高阶高斯积分节点的高精度数值计算  

High-precision Numerical Computation of High-degree Gauss-quadrature Nodes

作　　者：张庆礼[1] 王晓梅[2] 殷绍唐[1] 江海河[3] 

机构地区：[1]中国科学院安徽光学精密机械研究所,合肥230031 [2]中国科学院环境光学与技术重点实验室,合肥230031 [3]中国科学院合肥物质科学研究院,合肥230031

出　　处：《中国工程科学》2008年第2期35-40,共6页Strategic Study of CAE

基　　金：国家自然科学基金资助项目(60478025,50472104)

摘　　要：在工程数值计算、X射线衍射线形分析、光谱学等领域常使用高斯数值积分,高斯积分的节点及权重因子是数值积分的必须数据。研究了高次勒让德、拉盖尔和厄米多项式的零点,即高斯-勒让德、高斯-拉盖尔、高斯-厄米积分的节点的计算方法,给出了一种有效的高精度数值算法——搜索迭代方法(scan-iteration method,SIM)。根据勒让德、拉盖尔、厄米多项式的特点,对拉盖尔多项式、厄米多项式的定义稍做变化后,获得了计算多项式值的稳定递推关系。求它们的根时,先在一定范围内以一定的步长搜索根所在的区间,获得所有根的各自区间范围后,再通过常用的迭代方法如割线法、二分法进行求解。数值实验表明,这种方法是非常有效的,可获得高次勒让德、拉盖尔、厄米多项式的全部高精度根值。Gauss quadrature is used widely in many fields such as the engineering numerical computation, X-ray diffraction profile analysis, spectroscopy, and so on. The nodes and weight factors of Gauss-quadrature are essential data to the numerical integration. A method to compute the zeroes of the high-degree Legendre, Laguerre and Hermite polynomials, which are the nodes of Gauss-Legendre, Gauss-Laguerre and Gauss-Hermite Quadrature, respectively, is studied, and a very efficient algorithm scan-iteration method（SIM） is given. According to the properties of Legendre, Laguerre and Hermite polynomials, their definitions are modified a little, and the stable recursive relations to compute their value are obtained. To extract these polynomials, their root intervals are searched with a certain step within a certain range. After the intervals of all roots are obtained, the roots with the desired precision can be gotten by the general iteration methods such as secant or bisection method. Numerical experiments indicate that the method is very efficient and the high-precise roots of Legendre, Laguerre and Hermite polynomials can be extracted.

关 键 词：高斯积分 勒让德多项式 拉盖尔多项式 厄米多项式 求根 

分 类 号：O174.6[理学—数学] O241[理学—基础数学]