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<b>下一节：</b><a name="tex2html3560" href="node187.html">计算特征值的误差界限</a>
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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> 和 <span class="math-inline">F</span>，例如

<div class="math-display" id="eq:E2-pdp">(E,F) = \left[-r\widetilde x^*-\widetilde x r^*+\left(\widetilde\beta\widetilde x^* \widetilde A\widetilde x - \widetilde\alpha\widetilde x^*\widetilde B\widetilde x\right)\widetilde x\widetilde x^*\right]\cdot\left(\widetilde\beta I, -\widetilde\alpha I\right), \tag{5.40}</div>

使得 <span class="math-inline">(\widetilde\alpha,\widetilde\beta)</span> 和 <span class="math-inline">\widetilde x</span> 是 <span class="math-inline">\{A+E, B+F\}</span> 的一个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,F)</span>。
结果表明，对于谱范数 <span class="math-inline">\Vert\cdot\Vert _2</span>，最佳的 <span class="math-inline">(E,F)=(E_2,F_2)</span> 和对于弗罗贝尼乌斯范数 <span class="math-inline">\Vert\cdot\Vert _{F}</span>，最佳的 <span class="math-inline">(E,F)=(E_{F},F_{F})</span> 满足

<div class="math-display" id="eq:pdp-b2">\Vert(E_2, F_2)\Vert _2={\Vert r\Vert _2}, \quad
\Vert(E_{F}, F_F)\Vert_F = \sqrt{2\Vert r\Vert_2^2 - (\widetilde\beta\widetilde x^* A\widetilde x-\widetilde\alpha\widetilde x^* B\widetilde x)^2}. \tag{5.41}</div>

参见 [<a href="node421.html#kapj82">256</a>,<a href="node421.html#sun98">431</a>,<a href="node421.html#hihi98">473</a>]。<sup><a href="#footnote-cite">[1]</a></sup> 事实上，<span class="math-inline">(E_{F},F_{F})</span> 由 (<a href="node186.html#eq:E2-pdp">5.40</a>) 明确给出。
因此，如果 <span class="math-inline">\Vert r\Vert _2</span> 很小，计算得到的 <span class="math-inline">(\widetilde\alpha,\widetilde\beta)</span> 和 <span class="math-inline">\widetilde x</span> 是附近矩阵的精确特征值和特征向量。这种误差分析称为<em>后向误差分析</em>，矩阵 <span class="math-inline">(E,F)</span> 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\alpha,\widetilde\beta),\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+F\}</span> 的精确特征对，且 <span class="math-inline">\Vert(E,F)\Vert\le\tau</span>。
考虑到这些，可以对计算特征对 <span class="math-inline">((\widetilde\alpha,\widetilde\beta),\widetilde x)</span> 的算法的后向稳定性做出说明。
按照惯例，如果 <span class="math-inline">\tau = O(\epsilon_M \Vert(A,B)\Vert)</span>，算法被称为<em>后向稳定</em>。

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<li id="footnote-cite">Kahan、Parlett 和 Jiang[<a href="node421.html#kapj82">256</a>，定理1'] 仅给出了<span class="math-inline">\widetilde\beta E-\widetilde\alpha F\equiv Z</span>的最优范数。
  取<span class="math-inline">E=\widetilde\beta Z</span>和<span class="math-inline">F=-\widetilde\alpha Z</span>，可见<span class="math-inline">(E,F)</span>与<span class="math-inline">Z</span>具有相同的2-范数和Frobenius范数。</li>
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<b>下一节：</b><a name="tex2html3560" href="node187.html">计算特征值的误差界限</a>
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<b>上一节：</b><a name="tex2html3548" href="node185.html">残差向量</a>
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<address>
Susan Blackford
2000-11-20
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