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| Jens Lippmann, Marek Rouchal, Martin Wilck and others --> | ||
| <html> | ||
| <head> | ||
| <title>Locking.</title> | ||
| <title>锁</title> | ||
| <meta charset="utf-8"> | ||
| <meta name="description" content="Locking."> | ||
| <meta name="description" content="锁"> | ||
| <meta name="keywords" content="book, math, eigenvalue, eigenvector, linear algebra, sparse matrix"> | ||
| <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css" integrity="sha384-nB0miv6/jRmo5UMMR1wu3Gz6NLsoTkbqJghGIsx//Rlm+ZU03BU6SQNC66uf4l5+" crossorigin="anonymous"> | ||
| <script defer src="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.js" integrity="sha384-7zkQWkzuo3B5mTepMUcHkMB5jZaolc2xDwL6VFqjFALcbeS9Ggm/Yr2r3Dy4lfFg" crossorigin="anonymous"></script> | ||
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| href="node422.html"> | ||
| <button class="navigate">索引</button></a> | ||
| <br> | ||
| <b>下一节：</b><a name="tex2html2262" href="node101.html">Acceleration.</a> | ||
| <b>下一节：</b><a name="tex2html2262" href="node101.html">加速</a> | ||
| <b>上一级：</b><a name="tex2html2256" href="node98.html">子空间迭代法</a> | ||
| <b>上一节：</b><a name="tex2html2250" href="node99.html">Subspace Dimension.</a> | ||
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| <b>上一节：</b><a name="tex2html2250" href="node99.html">子空间维度</a> | ||
| <br> | ||
| <br> | ||
| <!--End of Navigation Panel--><h4><a name="SECTION001334020000000000000"></a> <a name="7529"></a> | ||
| <br> | ||
| 锁定机制。 | ||
| <!--End of Navigation Panel--> | ||
| <h4> | ||
| <a name="SECTION001334020000000000000"></a> | ||
| <a name="7529"></a> | ||
| 锁 | ||
| </h4> | ||
| 由于通过子空间迭代计算的每个近似特征值的收敛速度不同，通常的做法是一次提取一个并进行某种形式的缩小（deflation）。因此，一旦第一个特征向量收敛，就没有必要在后续迭代中继续将其与<span class="math-inline">A</span>相乘。实际上，我们可以冻结这个向量，只处理向量<span class="math-inline">v_2, \ldots,v_m</span>。然而，我们仍然需要在需要进行正交化时对冻结向量<span class="math-inline">v_1</span>进行后续正交化。这种策略的术语是<i>锁定</i>；也就是说，我们不再试图改进对<span class="math-inline">v_1</span>的锁定近似。 | ||
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| 由于通过子空间迭代计算的每个近似特征值的收敛速度不同，通常的做法是每次提取特征值，并进行某种形式的收缩。即，一旦第一个特征向量收敛，就没有必要在后续迭代中继续将其与<span class="math-inline">A</span>相乘。实际上，我们可以冻结这个向量，只处理向量<span class="math-inline">v_2, \ldots,v_m</span>。然而，我们仍然需要在需要进行正交化时对冻结向量<span class="math-inline">v_1</span>进行后续正交化。这种策略的术语是<i>锁定</i>；也就是说，我们不再试图改进对<span class="math-inline">v_1</span>的锁定近似。 | ||
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| <p> | ||
| 以下算法描述了一种实用的带缩小（锁定）的子空间迭代，用于计算<span class="math-inline">n_{\mathrm{ev}}</span>个主特征值。 | ||
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| <p> | ||
| <br> | ||
| <a name="symsubspaceiter"></a><img | ||
| width="620" height="302" align="BOTTOM" border="0" | ||
| src="img842.png" | ||
| alt="\begin{algorithm}{Subspace Iteration with Projection and Deflation for HEP | ||
| \inde... | ||
| ...igenvalues \\ | ||
| {\rm (11)} \> \> {\bf end while} | ||
| \end{tabbing}} | ||
| \end{algorithm}"> | ||
| <br> | ||
| <pre style="text-align: left;" id="symsubspaceiter"> | ||
| 算法 4.5 HEP带锁定和投影的子空间迭代 | ||
| (1) 取初始猜测矩阵 <span class="math-inline">Z</span>，对其做QR分解得到 <span class="math-inline">Z=V R</span>。 | ||
| (2) 令 <span class="math-inline">j = 0</span> | ||
| (3) while <span class="math-inline">j \le n_{ev}</span> do | ||
