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<title>收敛性质</title>
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<b>下一节：</b><a name="tex2html2341" href="node106.html">多重特征值</a>
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<h2>
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收敛性质
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基于正交多项式的特性，存在一个优美的理论来描述特征值何时收敛。三对角矩阵<span class="math-inline">T_{j}</span>的特征多项式是关于一个标量积的正交多项式，该标量积由起始向量<span class="math-inline">v</span>作为特征向量之和的展开定义。通过将这些未知的正交多项式替换为著名的切比雪夫多项式，可以得到<span class="math-inline">T_{j}</span>的特征值<span class="math-inline">\theta_i^{(j)}</span>与<span class="math-inline">A</span>的特征值<span class="math-inline">\lambda_i</span>之间的差异<span class="math-inline">\vert\theta_i^{(j)}-\lambda_i\vert</span>的界限，即所谓的Kaniel-Paige-Saad界限；参见[<a href="node421.html#parl80">353</a>]。
<p>
这一理论表明，我们会收敛到起始向量中表示的那些特征值，并且对于频谱两端的特征值，收敛速度更快。这些特征值与其他特征值的分离程度越好，它们的收敛速度就越快。
<p>
在实际情况下，我们通常只对最低的特征值感兴趣，幸运的是，这些特征值往往是最先收敛的。另一方面，最低特征值的相对分离度通常较差，因为这种分离是相对于整个频谱范围而言的，而不是相对于到原点的距离。
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<address>
Susan Blackford
2000-11-20
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