📄 p23：迭代法算例.htm
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<title>应用二分法求解非线性方程的计算过程显示页面</title>
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<h1>应用不动点算法解非线性方程的计算过程详解</h1>
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问题:<br>
记 F(x)=x*x*x-x-1, 求方程 F(x)=0<br>
在区间 [a,b]=[1,2] 内的近似解，<br>
采用的迭代格式为 T(x)=x-(x*x*x-x-1)/13<br>
要求解的绝对误差不超过 1e-8。<br>
<br>
为了帮助大家阅读计算过程，我们定义数组<br>
X[N],Y[N],Z[N] 保存中间结果。<br>
主要计算格式为：<br>
X[K+1]=T(X[K]);<br>
Y[K+1]=F(X[K+1]);<br>
R[K+1]=|X[K+1]-X[K]|;<br>
<br>
详细计算过程列表如下：</font>
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<table bgColor="#3399ff" border="4" style="COLOR: #ffff00; FONT-SIZE: 24pt" width="120%">
<tbody>
<tr>
<td align="middle" width="10%">K</td>
<td align="middle" width="30%">X[K]</td>
<td align="middle" width="30%">Y[K]</td>
<td align="middle" width="30%">R[K]</td>
</tr>
<tr>
<td align="middle" width="10%">0</td>
<td align="right" width="22%">+1.5</td>
<td align="right" width="22%">+0.875000E0</td>
<td align="right" width="22%">+0.1000E1</td>
</tr>
<tr>
<td align="middle" width="10%">1</td>
<td align="right" width="22%">+1.43269231</td>
<td align="right" width="22%">+0.508062E0</td>
<td align="right" width="22%">+0.6731E-1</td>
</tr>
<tr>
<td align="middle" width="10%">2</td>
<td align="right" width="22%">+1.39361059</td>
<td align="right" width="22%">+0.312991E0</td>
<td align="right" width="22%">+0.3908E-1</td>
</tr>
<tr>
<td align="middle" width="10%">3</td>
<td align="right" width="22%">+1.36953437</td>
<td align="right" width="22%">+0.199198E0</td>
<td align="right" width="22%">+0.2408E-1</td>
</tr>
<tr>
<td align="middle" width="10%">4</td>
<td align="right" width="22%">+1.35421147</td>
<td align="right" width="22%">+0.129262E0</td>
<td align="right" width="22%">+0.1532E-1</td>
</tr>
<tr>
<td align="middle" width="10%">5</td>
<td align="right" width="22%">+1.34426827</td>
<td align="right" width="22%">+0.849013E-1</td>
<td align="right" width="22%">+0.9943E-2</td>
</tr>
<tr>
<td align="middle" width="10%">6</td>
<td align="right" width="22%">+1.33773739</td>
<td align="right" width="22%">+0.561990E-1</td>
<td align="right" width="22%">+0.6531E-2</td>
</tr>
<tr>
<td align="middle" width="10%">7</td>
<td align="right" width="22%">+1.3334144</td>
<td align="right" width="22%">+0.373883E-1</td>
<td align="right" width="22%">+0.4323E-2</td>
</tr>
<tr>
<td align="middle" width="10%">8</td>
<td align="right" width="22%">+1.33053837</td>
<td align="right" width="22%">+0.249568E-1</td>
<td align="right" width="22%">+0.2876E-2</td>
</tr>
<tr>
<td align="middle" width="10%">9</td>
<td align="right" width="22%">+1.32861862</td>
<td align="right" width="22%">+0.166954E-1</td>
<td align="right" width="22%">+0.1920E-2</td>
</tr>
<tr>
<td align="middle" width="10%">10</td>
<td align="right" width="22%">+1.32733436</td>
<td align="right" width="22%">+0.111852E-1</td>
<td align="right" width="22%">+0.1284E-2</td>
</tr>
<tr>
<td align="middle" width="10%">11</td>
<td align="right" width="22%">+1.32647396</td>
<td align="right" width="22%">+0.750095E-2</td>
