不动点迭代法和牛顿迭代法
2024-05-29 06:57:14
不定点迭代法
方程的根
不动迭代法的概念
代码实现
import numpy
import numpy as np
from sympy import *
import math
import matplotlib.pyplot as plt
from sympy.simplify.fu import Ldef detfunction(x):return pow((x+1), 1/3)def erfen(point1, point2, min_area):min = point1max = point2isquit = Truewhile isquit:mid = point1+1/2*(point2-point1)if detfunction(mid) * detfunction(min) <= 0:max = midelif detfunction(mid)*detfunction(max) < 0:min = midif(math.fabs(max-min) < min_area):isquit = Falsereturn min, maxdef calu_result(accuary, point):minarea = 1# erfen()isquit = Truefirst = pointcount = 0 #次数while isquit:# 计算结果res = detfunction(first)if(abs(first-res) < accuary):first = resisquit = Falsefirst = rescount += 1return first, countdef draw(left, right):# x = np.arange(left-5,right+5,0.1)# x_v = x# y = []# for i in range(len(x)):# res =detfunction(x[i])# y.append(res)# plt.plot(x,x_v,label = "y=x")# plt.plot(x,y,label = "y = (x+1)^1/3")# # ax = gca()# plt.show()plt.figure(figsize=(12, 8), dpi=72)plt.subplot(1, 1, 1)X = np.linspace(-10,10, 2048, endpoint=True)X_val = XY_val = np.zeros(len(X))for i in range(len(X)):res = detfunction(X[i])Y_val[i] = resplt.plot(X, X_val, linewidth=1.5, linestyle="-", label="y =x")plt.plot(X, Y_val, linewidth=1.5, linestyle="-", label="y =(x+1)^1/3")plt.legend(loc='upper left')plt.xlim(-4.5, 4.5)# 设置纵轴的上下限plt.ylim(-4.5, 4.5)# 设置纵轴记号plt.yticks(np.linspace(-1, 1, 5, endpoint=True))ax = plt.gca()ax.spines['right'].set_color('none')ax.spines['top'].set_color('none')ax.xaxis.set_ticks_position('bottom')ax.spines['bottom'].set_position(('data', 0))ax.yaxis.set_ticks_position('left')ax.spines['left'].set_position(('data', 0))# savefig("sincosin.png",dpi=72) #以72dpi保存图像plt.show()point = float(input("请输入迭代法的初始值"))
accuary = float(input("请输入最大允许误差值"))
res, count = calu_result(accuary, point)
print(f"结果：{res}")
print(f"迭代次数：{count}")
ma = max(res, point)
small = min(res, point)
draw(small*0.8, ma*1.2)实验样例
函数的图像
样例2
牛顿法、牛顿简化法，牛顿下山法
牛顿法
迭代函数
牛顿简化法
由于牛顿法计算导数时比较困难，就用写于为f（x0)的导数的平行弦与x轴的交点来替代
牛顿下山法
代码实现
牛顿法
def calu_result(diff_function,function,accuary,point,x):is_quit = TrueInit_first = pointcount =0 #次数while is_quit:res1 = function.subs(x,Init_first)res2 = diff_function.subs(x,Init_first)temp = Init_first-(res1/res2) #每次迭代的xkif abs(temp-Init_first)<accuary:Init_first = temp #更新xkcount+=1return Init_first,countif(count>100):return temp,countInit_first = tempcount+=1
牛顿简化法
def calu_Simple_result(diff_function,function,accuary,point,x):is_quit = TrueInit_first = pointcount =0 #次数res2 = diff_function.subs(x,Init_first) #平行弦while is_quit:res1 = function.subs(x,Init_first)temp = Init_first-(res1/res2)if abs(temp-Init_first)<accuary:Init_first = tempcount+=1return Init_first,countif(count>100):return temp,countInit_first = tempcount+=1
牛顿下山法
