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<title>将传递残差误差转化为后向误差</title>
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<b>下一节：</b><a name="tex2html2999" href="node151.html">计算特征值的误差界</a>
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将残差误差转化为后向误差
</h4>
存在厄米矩阵 <span class="math-inline">E</span>，使得 <span class="math-inline">\widetilde\lambda</span> 和 <span class="math-inline">\widetilde x</span> 是 <span class="math-inline">A+E</span> 的一个精确特征值及其对应的特征向量，即
<div class="math-display">(A+E)\widetilde x=\widetilde\lambda\widetilde x.</div>
其中一种 <span class="math-inline">E</span> 的形式为
<div class="math-display" name="#eq:E2-H">E = -r\widetilde x^*-\widetilde x r^* +\left(\widetilde x^*A\widetilde x -\widetilde\lambda\right)\widetilde x\widetilde x^*. \tag{4.52}</div>
我们关注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>，最佳的 <span class="math-inline">E=E_2</span> 和对于 Frobenius 范数 <span class="math-inline">\Vert\cdot\Vert _{F}</span>，最佳的 <span class="math-inline">E=E_{F}</span> 满足
<div class="math-display" name="#eq:std-H-b2">\Vert E_2\Vert _2=\Vert r\Vert _2 \quad \mathrm{and} \quad\Vert E_F\Vert _F=\sqrt{\Vert r\Vert^2_2 - \left(\widetilde x^* A\widetilde x-\widetilde\lambda\right)^2}; \tag{4.53}</div>
参见，例如 [<a href="node421.html#kapj82">256</a>,<a href="node421.html#sun98">431</a>]。实际上，<span class="math-inline">E_{F}</span> 由 (<a href="node150.html#eq:E2-H">4.52</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\lambda</span> 和 <span class="math-inline">\widetilde x</span> 就是附近矩阵的一个精确特征对。这种类型的误差分析称为 <em>后向误差分析</em>，矩阵 <span class="math-inline">E</span> 是 <em>后向误差</em>。
<p>
我们称一个算法如果输出的近似特征对 <span class="math-inline">(\widetilde\lambda,\widetilde x)</span> 对于范数 <span class="math-inline">\Vert\cdot\Vert</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</span> 的一个精确特征对，且 <span class="math-inline">\Vert E\Vert\le\tau</span>。根据这个定义，可以对计算特征对 <span class="math-inline">(\widetilde\lambda,\widetilde x)</span> 的算法的数值稳定性进行说明。按照惯例，如果 <span class="math-inline">\tau = O(\epsilon_M \Vert A\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="tex2html2999" href="node151.html">计算特征值的误差界</a>
<b>上一级：</b><a name="tex2html2993" href="node148.html">稳定性与准确性评估</a>
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<address>
Susan Blackford
2000-11-20
</address>
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