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<title>隐式重启Lanczos R. Lehoucq and D. Sorensen</title>
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<h1>
<a name="SECTION001350000000000000000"></a>
<a name="sec:irlm"></a><a name="7954"></a>
隐式重启Lanczos方法
<br> <em>R. Lehoucq 和 D. Sorensen</em>
</h1>
<p>
厄米矩阵的Lanczos过程已在第<a href="node103.html#sec:lan">4.4</a>节中推导过。
这里，我们将讨论如何应用隐式重启技术。
我们的起点是一个<em><span class="math-inline">k</span>步Lanczos分解</em>（参见<a href="node103.html#herm-recursion">4.10</a>）：
<div class="math-display">A V_k = V_k T_k + r_k e_k^{\ast},</div>
其中，<span class="math-inline">V_k \in {\mathcal C}^{n \times k}</span>
的列是正交归一化的，<span class="math-inline"> V_k^{\ast} r_k = 0</span>，
而<span class="math-inline">T_k \in {\mathcal R}^{k \times k}</span>是实对称且三对角化的，其次对角线元素非负。
<span class="math-inline"> V_k </span>的列被称为<em>Lanczos向量</em>。
<a name="7967"></a>
对于隐式重启，确保<span class="math-inline"> V_k </span>的列达到完全正交精度至关重要。
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<li><a name="tex2html2509" href="node118.html">隐式重启</a>
<li><a name="tex2html2510" href="node119.html">位移的选择</a>
<li><a name="tex2html2511" href="node120.html">GEMV形式的Lanczos方法</a>
<li><a name="tex2html2512" href="node121.html">收敛性质</a>
<li><a name="tex2html2513" href="node122.html">计算成本与权衡</a>
<li><a name="tex2html2514" href="node123.html">收缩与停止规则</a>
<li><a name="tex2html2515" href="node124.html">正交收缩变换</a>
<ul>
<li><a name="tex2html2516" href="node125.html">锁定或清洗单个特征值</a>
<li><a name="tex2html2517" href="node126.html">锁定 <span class="math-inline">\theta </span></a>
<li><a name="tex2html2518" href="node127.html">清洗 <span class="math-inline">\theta </span></a>
<li><a name="tex2html2519" href="node128.html"><span class="math-inline">Q^* T Q</span> 的稳定性</a>
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<li><a name="tex2html2520" href="node129.html">锁和清洗的实施</a>
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<b>下一节：</b><a name="tex2html2508" href="node118.html">隐式重启</a>
<b>上一级：</b><a name="tex2html2502" href="node85.html">厄米特征值问题</a>
<b>上一节：</b><a name="tex2html2496" href="node116.html">L形膜片位移-逆结果</a>
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<address>
Susan Blackford
2000-11-20
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