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<!DOCTYPE html>
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<title>Stability and Accuracy Assessments Z. Bai and R. Li </title>
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<!--End of Navigation Panel--><h1><a name="SECTION001470000000000000000"></a><a name="sec:gsym_pert"></a><a name="18145"></a>
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稳定性与精度评估
<br> <em>Z. Bai 和 R. Li</em>
</h1>
<p>
对于一个厄米矩阵对 <span class="math-inline">\{A,B\}</span> 的广义特征值问题，其中 <span class="math-inline">A</span> 和 <span class="math-inline">B</span> 之一或它们的某种线性组合是正定的，这种情况在所有矩阵对广义特征值问题中占据独特地位，因为它在许多方面与第四章<a href="node85.html#chap:heig">4</a>讨论的标准厄米特征值问题相似。此类矩阵对被称为<em>厄米正定对</em>。我们将分别考虑以下两种情况：
<ol>
<li><span class="math-inline">B</span> 是正定且良态的，这意味着
<span class="math-inline">\kappa(B)\equiv \Vert B\Vert _2\Vert B^{-1}\Vert _2</span> 不太大。<sup><a href="#footnote-b-condition-number">[1]</a></sup>
</li>
<li>某些 <span class="math-inline">A</span> 和 <span class="math-inline">B</span> 的组合是正定且良态的。
</li>
</ol>
在本节中，我们仅回顾一些基本结果，这些结果可直接用于评估计算得到的特征值和特征向量的准确性。我们假设在成功计算结束后通常可获得残差向量，如果不可获得，之后也可以较低成本计算。
<p>
关于稠密广义厄米特征问题的计算特征值和特征向量的误差估计处理，请参见《LAPACK用户指南》第四章[<a href="node421.html#lapack">12</a>]。
<p>
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<li><a name="tex2html3417" href="node178.html">正定 <span class="math-inline">B</span></a>
<ul>
<li><a name="tex2html3418" href="node179.html">残差向量</a>
<li><a name="tex2html3419" href="node180.html">将残差误差转化为后向误差</a>
<li><a name="tex2html3420" href="node181.html">计算特征值的误差界限</a>
<li><a name="tex2html3421" href="node182.html">计算特征向量的误差界限</a>
<li><a name="tex2html3422" href="node183.html">关于聚集特征值的备注</a>
</ul>
<li><a name="tex2html3423" href="node184.html">某些 <span class="math-inline">A</span> 和 <span class="math-inline">B</span> 的组合是正定的</a>
<ul>
<li><a name="tex2html3424" href="node185.html">残差向量</a>
<li><a name="tex2html3425" href="node186.html">将残差误差转化为后向误差</a>
<li><a name="tex2html3426" href="node187.html">计算特征值的误差界限</a>
<li><a name="tex2html3427" href="node188.html">计算特征向量的误差界限</a>
<li><a name="tex2html3428" href="node189.html">关于聚集特征值的备注</a>
</ul></ul>
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<ol>
<li id="footnote-b-condition-number">究竟多大的数值才算“过大”，这是一个模糊的概念，通常需要根据具体情况来处理。一般而言，我们可能会认为，任何超过1000的条件数都可以被视为“过大”。</li>
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<b>下一节：</b><a name="tex2html3416" href="node178.html">正定 <span class="math-inline">B</span></a>
<b>上一级：</b><a name="tex2html3410" href="node155.html">广义厄米特征值问题</a>
<b>上一节：</b><a name="tex2html3406" href="node176.html">Jacobi-Davidson方法</a>
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<address>
Susan Blackford
2000-11-20
</address>
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