| (4) <span class="math-inline">\hat{Y} = [V_j, A^\text{iter}V_{p-j}]</span> | ||
| (5) 对 <span class="math-inline">\hat{Y}</span> 做正交化得到 <span class="math-inline">\hat{Y} = V R</span> （前 <span class="math-inline">j</span> 列保持不变） | ||
| (6) <span class="math-inline">H = V^\ast_{p-j} A V_{p-j}</span> | ||
| (7) 计算 <span class="math-inline">H</span> 的特征分解 <span class="math-inline">H = S \Theta S^\ast</span> | ||
| (8) 测试特征值 <span class="math-inline">\Theta</span> 的收敛性 | ||
| (9) <span class="math-inline">V = [V_j, V_{p-j} S]</span> | ||
| (10) 令 <span class="math-inline">j = j + i_\text{conv}</span>，其中 <span class="math-inline">i_\text{conv}</span> 是新收敛的特征值数量 | ||
| (11) end while | ||
| </pre> | ||
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| <p> | ||
| 现在我们描述一些实现细节。 | ||
| 下面我们讨论一些实现细节。 | ||
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| <p> | ||
| <dl> | ||
| <dt><strong>(1)</strong></dt> | ||
| <dd>初始启动矩阵<span class="math-inline">Z</span>应构建为在感兴趣的特征向量方向上占主导地位，以加速收敛。当没有此类先验信息时，随机矩阵与其他选择一样好。 | ||
| </dd> | ||
| <dt><strong>(4)</strong></dt> | ||
| <dd>迭代参数<span class="math-inline">\iter</span>应选择为在保持合理数值精度的同时最小化正交化成本。放大因子<span class="math-inline">(\lambda_1/\lambda_p)^{\iter}</span>，其中特征值<span class="math-inline">\lambda_i</span>按绝对值递减排序，给出了精度损失。鲁蒂斯豪瑟[<a href="node421.html#ruti69">381</a>]采取保守策略，仅允许放大因子为<span class="math-inline">10</span>，损失一位小数，而斯图尔特和詹宁斯[<a href="node421.html#stje81">426</a>]让算法运行至<span class="math-inline">\eps^{-1/2}</span>，即机器精度的一半，但不超过10次迭代。 | ||
| <dd>迭代参数 <span class="math-inline">\mathrm{iter}</span>（乘以 <span class="math-inline">A</span> 的次数）应选择为在保持合理数值精度的同时最小化正交化成本。放大因子<span class="math-inline">(\lambda_1/\lambda_p)^\text{iter}</span>，其中特征值<span class="math-inline">\lambda_i</span>按绝对值递减排序，给出了精度损失。鲁蒂斯豪瑟[<a href="node421.html#ruti69">381</a>]采取保守策略，仅允许放大因子为<span class="math-inline">10</span>，损失一位小数，而斯图尔特和詹宁斯[<a href="node421.html#stje81">426</a>]让算法运行至<span class="math-inline">\epsilon_M^{-1/2}</span>，即机器精度的一半，但不超过10次迭代。 | ||
| </dd> | ||
| </dl> | ||
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| href="node422.html"> | ||
| <button class="navigate">索引</button></a> | ||
| <br> | ||
| <b>下一节：</b><a name="tex2html2262" href="node101.html">Acceleration.</a> | ||
| <b>下一节：</b><a name="tex2html2262" href="node101.html">加速</a> | ||
| <b>上一级：</b><a name="tex2html2256" href="node98.html">子空间迭代法</a> | ||
| <b>上一节：</b><a name="tex2html2250" href="node99.html">Subspace Dimension.</a> | ||
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| <b>上一节：</b><a name="tex2html2250" href="node99.html">子空间维度</a> | ||
| <!--End of Navigation Panel--> | ||
| <address> | ||
| Susan Blackford | ||
|
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| Jens Lippmann, Marek Rouchal, Martin Wilck and others --> | ||
| <html> | ||
| <head> | ||
| <title>Acceleration.</title> | ||
| <title>加速</title> | ||
| <meta charset="utf-8"> | ||
| <meta name="description" content="Acceleration."> | ||
| <meta name="description" content="加速"> | ||
| <meta name="keywords" content="book, math, eigenvalue, eigenvector, linear algebra, sparse matrix"> | ||
| <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.css" integrity="sha384-nB0miv6/jRmo5UMMR1wu3Gz6NLsoTkbqJghGIsx//Rlm+ZU03BU6SQNC66uf4l5+" crossorigin="anonymous"> | ||
