数值分析实验之非线性方程求根（Python 现）

详细实验指导见上一篇，此处只写内容啦

实验内容：

 1. 用二分法求方程x³-3x-1=0在的所有根.要求每个根的误差小于0.001.

   提示与要求: (1) 利用精度找到迭代次数;

           (2) 由f(x)=3(x²-1)可取隔根区间[-2,-1].[-1,1].[1,2]);

           (3) 用程序求各隔根区间内的根.

 2. 用不动点迭代求: (1)x³+2x²+10x-20=0的所有根.

           或: (2)9x²-sinx-1=0在[0,1]上的一个根.

 3. 用Newton迭代法求解下列之一,准确到10^-5:

   (1) x³-x-1=0的所有根;

   (2) e^x+2^-x+2cosx-6=0位于[0,2]上的根.

实验代码： 

  • 牛顿迭代法

import math
x=0.5
n = 1
while n ==1 or abs(x-x1)>1e-5:
    x1 = x
    def F(x1):
        return 2**-x+2*math.cos(x)+math.exp(x)-6
    def F1(x1):
        return math.exp(x)-math.log(2)*(2**-x)-2*math.sin(x)
    x = x1 - F(x1)/F1(x1)
    print ('迭代步数:',n,'X1计算值=',x1,'X计算值=',x)
    n = n+1
else:
    print('方程的根=',(x))

运行结果：

   

 • 不动点法

import numpy as np

def f(x):
    return 9*x**2 - np.sin(x) - 1

def g1(x):
    return ((np.sin(x)+1)/9)**0.5

def g2(x):
    result = (abs(2 * x + 1))**(1 / 5)
    if (2 * x - 1) < 0:
        return -result
    return result

def getEpsilon(x, epsilon):
    maxY = minY = x[0]
    for each in x:
        maxY = max(f(each), maxY)
        minY = min((f(each), minY))
    epsilon = (maxY - minY) * epsilon
    return epsilon

def getInitialVal(x, N, step, epsilon):
    initalVal = []
    for i in range(N + 1):
        y1, y2, y3 = f(x - step), f(x), f(x + step)
        if (y1 * y2 < 0) and (i != 0):
            initalVal.append((x + x - step) / 2)
        if ((y2 - y1) * (y3 - y2) < 0) and (abs(y2) < epsilon):
            initalVal.append(x)
        x += step

    return initalVal

def findFixedPoint(initalVal, delta,epsilon):
    points = []
    for each in initalVal:
        if (abs(g1(each)) < 1):
            points.append(iteration(each, g1, delta,epsilon))
        else:
            points.append(iteration(each, g2, delta,epsilon))
    return points

def iteration(p1, g, delta,epsilon):
    while True:
        p2 = g(p1)
        err =abs(p2-p1)
        relerr = err/(abs(p2)+epsilon)
        if err<delta or relerr<delta:
            return p2
        p1 = p2
                    
def main():
    a, b, c = input().split(' ')
    a = float(a)
    b = float(b)
    c = int(c)
    delta = 10 ** (-c)
    N = 8
    epsilon = 0.01
    step = (b - a) / N
    x = np.arange(a, b + epsilon, epsilon)
    
    epsilon2 = getEpsilon(x,epsilon)
    initalVal = getInitialVal(a, N, step, epsilon2)
    ans = findFixedPoint(initalVal, delta,epsilon)

    for each in ans:
        print('方程的根为：%.6f' % each)
        
if __name__ == '__main__':
    main()

   运行结果：