實驗內容

分別用Newton切線法、Newton下山法和割線法求方$x^2-2x-e^x+2=0$的近似根，其對應函式影象如下圖所示。

演算法描述

Newton切線法

迭代公式

$x_{k+1}=x_k-\frac{f(x_k)}{f (x_k)}$

演算法流程圖

Newton下山法

在$x_k$和$x_{k+1}$之間找一個更好的點$\overline{x_{k+1}}$使得:$|f(\overline{x_{k+1}})|<|f(x_k)|$

迭代公式

$x_{k+1}=x_k-r\cdot\frac{f(x_k)}{f (x_k)}$

演算法流程圖

割線法

迭代公式

$x_{k+1}=x_k-\frac{x_k-x_0}{f(x_k )-f(x_0 ) } f(x_k )$

演算法流程圖

實驗結果

Python程式碼求解

#!/usr/bin/env python
# -*- coding: utf-8 -*-
 

# @Time    : 2018/10/14 19:59
# @Author  : Guojing Li
# @File    : SolveNonlinearEquation.py
# @Software: PyCharm
"""
Methods of solving non-linear equation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This module includes three Methods of solving my non-linear equation and a plot function.
You can choose a Methods to get the solution of  my non-linear equation.
This is my non-linear equation: x**2 - 2*x - exp(x) + 2 = 0
"""
 

import math
from matplotlib import pyplot as plt
import numpy as np

def plot_nonlinear_equation():
    """
    繪製函式f(x) = x**2 - 2*x - exp(x) + 2 和函式f(x) = 0的影象
    :return: None
    """
    X = np.arange(-3, 3, 0.01)
    Y = [f(x) for x in X]
    plt.plot(X, Y, linewidth='2', linestyle="-" )
    plt.hlines(0, -3, 3, colors='r')
    plt.xlim(-3, 3)
    plt.ylabel("f(x)")
    plt.xlabel("x")
    plt.legend(["f(x) = x**2 - 2*x - exp(x) + 2 ", "f(x) = 0"])
    plt.show()

def f(x):
    """
    :param x:
    :return: f(x)
    """
    return x**2 - 2*x - math.exp(x) + 2

def df(x):
    """
    :param x:
    :return: f(x)的導數
    """
    return 2*x - 2 - math.exp(x)

def tangent_method(x, ep=1e-05, max_iteration=500):
    """
    牛頓切線法求解非線性方程
    :param x: x初始值
    :param ep: 精度
    :param max_iteration: 最大迭代次數
    :return: 迭代次數, 方程的解
    """
    print("牛頓切線法求解方程，精度為", ep, ",最大迭代次數為", max_iteration)
    print("初始值：x =", x)
    for iteration in range(max_iteration):
        if (df(x) == 0):
            exit("導數為0，迭代失敗")
        x1 = x - f(x)/df(x)
        print("第", iteration+1, "次迭代，x =", x1)
        if(math.fabs(x1-x) <= ep):
            return iteration+1, x1
        else:
            x = x1
    exit("以迭代到上限值")

def downhill_method(x, ep=1e-05, max_iteration=500):
    """
    牛頓下山法求解非線性方程
    :param x: x初始值
    :param ep: 精度
    :param max_iteration: 最大迭代次數
    :return: 迭代次數, 方程的解
    """
    print("牛頓下山法求解方程，精度為", ep, ",最大迭代次數為", max_iteration)
    print("初始值：x =", x)
    for iteration in range(max_iteration):
        if (df(x) == 0):
            exit("導數為0，迭代失敗")
        r = 1
        x1 = x - r*f(x)/df(x)
        while (math.fabs(f(x1)) > math.fabs(f(x))):
            r = r / 2
            x1 = x - r*f(x)/df(x)
            if (math.fabs(f(x1)) < math.fabs(f(x))):
                print("下山調整後，r = ", r)
        x1 = x - r*f(x)/df(x)
        print("第", iteration + 1, "次迭代，x =", x1)
        if(math.fabs(x1-x) <= ep):
            return iteration+1, x1
        else:
            x = x1
    exit("以迭代到上限值")

def secant_method(x0, x1, ep=1e-05, max_iteration=500):
    """
    割線法求解非線性方程
    :param x0: x0初始值
    :param x1: x1初始值
    :param ep: 精度
    :param max_iteration: 最大迭代次數
    :return: 迭代次數, 方程的解
    """
    print("割線法求解方程，精度為", ep, ",最大迭代次數為", max_iteration)
    print("初始值：x0 =", x0, ", x1 = ", x1)
    for iteration in range(max_iteration):
        if(math.fabs(x1-x0) <= ep):
            return iteration, x1
        else:
            x2 = x1 - f(x1)*(x1-x0)/(f(x1)-f(x0))
            x0 = x1
            x1 = x2
            print("第", iteration+1, "次迭代，x0 =", x0, ", x1 =", x1)
    exit("以迭代到上限值")

if __name__ == "__main__":
    #繪製函式f(x) = x**2 - 2*x - exp(x) + 2 和函式f(x) = 0的影象函式的呼叫
     plot_nonlinear_equation()
    #牛頓切線法函式的呼叫
    count, result = tangent_method(1)
    print("共迭代%d次, 方程的解為%.17f" % (count, result))
    #牛頓下山法函式的呼叫
    count, result = downhill_method(1)
    print("共迭代%d次, 方程的解為%.17f" % (count, result))
    # #割線法函式的呼叫
    count, result = secant_method(0, 1)
    print("共迭代%d次, 方程的解為%.17f" % (count, result))