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node113.html
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<!DOCTYPE html>
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<title>数值示例</title>
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<b>下一节：</b><a name="tex2html2451" href="node114.html">L形膜片结果</a>
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<h2><a name="SECTION001346000000000000000"></a>
<a name="sec:laneg"></a>
数值示例
</h2>
我们将考虑应用于两个简单但典型的示例的Lanczos算法，使用的是LANSO软件。
<p>
第一个示例是计算L形膜的一些特征模态。采用网格间距<span class="math-inline">h=1/64</span>的标准五点有限差分近似，得到一个阶数为<span class="math-inline">n=2945</span>的稀疏对称正定矩阵。它将具有非常规则的稀疏带状结构，每行最多有五个非零元素。其特征值将位于区间<span class="math-inline">0<\lambda_i<8</span>内，对称分布在<span class="math-inline">\lambda=4</span>周围，并且在谱的两端分布更密集。
<p>
第二个测试示例来自信息检索应用，涉及一个术语-文档矩阵，其中每一列代表一个文档，每一行代表一个术语。元素<span class="math-inline">x_{i,j}</span>为1，如果术语<span class="math-inline">i</span>出现在文档<span class="math-inline">j</span>中，否则为零。一组主要奇异值的空间可用于发现某些文档之间的联系。具体的矩阵M<small>EDLINE</small>是大小为<span class="math-inline">7014\times 1033</span>的矩形矩阵，中等稀疏，填充了53,287个元素，即每行大约有8个元素。显式形成乘积<span class="math-inline">A=X^TX</span>不是一个好主意，因为这个矩阵将几乎全满，有910,755个元素（或<span class="math-inline">85\%</span>）非零。我们通过先计算<span class="math-inline">z=Xx</span>，然后计算<span class="math-inline">y=X^Tz</span>来实现乘积<span class="math-inline">y=Ax</span>。
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<li><a name="tex2html2452" href="node114.html">L形膜的结果</a>
<li><a name="tex2html2453" href="node115.html">Medline SVD的结果</a>
<li><a name="tex2html2454" href="node116.html">带位移-反转变换的L形膜结果</a>
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<b>上一节：</b><a name="tex2html2441" href="node112.html">可用的软件</a>
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<address>
Susan Blackford
2000-11-20
</address>
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