示例
已知积分公式如下
求[0.5，5]上积分，即求下图阴影部分面积
根据积分公式求源函数等于：
则确切解等与F(5)-F(0.5)=3.9002072872864524
当不知道源函数时使用以下方法可以求得积分面积
首先定义函数
def func(x):
return np.cos(np.pi) * np.exp(-x) + 1
1、通过迭代计算每个步长矩形面积求和
x = np.linspace(0.5, 5, 1000)
dx = (5-0.5)/1000
y = func(x)
area = np.sum(y * dx)结果等于：3.8994262124710404
2、通过scipy模块quad直接求积分
area, error = quad(func, 0.5, 5)quad的结果是 3.9002072872864515
3、使用蒙特卡洛模拟估算积分面积
原理是已知区间【0.5,5】上矩形面积，在矩形范围内随机打点，统计在阴影部分点的占比，从而估算出阴影部分面积。
k = 0
for i in range(N):
r_x = np.random.uniform(low=0.5, high=5, size=1)
r_y = np.random.uniform(low=0, high=1, size=1)
c_v = func(r_x)
if r_y <= c_v:
k += 1
area = k / N * 4.5蒙特卡洛的结果是 3.9393
所有代码如下
# @Time : 2020/8/26
# @Author : 大太阳小白
# @Software: PyCharm
# @blog：https://blog.csdn.net/weixin_41579863
import numpy as np
from scipy.integrate import quad
from matplotlib import pyplot as plt
def func(x):
return np.cos(np.pi) * np.exp(-x) + 1
def drawing_question(fig):
x = np.linspace(0, 6, 1000)
y = func(x)
fig = plt.figure()
ax1 = fig.add_subplot()
ax1.axis([np.min(x), np.max(x), 0, np.max(y)]) # 坐标范围
ax1.plot(x, y, label="$cos(π)e^{-x}+1$") # 画曲线，带图示
ax1.fill_between(x, y1=y, y2=0, where=(x >= 0.5) & (x <= 5), # 填充积分区域
facecolor='blue', alpha=0.2)
ax1.text(0.5 * (0.7 + 4), 0.4, r"$\int_{0.5}^5(cos(π)e^{-x}+1)\mathrm{d}x$",
horizontalalignment='center', fontsize=14) # 增加说明文本
ax1.legend() # 显示图示
plt.show()
def area_sum():
x = np.linspace(0.5, 5, 1000)
dx = (5-0.5)/1000
y = func(x)
area = np.sum(y * dx)
return area
def area_quad():
area, error = quad(func, 0.5, 5)
return area
def montecarlo(N=5000):
k = 0
for i in range(N):
r_x = np.random.uniform(low=0.5, high=5, size=1)
r_y = np.random.uniform(low=0, high=1, size=1)
c_v = func(r_x)
if r_y <= c_v:
k += 1
area = k / N * 4.5
return area
def montecarlo_show(N=5000):
k = 0
x0 = []
x1 = []
for i in range(N):
r_x = np.random.uniform(low=0.5, high=5, size=1)
r_y = np.random.uniform(low=0, high=1, size=1)
c_v = func(r_x)
if r_y <= c_v:
k += 1
x0.append([r_x, r_y])
else:
x1.append([r_x, r_y])
x = np.linspace(0, 6, 1000)
y = func(x)
x0 = np.array(x0)
x1 = np.array(x1)
fig = plt.figure()
ax1 = fig.add_subplot()
ax1.axis([np.min(x), np.max(x), 0, np.max(y)])
ax1.plot(x, y, label="$cos(π)e^{-x}+1$")
ax1.scatter(x0[:, 0], x0[:, 1])
ax1.scatter(x1[:, 0], x1[:, 1])
ax1.fill_between(x, y1=y, y2=0, where=(x >= 0.5) & (x <= 5),
facecolor='blue', alpha=0.2)
ax1.text(0.5 * (0.7 + 4), 0.4, r"$\int_{0.5}^5(cos(π)e^{-x}+1)\mathrm{d}x$",
horizontalalignment='center', fontsize=14) # 增加说明文本
ax1.legend() # 显示图示
plt.show()
if __name__ == '__main__':
# drawing_question()
original = lambda x: np.exp(-x) + x
print('期望值是：', original(5)-original(0.5))
print('基于矩形面积求和的结果是', area_sum())
print('quad的结果是', area_quad())
print('蒙特卡洛的结果是', montecarlo())
montecarlo_show(N=500) 本文地址：https://blog.csdn.net/weixin_41579863/article/details/108239755
