高斯-勒让德求积分的Python程序

理论知识：
勒让德多项式及性质
第四节 高斯(Gauss)求积公式
高斯-勒让德求积公式及Matlab实现

目标函数求积区间为[-1,1]时

代码：

#高斯-勒让德求积公式
from sympy import *
from scipy.special import perm,comb  #排列,组合

n = 2  #n次多项式正交，n越大精度越高(n=0,1,2,...)
x = symbols("x")

#勒让德多项式
def L(n):
    df = diff((x ** 2 - 1) ** (n + 1), x, n + 1)
    # Python内置阶乘函数factorial
    # L = factorial(n+1) / factorial(2*n+2) * df
    L = 1 /2**(n+1)/factorial(n+1) * df
    return L

#高斯点x求取
def Gauss_point(n):
    x_k_list = solve(L(n))   #求得零点解集
    return x_k_list

#求积系数A
def Quadrature_coefficient(x_k_list):
    A_list = []
    for x_k in x_k_list:
        A = 2/(1-x_k**2)/(diff(L(n),x,1).subs(x,x_k))**2
        A_list.append(A)
    return A_list
#需要求积的目标函数
def f(x):
    f = sqrt(x+1.5)
    return f

result = 0
x_k_list = Gauss_point(n)
A_list = Quadrature_coefficient(Gauss_point(n))
sum = len(A_list)
for i in range(sum):
    result += (A_list[i]*f(x_k_list[i])).evalf()
print(result)

结果：

2.39970807094290

目标函数求积区间不是[-1,1]时

代码：

#高斯-勒让德求积公式
from sympy import *
from scipy.special import perm,comb  #排列,组合
x,t = symbols("x,t")
#积分区间
a = 1
b = 2
#需要求积的目标函数
def f(x):
    f = 1/x
    return f

n = 2  #n次多项式正交，n越大精度越高(n=0,1,2,...)
#勒让德多项式
def L(n):
    df = diff((x ** 2 - 1) ** (n + 1), x, n + 1)
    # Python内置阶乘函数factorial
    L = 1 /2**(n+1)/factorial(n+1) * df
    return L

#高斯点x求取
def Gauss_point(n):
    x_k_list = solve(L(n))   #求得零点解集
    return x_k_list

#求积系数A
def Quadrature_coefficient(x_k_list):
    A_list = []
    for x_k in x_k_list:
        A = 2/(1-x_k**2)/(diff(L(n),x,1).subs(x,x_k))**2
        A_list.append(A)
    return A_list

result = 0
x_k_list = Gauss_point(n)
A_list = Quadrature_coefficient(Gauss_point(n))
sum = len(A_list)
#区间变换
if a == -1 and b == 1:
    for i in range(sum):
        result += (A_list[i] * f(x_k_list[i])).evalf()
    print(result)
#将求求粉公式中的区间(a,b)转换为[-1,1]
else:
    for i in range(sum):
        X = (b-a)/2 * x_k_list[i] + (a+b)/2  #区间变换
        result += (b-a)/2 * (A_list[i] * f(X)).evalf()
    print(result)

结果：

0.693121693121693