《用 Python 学微积分》笔记 2

《用 Python 学微积分》原文见参考资料 1。

13、大 O 记法

比较两个函数时，我们会想知道，随着输入值 x 的增长或减小，两个函数的输出值增长或减小的速度究竟谁快谁慢。通过绘制函数图，我们可以获得一些客观的感受。

比较 x!、ex、x3 和 log(x) 的变化趋势。

import numpy as np
import matplotlib.pyplot as plt

f = lambda x: x**2-2*x-4
l1 = lambda x: 2*x-8
l2 = lambda x: 6*x-20

x = np.linspace(0,5,100)

plt.plot(x,f(x),'black')
plt.plot(x[30:80],l1(x[30:80]),'blue', linestyle = '--')
plt.plot(x[66:],l2(x[66:]),'blue', linestyle = '--')

l = plt.axhline(y=0,xmin=0,xmax=1,color = 'black')
l = plt.axvline(x=2,ymin=2.0/18,ymax=6.0/18, linestyle = '--')
l = plt.axvline(x=4,ymin=6.0/18,ymax=10.0/18, linestyle = '--')

plt.text(1.9,0.5,r"$x_0$", fontsize = 18)
plt.text(3.9,-1.5,r"$x_1$", fontsize = 18)
plt.text(3.1,1.3,r"$x_2$", fontsize = 18)

plt.plot(2,0,marker = 'o', color = 'r' )
plt.plot(2,-4,marker = 'o', color = 'r' )
plt.plot(4,0,marker = 'o', color = 'r' )
plt.plot(4,4,marker = 'o', color = 'r' )
plt.plot(10.0/3,0,marker = 'o', color = 'r' )

plt.show()

View Code



如下定义牛顿迭代法：

def NewTon(f, s = 1, maxiter = 100, prt_step = False):
for i in range(maxiter):
# 相较于f.evalf(subs={x:s}),subs()是更好的将值带入并计算的方法。
s = s - f.subs(x,s)/f.diff().subs(x,s)
if prt_step == True:
print "After {0} iteration, the solution is updated to {1}".format(i+1,s)
return s

from sympy.abc import x
f = x**2-2*x-4
print NewTon(f, s = 2, maxiter = 4, prt_step = True)
# After 1 iteration, the solution is updated to 4
# After 2 iteration, the solution is updated to 10/3
# After 3 iteration, the solution is updated to 68/21
# After 4 iteration, the solution is updated to 3194/987
# 3194/987

Sympy 可以帮助我们求解方程：

import sympy
from sympy.abc import x
f = x**2-2*x-4
print sympy.solve(f,x)
# result is：[1 + sqrt(5), -sqrt(5) + 1]

参考资料：

[1] https://ryancheunggit.gitbooks.io/calculus-with-python/content/