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node116.html
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<!DOCTYPE html>
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<html>
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<title>L形膜片位移-逆结果</title>
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<meta name="keywords" content="book, math, eigenvalue, eigenvector, linear algebra, sparse matrix">
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<b>下一节：</b><a name="tex2html2494" href="node117.html">隐式重启Lanczos</a>
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<h4><a name="SECTION001346030000000000000">
L形膜片位移-逆结果</a>
</h4>
为了说明将Lanczos算法应用于移位-逆变换算子（参见<a href="node107.html#shift_invert">4.14节</a>）的效果，我们跟踪了在原点施加移位<span class="math-inline">\sigma=0</span>的L型薄膜矩阵的收敛情况。如图<a href="node114.html#Lmemb5SIres">4.4</a>所示，现在的图像与Medline的SVD示例非常相似，并且在经过<span class="math-inline">j=36</span>步后，我们获得了六个最小特征值的全部精度。在此过程中，我们采用了上述推荐的完全重正交化方法。即使将步数<span class="math-inline">j</span>减少超过一个数量级，我们也必须考虑到因式分解（参见<a href="node107.html#shift_invert_factor">4.16节</a>）需要一定时间，并且因子比原始矩阵<span class="math-inline">A</span>更密集，因此在每一步中应用它们的逆（参见<a href="node107.html#ulinvx">4.17节</a>）比直接Lanczos方法中应用原始矩阵<span class="math-inline">A</span>（参见<a href="node103.html#Ax_op">4.8节</a>）需要更多的计算工作。
<div style="text-align: center;">
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<img src="icon/Lmemb5SIres.png" alt="图4.4：移位反转L形膜片矩阵的残差估计。" id="Lmemb5SIres"/>
<figcaption>图4.4：移位反转L形膜片矩阵的残差估计。</figcaption>
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<br>
<b>下一节：</b><a name="tex2html2494" href="node117.html">隐式重启Lanczos</a>
<b>上一级：</b><a name="tex2html2488" href="node113.html">数值示例</a>
<b>上一节：</b><a name="tex2html2484" href="node115.html">Medline SVD结果</a>
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<address>
Susan Blackford
2000-11-20
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