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<h4><a name="SECTION001334030000000000000"></a>
加速方法
</h4>

若需要求解接近某一<i>位移</i> <span class="math-inline">\sigma</span> 的特征值，并且可以便捷地得到分解式
<span class="math-inline">A - \sigma \, I=LU</span>（参见第<a href="node385.html#sec:directsolvers">10.3</a>节），
那么可以将上述算法应用于

<span class="math-inline">\left(A - \sigma \, I\right)^{-1}</span>。接近 <span class="math-inline">\sigma</span> 的特征值将快速收敛。

<p>
此外，还可以采用<i>多项式加速</i>
<a name="7575"></a>
通过将幂 <span class="math-inline">A^{m} </span> 替换为多项式
<span class="math-inline">T_m [ (A - \sigma I) / \rho ] </span> 来加快计算速度，其中 <span class="math-inline">T_m</span>
是第一类切比雪夫多项式，次数为
<span class="math-inline">m</span>，而 <span class="math-inline">\sigma</span> 和 <span class="math-inline">\rho</span> 则提供了对感兴趣频谱部分的平移和缩放。理想情况下，应取
<span class="math-inline">\sigma=(\lambda_{p+1}+\lambda_n)/2</span> 作为中心，

<span class="math-inline">\rho=(\lambda_{p+1}-\lambda_n)/2</span> 作为包含不感兴趣特征值区间的半宽度，这些特征值的估计值应合理。我们假设特征值沿实轴有序排列，并且我们希望在一端找到 <span class="math-inline">p</span> 个特征值。

<p>
通过这些改进，子空间迭代法可能成为一种相当高效的方法，其优势在于易于编码和理解。然而，后续讨论的一些方法通常更受欢迎，因为它们往往能更快地找到特征值/特征向量。

<p>
本节内容大量引自 Demmel [<a href="node421.html#demmelbook">114</a>]，Golub 和 Van Loan [<a href="node421.html#golo96">198</a>]，
以及 Saad [<a href="node421.html#saad92">387</a>]。关于子空间迭代的进一步讨论，建议读者参考 Chatelin [<a href="node421.html#chat93">79</a>]，
Lehoucq 和 Scott [<a href="node421.html#lesc95">292</a>]，Stewart [<a href="node421.html#stew76">422</a>]，以及
Wilkinson [<a href="node421.html#wilk65">457</a>]。另请参阅 Bathe 和 Wilson [<a href="node421.html#bawi76">42</a>] 以及
Jennings [<a href="node421.html#jenn77">242</a>]
关于结构工程方法的论述。

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<address>
Susan Blackford
2000-11-20
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