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<title>将残差误差转化为后向误差</title>
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将残差误差转化为后向误差
</h4>
可以证明存在厄米矩阵 <span class="math-inline">E</span>，例如
<div class="math-display" id="eq:E2-pdpBdef">E = -r\widetilde x^*-\widetilde x r^*+\left(\widetilde x^*A\widetilde x-\widetilde\lambda \widetilde x^*B\widetilde x \right)\widetilde x\widetilde x^*, \tag{5.29}</div>
使得 <span class="math-inline">\widetilde\lambda</span> 和 <span class="math-inline">\widetilde x</span> 是 <span class="math-inline">\{A+E,B\}</span> 的精确特征值及其对应的特征向量。
我们感兴趣的是那些范数尽可能小的矩阵 <span class="math-inline">E</span>。
事实证明，对于谱范数 <span class="math-inline">\Vert\cdot\Vert _2</span> 而言，最佳的 <span class="math-inline">E=E_2</span> 以及对于 Frobenius 范数 <span class="math-inline">\Vert\cdot\Vert _{F}</span> 而言，最佳的 <span class="math-inline">E=E_{F}</span> 满足
<div class="math-display" id="eq:pdpBdef-b2">\Vert E_2\Vert _2=\Vert r\Vert _2, \quad \Vert E_{F}\Vert_F = \sqrt{2\Vert r\Vert^2_2- (\widetilde x^* A\widetilde x-\widetilde\lambda\widetilde x^* B\widetilde x)^2}. \tag{5.30}</div>
参见 [<a href="node421.html#kapj82">256</a>,<a href="node421.html#sun98">431</a>,<a href="node421.html#hihi98">473</a>]。
实际上，<span class="math-inline">E_{F}</span> 由 (<a href="node180.html#eq:E2-pdpBdef">5.29</a>) 明确给出。
因此，如果 <span class="math-inline">\Vert r\Vert _2</span> 很小，那么计算得到的 <span class="math-inline">\widetilde\lambda</span> 和 <span class="math-inline">\widetilde x</span> 就是 <em>邻近</em> 矩阵的精确特征值和特征向量。
此类误差分析被称为 <em>后向误差分析</em>，而矩阵 <span class="math-inline">E</span> 则是 <em>后向误差</em>。
<p>
我们称一个算法在范数 <span class="math-inline">\Vert\cdot\Vert</span> 下对于近似特征对 <span class="math-inline">(\widetilde\lambda,\widetilde x)</span> 是 <em><span class="math-inline">\tau</span>-后向稳定</em> 的，如果它是 <span class="math-inline">\{A+E,B\}</span> 的精确特征对且 <span class="math-inline">\Vert E\Vert\le\tau</span>。
基于这些概念，可以对计算特征对 <span class="math-inline">(\widetilde\lambda,\widetilde x)</span> 的算法的后向稳定性进行论述。
按照惯例，如果 <span class="math-inline">\tau = O(\epsilon_M \Vert(A,B)\Vert)</span>，其中 <span class="math-inline">\epsilon_M</span> 是机器精度，则称该算法为 <em>后向稳定</em> 的。
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<b>下一节：</b><a name="tex2html3475" href="node181.html">计算特征值的误差界限</a>
<b>上一级：</b><a name="tex2html3469" href="node178.html">正定 <span class="math-inline">B</span></a>
<b>上一节：</b><a name="tex2html3463" href="node179.html">残差向量</a>
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<address>
Susan Blackford
2000-11-20
</address>
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