目录

0、Gauss型积分通用形式

1、Gauss–Legendre quadrature勒让德

2、Gauss–Laguerre quadrature拉盖尔——积分区间[0，inf]

3、Chebyshev–Gauss quadrature切比雪夫

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0、Gauss型积分通用形式

The integration problem can be expressed in a slightly more general way by introducing a positive weight function ω into the integrand（被积函数）, and allowing an interval other than（除了，不同于） [−1, 1]. That is, the problem is to calculate

                                                                                              

for some choices of a, b, and ω. For a = −1, b = 1, and ω(x) = 1, the problem is the same as that considered above（勒让德问题）. Other choices lead to other integration rules. Some of these are tabulated（列表） below.

                                             

1、Gauss–Legendre quadrature勒让德——积分区间[-1，1]

       The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as

                                                             

       which is exact for polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f(x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1]. The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint singularities.（端点奇点）

（1）基本概念

       

        注：P0没有根（与x轴无交点），P1有1个根（与x轴有一个交点），P2有2个根（与x轴有两个交点），。。。，Pn有n个根

                   

（2）不属于[-1,1]区间的，我们可以通过下面的积分区间转换

        

2、Gauss–Laguerre quadrature拉盖尔——积分区间[0，inf]

       

注：如果不符合上述被积函数或者区间形式，都可以凑成。

                                                

广义拉盖尔多项式：

(1) The simple Laguerre polynomials are the special case α = 0 of the generalized Laguerre polynomials: 

              

(2) Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as

            

When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression:

          

（3） The closed form for these generalized Laguerre polynomials of degree n is

          

3、Chebyshev–Gauss quadrature切比雪夫——积分区间[-1，1]

 

形式：

         and 

结果：

                       

如果区间不是[-1,1]，首先将区间转换成[-1,1]，实际上就是：（第一种比较常用）

          

In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial minimizes the effect of Runge's phenomenon.