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<!DOCTYPE html>
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<title>可用算法综述</title>
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可用算法概览</a>
</h4>
<p>
在许多情况下，我们可以将矩阵束（参见<a href="node156.html#gsymeig">5.1</a>节）转化为第四章<a href="node85.html#chap:heig">4</a>讨论的标准厄米特征值问题(HEP)，或在某些情况下转化为第七章<a href="node203.html#chap:nep">7</a>讨论的标准非厄米特征值问题(NHEP)，并使用这两章中描述的任何算法。我们在§<a href="node163.html#sec:dense_gsym">5.2</a>中简要回顾了此类转换。
<p>
本章的主体部分专门介绍针对广义厄米矩阵束（参见<a href="node156.html#gsymeig">5.1</a>节）的特殊形式的算法：
<dl>
<dt><strong><i>幂法与逆迭代</i></strong></dt>
<dd>§<a href="node165.html#sec_power_gsym">5.4</a>介绍的基本迭代方法。它从一个适当的初始向量开始，在每次迭代中进行一次矩阵-向量乘法和一次线性系统求解。由于这两种操作都需要，它不如标准情况下的方法吸引人，但由于其简单性以及为了阐明它与后续描述的更复杂方案的关系，我们在此将其包括进来。
<p>
</dd>
<dt><strong><i>Lanczos方法</i></strong></dt>
<dd>§<a href="node169.html#sec:lan_gsym">5.5</a>构建了一个<span class="math-inline">B</span>正交基，该基由矩阵算子表示为三对角矩阵<span class="math-inline">T</span>的Krylov序列向量组成，<span class="math-inline">T</span>的特征值产生原始矩阵束（参见<a href="node156.html#gsymeig">5.1</a>节）的几个特征值的Ritz近似。我们将在§<a href="node169.html#sec:lan_gsym">5.5</a>中描述两种变体，一种直接迭代需要进行<span class="math-inline">A</span>的乘法运算和<span class="math-inline">B</span>的系统求解，另一种位移-逆迭代需要进行<span class="math-inline">(A-\sigma B)</span>的系统求解和<span class="math-inline">B</span>的乘法运算。
<p>
</dd>
<dt><strong><i>Jacobi-Davidson方法</i></strong></dt>
<dd>§<a href="node176.html#sec:jacdav_gsym">5.6</a>更新一系列子空间，操作预处理的位移矩阵。它使用<span class="math-inline">B</span>正交性，不需要线性系统的求解。这使得它在矩阵太大以至于无法进行线性系统求解的情况下（如本章描述的其他算法所需）仍然适用。
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<address>
Susan Blackford
2000-11-20
</address>
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