|
|
# -*- coding: utf-8 -*-
|
|
|
"""
|
|
|
Created on Sun Dec 26 16:44:41 2021
|
|
|
|
|
|
@author: hzh
|
|
|
"""
|
|
|
# 求方程 f(x) = x3-5x2+10x-80 = 0的根（二分法）
|
|
|
# 绘图
|
|
|
import numpy as np
|
|
|
import matplotlib.pyplot as plt
|
|
|
|
|
|
x = np.linspace(-20, 20, 200)
|
|
|
y = x ** 3 - 5 * x ** 2 + 10 * x - 80
|
|
|
# 设置横轴纵轴刻度范围
|
|
|
plt.axis([-20, 20, -500, 500])
|
|
|
plt.grid('on') # 绘制网格
|
|
|
plt.plot(x, y)
|
|
|
plt.show()
|
|
|
|
|
|
# 二分法实现
|
|
|
E = 1e-10
|
|
|
x1, x2 = 0, 10
|
|
|
fx = 1
|
|
|
while abs(fx) > E:
|
|
|
x = (x1 + x2) / 2
|
|
|
fx = x ** 3 - 5 * x ** 2 + 10 * x - 80
|
|
|
if fx >= 0:
|
|
|
x2 = x
|
|
|
else:
|
|
|
x1 = x
|
|
|
print('x=', x, 'fx=', fx)
|
|
|
|
|
|
|
|
|
# 例2：求x2-2=0的根，即求2的平方根
|
|
|
def f(x):
|
|
|
return x ** 2 - 2
|
|
|
|
|
|
|
|
|
left = 1;
|
|
|
right = 2;
|
|
|
# left=-2;right=-1;
|
|
|
n = 0 # n用来统计二分次数
|
|
|
err = 1e-6 # 误差
|
|
|
while True:
|
|
|
n += 1;
|
|
|
middle = (left + right) * 0.5
|
|
|
if abs(f(middle)) < err:
|
|
|
print('f(x)=%.8f x=%.8f n=%d' % (f(middle), middle, n))
|
|
|
break
|
|
|
if f(middle) * f(left) > 0:
|
|
|
left = middle # 中点变为左边界
|
|
|
# right=middle #中点变为右边界
|
|
|
else:
|
|
|
right = middle # 中点变为右边界
|
|
|
# left=middle #中点变为左边界
|
|
|
'''
|
|
|
求方程f(x)=x3-11.1x2+38.8x-41.77的在[0,10]内的根，
|
|
|
请先画图，再用二分法求根。
|
|
|
'''
|
|
|
|
|
|
|
|
|
# 2牛顿迭代法
|
|
|
def f(x):
|
|
|
return x ** 2 - 2
|
|
|
|
|
|
|
|
|
def df(x):
|
|
|
return x * 2
|
|
|
|
|
|
|
|
|
x = 4;
|
|
|
err = 1e-6;
|
|
|
n = 0;
|
|
|
while True:
|
|
|
if abs(f(x)) < err:
|
|
|
print('fx=%.8f x=%.8f n=%d' % (f(x), x, n))
|
|
|
break
|
|
|
x = x - f(x) / df(x) #
|
|
|
n += 1
|
|
|
|
|
|
'''习题 利用Newton迭代法求下列方程的根：
|
|
|
-sin(x)ex+5=0 位于0~10之间的根
|
|
|
x3-11.1x2+38.8x-41.77=0 位于0~10之间的根
|
|
|
xex-1=0 位于0~10之间的根
|
|
|
'''
|
|
|
|
|
|
# 牛顿割线法求方程的根
|
|
|
x0 = 1;
|
|
|
x1 = 2;
|
|
|
err = 1e-6;
|
|
|
n = 0
|
|
|
|
|
|
|
|
|
def f(x):
|
|
|
return x ** 2 - 2
|
|
|
|
|
|
|
|
|
for x0, x1 in [(-1, -2), (1, 2)]:
|
|
|
while True:
|
|
|
if f(x1) - f(x0) == 0:
|
|
|
print('False')
|
|
|
break
|
|
|
x = x1 - f(x1) / (f(x1) - f(x0)) * (x1 - x0)
|
|
|
n += 1
|
|
|
if abs(f(x)) < err:
|
|
|
print('fx=%.8f x=%.8f n=%d' % (f(x), x, n))
|
|
|
break
|
|
|
x0 = x1
|
|
|
x1 = x
|
|
|
'''
|
|
|
习题 利用Newton割线法求下列方程的根：
|
|
|
x3-3=0 位于0~10之间的根
|
|
|
x3-11.1x2+38.8x-41.77=0 位于0~10之间的根
|
|
|
xex-1=0 位于0~10之间的根
|
|
|
'''
|
|
|
################begin##################
|
|
|
##习题 利用Newton割线法求下列方程的根：
|
|
|
##x**3-3=0 位于0~10之间的根
|