<td align="right" width="22%">+0.8604E-3</td>
</tr>
<tr>
<td align="middle" width="10%">12</td>
<td align="right" width="22%">+1.32589696</td>
<td align="right" width="22%">+0.503354E-2</td>
<td align="right" width="22%">+0.5770E-3</td>
</tr>
<tr>
<td align="middle" width="10%">13</td>
<td align="right" width="22%">+1.32550976</td>
<td align="right" width="22%">+0.337926E-2</td>
<td align="right" width="22%">+0.3872E-3</td>
</tr>
<tr>
<td align="middle" width="10%">14</td>
<td align="right" width="22%">+1.32524982</td>
<td align="right" width="22%">+0.226933E-2</td>
<td align="right" width="22%">+0.2599E-3</td>
</tr>
<tr>
<td align="middle" width="10%">15</td>
<td align="right" width="22%">+1.32507526</td>
<td align="right" width="22%">+0.152426E-2</td>
<td align="right" width="22%">+0.1746E-3</td>
</tr>
<tr>
<td align="middle" width="10%">16</td>
<td align="right" width="22%">+1.32495801</td>
<td align="right" width="22%">+0.102395E-2</td>
<td align="right" width="22%">+0.1173E-3</td>
</tr>
<tr>
<td align="middle" width="10%">17</td>
<td align="right" width="22%">+1.32487924</td>
<td align="right" width="22%">+0.687920E-3</td>
<td align="right" width="22%">+0.7877E-4</td>
</tr>
<tr>
<td align="middle" width="10%">18</td>
<td align="right" width="22%">+1.32482632</td>
<td align="right" width="22%">+0.462192E-3</td>
<td align="right" width="22%">+0.5292E-4</td>
</tr>
<tr>
<td align="middle" width="10%">19</td>
<td align="right" width="22%">+1.32479077</td>
<td align="right" width="22%">+0.310545E-3</td>
<td align="right" width="22%">+0.3555E-4</td>
</tr>
<tr>
<td align="middle" width="10%">20</td>
<td align="right" width="22%">+1.32476688</td>
<td align="right" width="22%">+0.208660E-3</td>
<td align="right" width="22%">+0.2389E-4</td>
</tr>
<tr>
<td align="middle" width="10%">21</td>
<td align="right" width="22%">+1.32475083</td>
<td align="right" width="22%">+0.140204E-3</td>
<td align="right" width="22%">+0.1605E-4</td>
</tr>
<tr>
<td align="middle" width="10%">22</td>
<td align="right" width="22%">+1.32474005</td>
<td align="right" width="22%">+0.942078E-4</td>
<td align="right" width="22%">+0.1078E-4</td>
</tr>
<tr>
<td align="middle" width="10%">23</td>
<td align="right" width="22%">+1.3247328</td>
<td align="right" width="22%">+0.633020E-4</td>
<td align="right" width="22%">+0.7247E-5</td>
</tr>
<tr>
<td align="middle" width="10%">24</td>
<td align="right" width="22%">+1.32472793</td>
<td align="right" width="22%">+0.425354E-4</td>
<td align="right" width="22%">+0.4869E-5</td>
</tr>
<tr>
<td align="middle" width="10%">25</td>
<td align="right" width="22%">+1.32472466</td>
<td align="right" width="22%">+0.285815E-4</td>
<td align="right" width="22%">+0.3272E-5</td>
</tr>
<tr>
<td align="middle" width="10%">26</td>
<td align="right" width="22%">+1.32472246</td>
<td align="right" width="22%">+0.192053E-4</td>
<td align="right" width="22%">+0.2199E-5</td>
</tr>
<tr>
<td align="middle" width="10%">27</td>
<td align="right" width="22%">+1.32472098</td>
<td align="right" width="22%">+0.129050E-4</td>
<td align="right" width="22%">+0.1477E-5</td>
