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<!DOCTYPE html>
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<html>
<head>
<title>收缩</title>
<meta charset="utf-8">
<meta name="description" content="收缩">
<meta name="keywords" content="book, math, eigenvalue, eigenvector, linear algebra, sparse matrix">
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收缩
</h4>
<p>
当一个Ritz值足够接近某个特征值时，当前子空间的剩余部分将已经包含丰富的邻近特征对成分，因为我们已经在每一步中选择了与期望特征值接近的Ritz向量。我们可以利用这些信息作为计算下一个特征向量的子空间基础。为了防止已计算的特征向量重新进入计算过程，我们在Jacobi-Davidson算法中使新的搜索向量显式地与已计算的特征向量正交。这种技术被称为显式<em>收缩</em>。我们将在稍详细地讨论这一点。
<p>
设<span class="math-inline">\widetilde{x}_1,\ldots,\widetilde{x}_{k-1}</span>表示已接受的特征向量近似，并假设这些向量是正交归一的。矩阵<span class="math-inline">\widetilde{X}_{k-1}</span>的列向量为<span class="math-inline">\widetilde{x}_j</span>。为了找到下一个特征向量<span class="math-inline">\widetilde{x}_k</span>，我们将Jacobi-Davidson算法应用于收缩矩阵 <sup><a href="#footnote-deflated-matrix">1</a></sup>
<div class="math-display">(I-\widetilde{X}_{k-1}\widetilde{X}_{k-1}^\ast)\,A\,(I-\widetilde{X}_{k-1}\widetilde{X}_{k-1}^\ast) ,</div>
这将导致一个修正方程，形式如下
<div class="math-display" name="#eq:jddefl">{P}_{m}(I-\widetilde{X}_{k-1}\widetilde{X}_{k-1}^\ast)(A-\theta^{(m)}_j I)(I-\widetilde{X}_{k-1}\widetilde{X}_{k-1}^\ast){P}_{m} t_j^{(m)} = -r_j^{({m})}, \tag{4.50}</div>
其中<span class="math-inline">{P}_{m}\equiv (I-{u}_j^{(m)}{{u}_j^{(m)}}^{\ast})</span>，需要为每个新的特征向量近似<span class="math-inline">{u}_j^{(m)}</span>求解修正<span class="math-inline">t_j^{(m)}</span>，对应的Ritz值为<span class="math-inline">\theta_j^{(m)}</span>。在[<a href="node421.html#fosv98">172</a>]中，通过数值证据表明，对于修正方程，显式地对由<span class="math-inline">\widetilde{X}_{k-1}</span>表示的向量进行收缩是非常推荐的，但在计算投影矩阵时（将<span class="math-inline">A</span>投影到由连续近似<span class="math-inline">{v}_j</span>张成的子空间上，以寻找第<span class="math-inline">k</span>个特征向量），并不需要包含这种收缩。投影矩阵可以计算为<span class="math-inline">{V}_{m}^\ast A{V}_{m}</span>，而不会显著损失精度。
<p>
<hr>
<ol>
<li id="footnote-deflated-matrix">
在利用近似特征向量进行收缩时，若计算出的特征值与剩余特征值之间有良好的分离度，则可能会在特征值上引入数量级为 <span class="math-inline">\epsilon^2</span> 的误差[<a href="node421.html#parl80">353</a>，第5.1节]。
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<address>
Susan Blackford
2000-11-20
</address>
</body>
</html>