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<td width="1000" height="432" valign="top"><SPAN class=addtime>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 20pt; mso-line-height-rule: exactly"><hr></P>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 20pt; mso-line-height-rule: exactly"><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">线性方程组的解法有两类：<SPAN
lang=EN-US><o:p></o:p></SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 0cm; LINE-HEIGHT: 20pt; mso-list: l73 level1 lfo10; mso-line-height-rule: exactly; tab-stops: list 31.5pt"><SPAN
lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">a)<SPAN
style="FONT: 7pt 'Times New Roman'">???????????? </SPAN></SPAN><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">直接法<SPAN
lang=EN-US><o:p></o:p></SPAN></SPAN></P>
</SPAN> <P class=MsoBodyTextIndent3 style2
style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 20pt; mso-line-height-rule: exactly"><span class="addtime"><FONT
face=宋体>所谓的直接法，就是经过有限此的运算即可求得方程组的精确解的方法（如果没有舍入误差的话）。他的优点是可以直接得到方程组得精确解，适合与系数矩阵阶数不太高得时候，但是程序较复杂。</FONT></span></P>
<SPAN class=addtime>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 20pt; mso-line-height-rule: exactly"><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">如：高斯消元法（<SPAN
lang=EN-US>Gauss）、LU分解法等等。<o:p></o:p></SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 0cm; LINE-HEIGHT: 20pt; mso-list: l73 level1 lfo10; mso-line-height-rule: exactly; tab-stops: list 31.5pt"><SPAN
lang=EN-US style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">b)<SPAN
style="FONT: 7pt 'Times New Roman'">???????????? </SPAN></SPAN><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">迭代法<SPAN
lang=EN-US><o:p></o:p></SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 20pt; mso-line-height-rule: exactly"><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">所谓的迭代法，与直接法相反，它将求解方程组的问题转化为一个无限的求解系列，其极限就是方程组的解。所以在有限步内，迭代法是得不到精确解得。它的优点是使用于系数矩阵阶数太高的问题，但是工作量比较大。<SPAN
lang=EN-US><o:p></o:p></SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 20pt; mso-line-height-rule: exactly"><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">如：雅克比迭代（<SPAN
lang=EN-US>Jacobi）、高斯－赛德尔迭代（Gauss－Seidel）等</SPAN></SPAN>。</P>
</SPAN>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; LINE-HEIGHT: 20pt; mso-char-indent-count: 2.0; mso-char-indent-size: 12.0pt; mso-line-height-rule: exactly"><span class="addtime style2">高斯消元法的步骤分为两个过程：消元和回代。这个方法在解线性方程组中是常用的一种方法。但是，当它用于程序设计中时可能会存在一个不稳定、不可靠的问题，不能对所有的系数矩阵都能适用。它是依次地对系数矩阵的对角变量进行消元。这就要求这个系数矩阵必须是严格对角占优的。只有这样才能保证消元过程中高斯消元的主元不等<FONT
face="Times New Roman">0</FONT>。<o:p></o:p></span></P>
<P class=MsoNormal style2
style="MARGIN: 0cm 0cm 0pt; LINE-HEIGHT: 20pt; mso-line-height-rule: exactly"><span class="addtime">如果，在程序设计中直接用高斯消元法将极可能发生除零中断。而且，算法无法依次消元。<o:p></o:p></span></P>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; LINE-HEIGHT: 20pt; mso-char-indent-count: 2.0; mso-char-indent-size: 12.0pt; mso-line-height-rule: exactly"><span class="addtime style2">要实现一个计算稳定，能应对所有“严格对角占优”和“非严格对角占优”的系数矩阵，并且计算灵巧、高效，必须对高斯消元法进行一些改进。</span><span class="addtime style1"><o:p></o:p></span><span class="addtime"><o:p></o:p></span></P>
<P class=MsoNormal
style="MARGIN: 0cm 0cm 0pt; TEXT-INDENT: 24pt; LINE-HEIGHT: 20pt; mso-char-indent-count: 2.0; mso-char-indent-size: 12.0pt; mso-line-height-rule: exactly"><span class="addtime"><o:p><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">雅克比迭代（<SPAN
lang=EN-US>Jacobi）、高斯－赛德尔迭代（Gauss－Seidel）</SPAN></SPAN></o:p></span><span class="style2">具体程序实现如下：</span></P>
<DIV id=td_content>
<DIV>
<p>//<span class="MsoNormal"><span class="addtime"><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体">雅克比迭代</SPAN></span></span></p>
<p><FONT size=3>#include <iostream></FONT></p>
</DIV>
<DIV><FONT size=3>#include <string></FONT></DIV>
<DIV><FONT size=3>#include <vector></FONT></DIV>
<DIV><FONT size=3>#include <iomanip></FONT></DIV>
<DIV><FONT size=3>using namespace std;</FONT></DIV>
<DIV>?</DIV>
<DIV><FONT size=3>const int n=3;<SPAN>????????? //</SPAN>设置方程组的维</FONT></DIV>
<DIV>?</DIV>
<DIV><FONT size=3>float a[n][n],x[n],b[n];</FONT></DIV>
<DIV>?</DIV>
<DIV><FONT size=3>void input_data() //输入方程组的相关数据</FONT></DIV>
<DIV><FONT size=3>{<SPAN>?? </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<"</SPAN>输入方程组的系数矩阵<SPAN>a["<<n<<"]["<<n<<"]:"<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? for(int i=0;i<n;i++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? for(int j=0;j<n;j++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? cin>>a[i][j];</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<"</SPAN>输入<SPAN>x[1,...,"<<n<<"]的初值:"<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? for(i=0;i<n;i++)??????? </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? cin>>x[i];</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<"</SPAN>输入<SPAN>b[1,...,"<<n<<"]的初值:"<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? for(i=0;i<n;i++)??????? </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? cin>>b[i];?</SPAN></FONT></DIV>