</tr>
<tr>
<td align="middle" width="10%">28</td>
<td align="right" width="22%">+1.32471999</td>
<td align="right" width="22%">+0.867149E-5</td>
<td align="right" width="22%">+0.9927E-6</td>
</tr>
<tr>
<td align="middle" width="10%">29</td>
<td align="right" width="22%">+1.32471932</td>
<td align="right" width="22%">+0.582681E-5</td>
<td align="right" width="22%">+0.6670E-6</td>
</tr>
<tr>
<td align="middle" width="10%">30</td>
<td align="right" width="22%">+1.32471888</td>
<td align="right" width="22%">+0.391533E-5</td>
<td align="right" width="22%">+0.4482E-6</td>
</tr>
<tr>
<td align="middle" width="10%">31</td>
<td align="right" width="22%">+1.32471857</td>
<td align="right" width="22%">+0.263091E-5</td>
<td align="right" width="22%">+0.3012E-6</td>
</tr>
<tr>
<td align="middle" width="10%">32</td>
<td align="right" width="22%">+1.32471837</td>
<td align="right" width="22%">+0.176784E-5</td>
<td align="right" width="22%">+0.2024E-6</td>
</tr>
<tr>
<td align="middle" width="10%">33</td>
<td align="right" width="22%">+1.32471824</td>
<td align="right" width="22%">+0.118790E-5</td>
<td align="right" width="22%">+0.1360E-6</td>
</tr>
<tr>
<td align="middle" width="10%">34</td>
<td align="right" width="22%">+1.32471814</td>
<td align="right" width="22%">+0.798212E-6</td>
<td align="right" width="22%">+0.9138E-7</td>
</tr>
<tr>
<td align="middle" width="10%">35</td>
<td align="right" width="22%">+1.32471808</td>
<td align="right" width="22%">+0.536360E-6</td>
<td align="right" width="22%">+0.6140E-7</td>
</tr>
<tr>
<td align="middle" width="10%">36</td>
<td align="right" width="22%">+1.32471804</td>
<td align="right" width="22%">+0.360408E-6</td>
<td align="right" width="22%">+0.4126E-7</td>
</tr>
<tr>
<td align="middle" width="10%">37</td>
<td align="right" width="22%">+1.32471801</td>
<td align="right" width="22%">+0.242176E-6</td>
<td align="right" width="22%">+0.2772E-7</td>
</tr>
<tr>
<td align="middle" width="10%">38</td>
<td align="right" width="22%">+1.324718</td>
<td align="right" width="22%">+0.162731E-6</td>
<td align="right" width="22%">+0.1863E-7</td>
</tr>
<tr>
<td align="middle" width="10%">39</td>
<td align="right" width="22%">+1.32471798</td>
<td align="right" width="22%">+0.109347E-6</td>
<td align="right" width="22%">+0.1252E-7</td>
</tr>
<tr>
<td align="middle" width="10%">40</td>
<td align="right" width="22%">+1.32471797</td>
<td align="right" width="22%">+0.734759E-7</td>
<td align="right" width="22%">+0.8411E-8</td>
</tr>
</tbody>
</table>
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<strong><font color="#ffcc00" face="楷体_GB2312" size="6"><br>
附：算法的C语言程序如下：
<pre>
#define F(x) x*x*x-x-1
#define G(x) x-(x*x*x-x-1)/13
static float a = 1;
static float b = 2;
static float EPS = 1e-8;
static float X[51],Y[51],R[51];
static int N=51;
P23_BDD()
{ int K;
X[0]=(a+b)/2.0;
Y[0]=F(X[0]);
R[0]=(b-a);
for(K=0;K<N;K++)
{ X[K+1]=T(X[K]);
Y[K+1]=F(X[K]);
R[K+1]=fabs(X[K+1]-X[K]);
if(R[K+1]<EPS) break;
}
return ;
}</pre>
</font></strong>
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