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<!--End of Navigation Panel--><h1><a name="SECTION001360000000000000000"></a> <a name="rwf:bandsym"></a><a name="8295"></a>
带状Lanczos方法
<br> <em>R. Freund</em>
</h1>
<p>
标准的厄米Lanczos算法利用由矩阵<span class="math-inline">A</span>和一个初始向量<span class="math-inline">b</span>生成的Krylov子空间来近似求解厄米特征值问题<span class="math-inline">Ax = \lambda x</span>。然而，在某些情况下，使用一组<span class="math-inline">p</span>个初始向量而非单一初始向量更为可取。例如，对于具有多个或紧密聚集特征值的矩阵，为了获得对应于这<span class="math-inline">p</span>个特征值的特征空间的基向量，需要使用由<span class="math-inline">A</span>和一组<span class="math-inline">p</span>个初始向量生成的块Krylov子空间。
<p>
一个重要的应用场景是<span class="math-inline">m</span>输入<span class="math-inline">p</span>输出的线性动态系统的降阶建模，其中多个初始向量作为问题的一部分给出；详见§<a href="node260.html#rwf:reduced_order">7.10.4</a>节。尽管这类应用通常涉及非厄米方阵<span class="math-inline">A</span>、具有<span class="math-inline">m</span>列的矩形“输入”矩阵<span class="math-inline">B</span>和具有<span class="math-inline">p</span>列的矩形“输出”矩阵<span class="math-inline">C</span>，但当<span class="math-inline">A</span>为厄米矩阵且输入和输出矩阵<span class="math-inline">B</span>与<span class="math-inline">C</span>相同时，具有特殊重要性。例如，这种情况在VLSI电路互连建模的背景下出现；参见例如[<a href="node421.html#freu98">176</a>]。对于这种特殊情况，需要将厄米Lanczos算法扩展到多个初始向量，即<span class="math-inline">B=C</span>的<span class="math-inline">p</span>列。
<p>
最后，当计算矩阵-矩阵乘积<span class="math-inline">AV</span>（其中<span class="math-inline">V</span>是具有<span class="math-inline">p</span>列的矩阵）比依次计算<span class="math-inline">p</span>个向量的矩阵-向量乘积<span class="math-inline">Av</span>更经济时，利用块Krylov子空间也是有益的。基于块Krylov子空间的Lanczos方法允许将所有必要的<span class="math-inline">A</span>乘法计算为块大小为<span class="math-inline">p</span>的矩阵-矩阵乘积<span class="math-inline">AV</span>，而在标准厄米Lanczos算法中，<span class="math-inline">A</span>的乘法必须计算为一连串的矩阵-向量乘积<span class="math-inline">Av</span>。
<p>
在本节中，我们描述了一种用于厄米特征值问题的带状Lanczos方法，
<div class="math-display" name="#rwf:AAA">A x = \lambda x, \tag{4.25}</div>
该方法基于由矩阵<span class="math-inline">A</span>和一组<span class="math-inline">p</span>个初始向量生成的块Krylov子空间，
<div class="math-display" name="#rwf:AAB">b_1,b_2,\ldots,b_p. \tag{4.26}</div>
矩阵<span class="math-inline">A</span>和初始向量(<a href="node131.html#rwf:AAB">4.26</a>)生成了块Krylov序列
<div class="math-display" name="#rwf:ABA">b_1,b_2,\ldots,b_p,A b_1, A b_2, \ldots, A b_p,\ldots,A^{k-1} b_1, A^{k-1} b_2, \ldots, A^{k-1} b_p, \ldots. \tag{4.27}</div>
带状Lanczos算法的目标是构造正交向量，
<div class="math-display" name="#rwf:ABB">v_1,v_2,\ldots,v_j, \tag{4.28}</div>
这些向量构成了块Krylov序列(<a href="node131.html#rwf:ABA">4.27</a>)中前<span class="math-inline">j</span>个线性无关向量所张成子空间的基。
<p>
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<li><a name="tex2html2718" href="node132.html">收缩的必要性</a>
<li><a name="tex2html2719" href="node133.html">基本性质</a>
<li><a name="tex2html2720" href="node134.html">算法</a>
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<address>
Susan Blackford
2000-11-20
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