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node114.html
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<!DOCTYPE html>
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<html>
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<title>L形膜片结果</title>
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<h4><a name="SECTION001346010000000000000">
L形膜片的结果。</a>
</h4>
对于直接应用Lanczos算法的情况，极值特征值首先收敛，并且在这种情况下，收敛在谱的两端都非常相似。
<p>
<div style="text-align: center;">
<a name="7915"></a>
<img src="icon/LmembLanRes.png" alt="图4.1：L形膜片矩阵的残差估计。" id="LmembLanRes"/>
<figcaption>图4.1：L形膜片矩阵的残差估计。</figcaption>
</div>
<p>
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<img src="icon/LmembLanRitz.png" alt="图4.2：L形膜片矩阵的Ritz值。" id="LmembLanRitz"/>
<figcaption>图4.2：L形膜片矩阵的Ritz值。</figcaption>
</div>
<p>
我们在图<a href="node114.html#LmembLanRes">4.1</a>中绘制了作为Lanczos步数<span class="math-inline">j</span>函数的六个最大特征值的估计残差（<a href="node103.html#estimate_residual">4.13</a>）。这些曲线显示了每一步的残差估计（<a href="node103.html#estimate_residual">4.13</a>），并且仅在实际计算后为了说明目的而计算。LANSO算法调用QL算法来计算特征值和特征向量的最后元素以测试收敛性（<a href="node103.html#estimate_residual">4.13</a>），在图中用虚线垂直线标记的迭代中。选择性正交化在我们用虚线标记的步骤中触发了重新正交化，总共在这300步中只发生了三次。这是典型的情况，收敛缓慢：正交性一直保持到第一个Ritz值收敛。
<p>
需要注意的是，大约需要150步才能将主导特征值的残差降低到<span class="math-inline">10^{-4}</span>，并且还需要另外150步才能达到完全的机器精度。观察图<a href="node114.html#LmembLanRitz">4.2</a>中绘制的Ritz值<span class="math-inline">\theta^{(j)}_i</span>（<a href="node103.html#T-eigenvalues">4.11</a>）与<span class="math-inline">j</span>的关系，我们可以看到最大的Ritz值随着<span class="math-inline">j</span>增长直到大约第60步时趋于稳定。通常一个Ritz值会开始接近一个特征值，但随后会移动到另一个尚未找到的特征值。这在第60到80步的第三个Ritz值和第150到210步的第六个Ritz值中表现得更为明显。这种现象在图<a href="node114.html#LmembLanRes">4.1</a>中以不太明显的方式出现，其中第三和第六条曲线在Ritz值改变归属的步骤处出现了平台。直接使用Lanczos方法的用户在决定是否所有最大特征值都真正收敛时应谨慎。
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<br>
<b>下一节：</b><a name="tex2html2468" href="node115.html">Medline SVD 结果</a>
<b>上一级：</b><a name="tex2html2462" href="node113.html">数值示例</a>
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<address>
Susan Blackford
2000-11-20
</address>
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