<DIV><FONT size=3>}</FONT></DIV>
<DIV>?</DIV>
<DIV><FONT size=3>void output_equation()?//输出方程组</FONT></DIV>
<DIV><FONT size=3>{</FONT></DIV>
<DIV><FONT size=3><SPAN>??? for(int i=0;i<n;i++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? {</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? for(int j=0;j<n;j++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? {</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? if(j>0&&a[i][j]>0)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????????? cout<<"+";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? cout<<a[i][j]<<"?x["<<j+1<<"]?";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? }</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? cout<<"="<<b[i]<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? }</SPAN></FONT></DIV>
<DIV><FONT size=3>}</FONT></DIV>
<DIV>?</DIV>
<DIV><FONT size=3>void main ()</FONT></DIV>
<DIV><FONT size=3>{</FONT></DIV>
<DIV><FONT size=3><SPAN>??? input_data();</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? int counter;??????????? //</SPAN>设置计算时迭代的次数</FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<"</SPAN>输入迭代的次数<SPAN>"<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cin>>counter;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? system("cls");</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? output_equation();</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? int k=0,i=0;??????????? //k</SPAN>用来统计迭代的次数<SPAN>;i用来表示x的序列</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? float m_num1;???? //m_x</SPAN>用来保存<SPAN>x的原先值;m_num1用来计算j!=i的数据之和</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<"k\t";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? for(int j=0;j<n;j++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? cout<<"x["<<j+1<<"]\t\t\t";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<k<<"\t";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? k++;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout << setiosflags(ios::fixed);?? //</SPAN>和下一句结合起来用来设定小数位的位数</FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout << setprecision(10);</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? float x_next[n];</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? for(j=0;j<n;j++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? {</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? cout<<x[j]<<"\t\t";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? }</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? cout<<endl; </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? do</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? {?????? </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? m_num1=0;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? for(j=0;j<n;j++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? {</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? if(j!=i)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????????? m_num1=m_num1+a[i][j]*x[j];???? </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? }</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? x_next[i]=(b[i]-m_num1)/a[i][i];??????? </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? if(i<n-1)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? i++;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? else</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? {</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? for(j=0;j<n;j++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????????? x[j]=x_next[j];</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? cout<<k<<"\t";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? for(j=0;j<n;j++)</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????????? cout<<x[j]<<"\t\t";</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? cout<<endl;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? i=0;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????????? k++;</SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??????? };????? </SPAN></FONT></DIV>
<DIV><FONT size=3><SPAN>??? }while(k<=counter);</SPAN></FONT></DIV>
<DIV>
<p><FONT size=3>}</FONT></p>
<p> </p>
<p><font size="3">//<span class="MsoNormal"><span class="addtime"><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 宋体"><SPAN
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