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<p ALIGN="center"><b><font color="#FFFF00" size="6">5.2.3 <font FACE="??ì?,SimSun" LANG="ZH-CN">带状矩阵</font></font></b></p>
<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b> n<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">阶矩阵</font>A<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">称为带状矩阵，如果存在最小正数</font>m
<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">，满足当∣</font>i-j<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">∣≥</font>m
<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">时，</font>a<sub>ij</sub>
=0<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">，这时称</font>w=2n-1<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">为矩阵</font>A<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">的带宽。如图</font>5.10(a)<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">是一个</font>w=3(m=2)<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">的带状矩阵。带状矩阵也称为对角矩阵。由图</font>5.10(a)<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">可看出，在这种矩阵中，所有非零元素都集中在以主对角线为中心的带状区域中，即除了主对角线和它的上下方若干条对角线的元素外，所有其他元素都为零</font>(<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">或同一个常数</font>c)<font FACE="??ì?,SimSun" LANG="ZH-CN">。</font></b></font></p>
<p><font size="5" color="#FFFFFF"><b><font FACE="??ì?,SimSun" LANG="ZH-CN">带状矩阵</font>A<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">也可以采用压缩存储。一种压缩方法是将</font>A<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">压缩到一个</font>n<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">行</font>w<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">列的二维数组</font>B<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">中，如图</font>5.10(b)<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">所示，当某行非零元素的个数小于带宽</font>w<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">时，先存放非零元素后补零。那么</font>a<sub>ij</sub>
<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">映射为</font>b<sub>
i<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">′</font>j<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">′</font></sub><font FACE="??ì?,SimSun" LANG="ZH-CN">，映射关系为：
</font>
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<font FACE="??ì?,SimSun" LANG="ZH-CN">
另一种压缩方法是将带状矩阵压缩到向量</font>C<font FACE="??ì?,SimSun" LANG="ZH-CN" SIZE="3">中去，按以行为主序，顺序的存储其非零元素，如图</font>5.10(c)<font FACE="??ì?,SimSun" LANG="ZH-CN">所示，按其压缩规律，找到相应的映象函数。
</font></b></font></p>
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<p ALIGN="JUSTIFY"></font><font size="5" color="#FFFFFF"><b><font FACE="??ì?,SimSun" LANG="ZH-CN">如当</font>w=3<font FACE="??ì?,SimSun" LANG="ZH-CN">时，映象函数为：
</font></b></font></p>
<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>k=2*i+j-3</b></font></p>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>0</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>1</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>2</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>3</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>4</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>5</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>6</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>7</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>8</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>9</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>10</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>11</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>12</b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>11</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>A<sub>12</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>21</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>22</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>23</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>32</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>33</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>34</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>43</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>44</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>45</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>54</sub></b></font></td>
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<p ALIGN="JUSTIFY"><font size="5" color="#FFFFFF"><b>a<sub>55</sub></b></font></td>
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