# 内联显示matplotlib输出
%matplotlib inline
# 导入相关模块
import math
import matplotlib
import numpy as np
import matplotlib.pyplot as plt
from prettytable import PrettyTable
# 绘制f(x)的图像
e = math.e
x = np.arange(0, 1, 0.01)
def f(x):
y = e**x + 4*x**3 + 2*x**2 + x - 2
return y
plt.figure(facecolor='white',edgecolor='white')
plt.xticks(np.arange(min(x), max(x)+1, 0.1))
plt.title('$ f(x) = e^x + 4*x^3 + 2*x^2 + x - 2 $')
plt.grid(True)
plt.plot(x, f(x))
plt.show()
output_8_0.png
2. 二分法求解 f(x)在[0, 1]区间上的根
# 二分法求根
class FError(Exception):
pass
def dichotomy(f, interval, error, m):
'''二分法求方程根
Args:
f: 方程
interval: 有根区间数组
error: 允许的迭代误差
m: 最大迭代次数
Returns:
如果在最大迭代次数内得到了满足要求的根, 则返回该值
Raises:
参数错误,超出最大迭代次数
'''
n = 0
c = 0
x = 0
temp = 0
# try:
if 2 == len(interval):
(a, b) = interval
if (a > b):
print(f'[{a}, {b}]区间无解')
return
if f(a)==0:
return a
if f(b)==0:
return b
if (f(a)*f(b)>0):
print('f(a), f(b)同号')
return
table = PrettyTable(['迭代次数', 'a', 'b', 'x', 'f(x)'])
while (n<m):
plt.plot([a, b], [f(a), f(b)])
if (b-a) < error:
break
# 取a,b中点x
x = (a+b)/2
y = f(x)
table.add_row([n+1, a, b, x, y])
# 若f(x)=0,返回x
if f(x)==0:
break
# 取函数值异号两项作为新区间
elif (f(x)*f(a)<0):
b = x
elif (f(x)*f(b)<0):
a = x
n += 1
print(table)
x = np.arange(0, 1, 0.01)
plt.figure(facecolor='white',edgecolor='white')
plt.xticks(np.arange(min(x), max(x)+1, 0.1))
plt.title('$ f(x) = e^x + 4*x^3 + 2*x^2 + x - 2 $')
plt.grid(True)
plt.plot(x, f(x))
dichotomy(f, [0, 1], 10**-4, 20)
plt.show()
+----------+-----------------+----------------+------------------+------------------------+
| 迭代次数 | a | b | x | f(x) |
+----------+-----------------+----------------+------------------+------------------------+
| 1 | 0 | 1 | 0.5 | 1.1487212707001282 |
| 2 | 0 | 0.5 | 0.25 | -0.2784745833122586 |
| 3 | 0.25 | 0.5 | 0.375 | 0.32217891461820125 |
| 4 | 0.25 | 0.375 | 0.3125 | -0.0032792463262036886 |
| 5 | 0.3125 | 0.375 | 0.34375 | 0.15277974586321097 |
| 6 | 0.3125 | 0.34375 | 0.328125 | 0.07313118351443926 |
| 7 | 0.3125 | 0.328125 | 0.3203125 | 0.03452725444645388 |
| 8 | 0.3125 | 0.3203125 | 0.31640625 | 0.015525081793443807 |
| 9 | 0.3125 | 0.31640625 | 0.314453125 | 0.006098281683223128 |
| 10 | 0.3125 | 0.314453125 | 0.3134765625 | 0.0014033704793536828 |
| 11 | 0.3125 | 0.3134765625 | 0.31298828125 | -0.0009394732466017075 |
| 12 | 0.31298828125 | 0.3134765625 | 0.313232421875 | 0.00023156460100803855 |
| 13 | 0.31298828125 | 0.313232421875 | 0.3131103515625 | -0.000354050303566833 |
| 14 | 0.3131103515625 | 0.313232421875 | 0.31317138671875 | -6.126684935581217e-05 |
+----------+-----------------+----------------+------------------+------------------------+
output_10_1.png
