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node185.html
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<!DOCTYPE html>
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<html>
<head>
<title>残差向量</title>
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<meta name="keywords" content="book, math, eigenvalue, eigenvector, linear algebra, sparse matrix">
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<br>
<b>下一节：</b><a name="tex2html3546" href="node186.html">将残差误差转化为后向误差</a>
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<h4><a name="SECTION001472010000000000000"></a>
<a name="18319"></a>
残差向量
</h4>
假设
<span class="math-inline">(\widetilde\alpha,\widetilde\beta)</span> 表示一个计算得到的特征值，而
<span class="math-inline">\widetilde x</span> 是其对应的计算特征向量。为了简化，我们对其进行归一化处理，使得
<div class="math-display">\Vert\widetilde x\Vert _2 = 1, \quad \vert\widetilde\alpha\vert^2+\vert\widetilde\beta\vert^2=1.</div>
对应的<em>残差向量</em>或<em>残差误差</em>定义为
<div class="math-display">r = \widetilde\beta A\widetilde x - \widetilde\alpha B\widetilde x.</div>
理想情况下，我们希望 <span class="math-inline">r=0</span>，但实际上 <span class="math-inline">\Vert r\Vert _2</span> 很小。可以想象，小的残差误差意味着计算得到的
<span class="math-inline">(\widetilde\alpha,\widetilde\beta)</span> 和 <span class="math-inline">\widetilde x</span> 具有良好的精度。我们感兴趣的是，在给定 <span class="math-inline">r</span> 的情况下，如何准确评估计算得到的
<span class="math-inline">(\widetilde\alpha,\widetilde\beta)</span> 和 <span class="math-inline">\widetilde x</span> 的精度。
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<br>
<b>下一节：</b><a name="tex2html3546" href="node186.html">将残差误差转化为后向误差</a>
<b>上一级：</b><a name="tex2html3540" href="node184.html">某些<span class="math-inline">A</span>与<span class="math-inline">B</span>的组合是正定矩阵</a>
<b>上一节：</b><a name="tex2html3534" href="node184.html">某些<span class="math-inline">A</span>与<span class="math-inline">B</span>的组合是正定矩阵</a>
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<address>
Susan Blackford
2000-11-20
</address>
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