数值积分——梯形公式和Simpson公式

梯形公式：

$$I(f)={\int }_a^{b}f(x)dx\approx \frac{b-a}{2}[f(a)+f(b)]$$

def Trapezoid(a,b):
    y = (a - b) * ((1/2) * f(a) + (1/2) * f(b))
    return y

Simpson 公式：

$$I(f)={\int }_a^{b}f(x)dx\approx (b-a)[\frac{1}{6}f(a)+\frac{4}{6}f(\frac{a+b}{2})+\frac{1}{6}f(b)]$$

def Simpson(a,b):
    y = (a-b)*((1/6)*f(a) + (4/6)*f((a+b)/2) + (1/6) * f(b))
    return y

测试一下：

def f(x):
    f = np.exp(-x)
    return f

print(Trapezoid(0.5,0.1))
print(Simpson(0.5,0.1))

结果为：

0.3022736155497186
0.298309397365031

复化梯形公式：

$$\begin{gathered} T_n={\int }_a^{b}f(x)dx\approx \frac{h}{2}[f(a)+f(b)+2\sum _{k=1}^{n-1}f(x_k)] \\ h=\frac{b-a}{n},x_k=a+kh,(k=0,1,\dots ,n) \end{gathered}$$

def complex_Trapezoid(n,a,b):
    h = (b-a) / n
    temp_s= 0
    for k in range(1,n):
        x_k = a + k*h
        temp_s += f(x_k)
    I = h/2 * (f(a) + f(b) + 2*temp_s)
    return I

复化Simpson公式：

$$\begin{gathered} {\int }_a^{b}f(x)dx\approx \frac{h}{3}[f(a)+f(b)+2\sum _{k=1}^{m-1}f(x_{2k})+4\sum _{k=1}^{m}f(x_{2k-1})] \\ n=2m,h=\frac{b-a}{n},x_k=a+kh,(k=0,1,\dots ,n) \end{gathered}$$

def complex_simpson(n,a,b):
    '''
    n:划分[a,b]区间成n等分
    a: 积分下限
    b: 积分上限
    '''
    h = (b-a) / n        
    temps_odd = 0
    temps_even = 0
    for k in range(1,n):
        if k % 2 == 0:
            x_k = a + k*h
            temps_even += 2 * f(x_k)
        else:
            x_k = a + k*h
            temps_odd += 4 * f(x_k)
    I = (h/3) * (f(a) + f(b) + temps_even + temps_odd)
    return I

测试一下：

def f(x):
    if x == 0:
        return 1
    else:
        f = np.sin(x) / x
        return f

print(complex_Trapezoid(8,0,1))
print(complex_simpson(8,0,1))

结果为：

0.9456908635827014
0.9460833108884719