def mountain_result(diff_function,function,accuary,point,x):is_quit = TrueInit_first = pointsecond =pointcount =0weight = 1is_mul = False#计算下山因子while is_quit!=False:res1 = function.subs(x,second)res2 = diff_function.subs(x,second)temp = second-(res1/res2)if abs(detfunction(temp))-abs(detfunction(second)) <0:break#每次减半，直到合适的下山因子weight = weight/2second = temp# print(weight)while is_quit:#根据下山因子计算res1 = function.subs(x,Init_first)res2 = diff_function.subs(x,Init_first)temp = Init_first - weight*(res1/res2)if abs(temp-Init_first)<accuary:Init_first = tempreturn Init_first,count,weight#设定一个，100轮还没有结束，也退出，相当于一个最大迭代次数if(count>100):return temp,count,weightInit_first = tempcount+=1
三种方法实验结果
样例1
第一行为求导后的表达式，后面的为迭代次数和结果
当取初值为0.6时，应该可能出现，离根比较远的情况，此时牛顿法收敛法比较慢，而牛顿简化法已经不满足收敛条件了，而牛顿下山法可以保证它的收敛性，慢慢逼近根
全部代码
不动点迭代法
import numpy
import numpy as np
from sympy import *
import math
import matplotlib.pyplot as plt
from sympy.simplify.fu import Ldef detfunction(x):return 1/exp(x)def erfen(point1, point2, min_area):min = point1max = point2isquit = Truewhile isquit:mid = point1+1/2*(point2-point1)if detfunction(mid) * detfunction(min) <= 0:max = midelif detfunction(mid)*detfunction(max) < 0:min = midif(math.fabs(max-min) < min_area):isquit = Falsereturn min, maxdef calu_result(accuary, point):minarea = 1# erfen()isquit = Truefirst = pointcount = 0 #次数while isquit:# 计算结果res = detfunction(first)if(abs(first-res) < accuary):first = resisquit = Falsefirst = rescount += 1return first, countdef draw(left, right):# x = np.arange(left-5,right+5,0.1)# x_v = x# y = []# for i in range(len(x)):# res =detfunction(x[i])# y.append(res)# plt.plot(x,x_v,label = "y=x")# plt.plot(x,y,label = "y = (x+1)^1/3")# # ax = gca()# plt.show()plt.figure(figsize=(12, 8), dpi=72)plt.subplot(1, 1, 1)X = np.linspace(-10,10, 2048, endpoint=True)X_val = XY_val = np.zeros(len(X))for i in range(len(X)):res = detfunction(X[i])Y_val[i] = resplt.plot(X, X_val, linewidth=1.5, linestyle="-", label="y =x")plt.plot(X, Y_val, linewidth=1.5, linestyle="-", label="y =1/e^x")plt.legend(loc='upper left')plt.xlim(-4.5, 4.5)# 设置纵轴的上下限plt.ylim(-4.5, 4.5)# 设置纵轴记号plt.yticks(np.linspace(-1, 1, 5, endpoint=True))ax = plt.gca()ax.spines['right'].set_color('none')ax.spines['top'].set_color('none')ax.xaxis.set_ticks_position('bottom')ax.spines['bottom'].set_position(('data', 0))ax.yaxis.set_ticks_position('left')ax.spines['left'].set_position(('data', 0))# savefig("sincosin.png",dpi=72) #以72dpi保存图像plt.show()point = float(input("请输入迭代法的初始值"))
accuary = float(input("请输入最大允许误差值"))
res, count = calu_result(accuary, point)
print(f"结果：{res}")
print(f"迭代次数：{count}")
ma = max(res, point)
small = min(res, point)
draw(small*0.8, ma*1.2)牛顿迭代法
import numpy
import numpy as np
from sympy import *
import math
import matplotlib.pyplot as plt
from sympy.simplify.fu import Ldef draw(left, right):# x = np.arange(left-5,right+5,0.1)# x_v = x# y = []# for i in range(len(x)):# res =detfunction(x[i])# y.append(res)# plt.plot(x,x_v,label = "y=x")# plt.plot(x,y,label = "y = (x+1)^1/3")# # ax = gca()# plt.show()plt.figure(figsize=(12, 8), dpi=72)plt.subplot(1, 1, 1)X = np.linspace(-10,10, 2048, endpoint=True)X_val = XY_val = np.zeros(len(X))for i in range(len(X)):res = detfunction(X[i])Y_val[i] = resplt.plot(X, X_val, linewidth=1.5, linestyle="-", label="y =x")plt.plot(X, Y_val, linewidth=1.5, linestyle="-", label="y =(x+1)^3-x-1")plt.legend(loc='upper left')plt.xlim(-4.5, 4.5)# 设置纵轴的上下限plt.ylim(-4.5, 4.5)# 设置纵轴记号plt.yticks(np.linspace(-1, 1, 5, endpoint=True))ax = plt.gca()ax.spines['right'].set_color('none')ax.spines['top'].set_color('none')ax.xaxis.set_ticks_position('bottom')ax.spines['bottom'].set_position(('data', 0))ax.yaxis.set_ticks_position('left')ax.spines['left'].set_position(('data', 0))# savefig("sincosin.png",dpi=72) #以72dpi保存图像# 在屏幕上显示plt.show()def getdiff(function,x):return diff(function,x)def detfunction(x):return pow(x,3)-x-1def calu_Simple_result(diff_function,function,accuary,point,x):print("牛顿简化法")is_quit = TrueInit_first = pointcount =0res2 = diff_function.subs(x,Init_first)while is_quit:res1 = function.subs(x,Init_first)temp = Init_first-(res1/res2)print(f"第{count}的值{temp}")if abs(temp-Init_first)<accuary:Init_first = tempcount+=1return Init_first,countif(count>100 or abs(temp-Init_first)>100000):return temp,countInit_first = tempcount+=1def calu_result(diff_function,function,accuary,point,x):print("牛顿法")is_quit = TrueInit_first = pointcount =0 while is_quit:res1 = function.subs(x,Init_first)res2 = diff_function.subs(x,Init_first)temp = Init_first-(res1/res2)print(f"第{count}的值：{temp}")if abs(temp-Init_first)<accuary:Init_first = tempcount+=1return Init_first,countif(count>100):return temp,countInit_first = tempcount+=1def mountain_result(diff_function,function,accuary,point,x):print("牛顿下山法")is_quit = TrueInit_first = pointsecond =pointcount =0weight = 1is_mul = Falsewhile is_quit!=False:res1 = function.subs(x,second)res2 = diff_function.subs(x,second)temp = second-(res1/res2)if abs(detfunction(temp))-abs(detfunction(second)) <0:breakweight = weight/2second = temp# print(weight)while is_quit:res1 = function.subs(x,Init_first)res2 = diff_function.subs(x,Init_first)temp = Init_first - weight*(res1/res2)print(f"第{count}的值{temp}")if abs(temp-Init_first)<accuary:Init_first = tempreturn Init_first,count,weightif(count>100):return temp,count,weightInit_first = tempcount+=1x = symbols('x')
function = detfunction(x)
diff_function = getdiff(function,x)
print(diff_function)
point = float(input("请输入迭代法的初始值"))
accuary = float(input("请输入最大允许误差值"))
res, count = calu_result(diff_function,function,accuary, point,x)
res2,count2 = calu_Simple_result(diff_function,function,accuary, point,x)
res3,count3,weight = mountain_result(diff_function,function,accuary, point,x)
if count!=100:print(f"牛顿法结果：{res}")print(f"迭代次数：{count}")
# else:
# print(f"迭代次数：{count}")
# print(f"牛顿法结果：{res},达到最大迭代次数")if count2!=100:print(f"牛顿简化法结果：{res2}")print(f"迭代次数：{count2}")
# else:
# print(f"牛顿简化法结果：{res2}")
# print(f"迭代次数：{count2},达到最大迭代次数")if count3!=100:print("下山因子为：{}".format(weight))print(f"牛顿下山法结果：{res3}")print(f"迭代次数：{count3}")
ma = max(res, point)
small = min(res, point)
draw(small*0.8, ma*1.2)
总结
本文是学习的一些笔记，创作不易，点个爱心是缘分，不点是本分，如果有问题可以私信或者在下面评论，大家可以一起学习！！！
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