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<!DOCTYPE html>
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<html>
<head>
<title>收敛性质</title>
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<br>
<b>下一节：</b><a name="tex2html3350" href="node173.html">多重特征值</a>
<b>上一级：</b><a name="tex2html3344" href="node169.html">Lanczos方法</a>
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收敛性质
</h4>
收敛性受标准情况下的同一正交多项式理论支配，例如，参见[<a href="node421.html#parl80">353</a>]。
<p>
该理论指出，我们能收敛到起始向量中表示的那些特征值，且在谱的两端的特征值收敛更快。这些特征值与剩余特征值的分离程度越好，它们的收敛速度就越快。
<p>
在实际应用中，我们往往只对最低特征值感兴趣，而直接迭代算法[<a href="node170.html#Gen_Herm_Lanczos">5.4</a>]中这些特征值是首先收敛的，这是有益的。另一方面，最低特征值的相对分离度常常不佳——记住，分离度是相对于整个谱的分布，而非与原点的距离。
<p>
在这些情况下，当我们希望获得指定范围内的特征值，例如
<span class="math-inline">\alpha \le \lambda \le \beta</span>，采用适当选择的位移<span class="math-inline">\sigma</span>的位移-反演算法[<a href="node171.html#si-Gen_Herm_Lanczos">5.5</a>]，例如在区间
<span class="math-inline">\alpha \le \sigma \le \beta</span>内，具有极大的优势。
<p>
在广义情况下，位移-反演策略比标准情况更具动力，因为我们无论如何都需解一个系统，无论是在步骤4还是步骤9。位移-反演算子<span class="math-inline">C</span>（[<a href="node171.html#gen_shift_invert">5.16</a>]），通常具有远为更好的分离特征值，仅需<span class="math-inline">j=50</span>而非数百次迭代就能得到10个特征值。
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<br>
<b>下一节：</b><a name="tex2html3350" href="node173.html">多重特征值</a>
<b>上一级：</b><a name="tex2html3344" href="node169.html">Lanczos方法</a>
<b>上一节：</b><a name="tex2html3338" href="node171.html">带SI的Lanczos算法</a>
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<address>
Susan Blackford
2000-11-20
</address>
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</html>