrature, a Julia package that implements two approaches for computing the Gauss quadrature nodes and weights: the moment determinants method and the Stieltjes procedure. The principal contribution is an implementation of the ill-conditioned moment determinants method with adaptively chosen working precision. Comparisons between computations at increasing precisions provide practical error indicators used to target absolute and relative errors in the Gauss rule nodes and weights, respectively, of about $10^{-16}$. The package also adapts the working precision and number of auxiliary quadrature nodes used for the Stieltjes procedure. Numerical examples using scaled chi and Weibull probability density functions as weight functions show close agreement between the results obtained by the two approaches. The package is intended particularly for statistical applications in which a single custom Gauss rule is computed once and then reused to approximate many weighted integrals having the same weight function but different computationally expensive integrands.
CommentsThis is greatly revised version that now includes extensive reporting of numerical results