| <script defer src="https://cdn.jsdelivr.net/npm/[email protected]/dist/katex.min.js" integrity="sha384-7zkQWkzuo3B5mTepMUcHkMB5jZaolc2xDwL6VFqjFALcbeS9Ggm/Yr2r3Dy4lfFg" crossorigin="anonymous"></script> | ||
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| <br> | ||
| <b>下一节：</b><a name="tex2html2274" href="node102.html">可用的软件</a> | ||
| <b>上一级：</b><a name="tex2html2268" href="node98.html">子空间迭代法</a> | ||
| <b>上一节：</b><a name="tex2html2264" href="node100.html">Locking.</a> | ||
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| <b>上一节：</b><a name="tex2html2264" href="node100.html">锁</a> | ||
| <br> | ||
| <br> | ||
| <!--End of Navigation Panel--><h4><a name="SECTION001334030000000000000"> | ||
| 加速方法</a> | ||
| <!--End of Navigation Panel--> | ||
| <h4><a name="SECTION001334030000000000000"></a> | ||
| 加速方法 | ||
| </h4> | ||
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| 若需要求解接近某一<i>位移</i> <span class="math-inline">\sigma</span> 的特征值，并且可以便捷地得到分解式 | ||
| <span class="math-inline">A - \sigma \, I=LU</span>（参见第§<a href="node385.html#sec:directsolvers">10.3</a>节）， | ||
| <span class="math-inline">A - \sigma \, I=LU</span>（参见第<a href="node385.html#sec:directsolvers">10.3</a>节）， | ||
| 那么可以将上述算法应用于 | ||
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| <span class="math-inline">\left(A - \sigma \, I\right)^{-1}</span>。接近 <span class="math-inline">\sigma</span> 的特征值将快速收敛。 | ||
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| 通过这些改进，子空间迭代法可能成为一种相当高效的方法，其优势在于易于编码和理解。然而，后续讨论的一些方法通常更受欢迎，因为它们往往能更快地找到特征值/特征向量。 | ||
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| <p> | ||
| 本节内容大量引自 Demmel [<a | ||
| href="node421.html#demmelbook">114</a>]，Golub 和 Van Loan [<a | ||
| href="node421.html#golo96">198</a>]， | ||
| 以及 Saad [<a | ||
| href="node421.html#saad92">387</a>]。关于子空间迭代的进一步讨论，建议读者参考 Chatelin [<a | ||
| href="node421.html#chat93">79</a>]， | ||
| Lehoucq 和 Scott [<a | ||
| href="node421.html#lesc95">292</a>]，Stewart [<a | ||
| href="node421.html#stew76">422</a>]，以及 | ||
| Wilkinson [<a | ||
| href="node421.html#wilk65">457</a>]。另请参阅 Bathe 和 Wilson [<a | ||
| href="node421.html#bawi76">42</a>] 以及 | ||
| Jennings [<a | ||
| href="node421.html#jenn77">242</a>] | ||
| 本节内容大量引自 Demmel [<a href="node421.html#demmelbook">114</a>]，Golub 和 Van Loan [<a href="node421.html#golo96">198</a>]， | ||
| 以及 Saad [<a href="node421.html#saad92">387</a>]。关于子空间迭代的进一步讨论，建议读者参考 Chatelin [<a href="node421.html#chat93">79</a>]， | ||
| Lehoucq 和 Scott [<a href="node421.html#lesc95">292</a>]，Stewart [<a href="node421.html#stew76">422</a>]，以及 | ||
| Wilkinson [<a href="node421.html#wilk65">457</a>]。另请参阅 Bathe 和 Wilson [<a href="node421.html#bawi76">42</a>] 以及 | ||
| Jennings [<a href="node421.html#jenn77">242</a>] | ||
| 关于结构工程方法的论述。 | ||
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| <p> | ||
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| <br> | ||
| <b>下一节：</b><a name="tex2html2274" href="node102.html">可用的软件</a> | ||
| <b>上一级：</b><a name="tex2html2268" href="node98.html">子空间迭代法</a> | ||
| <b>上一节：</b><a name="tex2html2264" href="node100.html">Locking.</a> | ||
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| <b>上一节：</b><a name="tex2html2264" href="node100.html">锁</a> | ||
| <!--End of Navigation Panel--> | ||
| <address> | ||
| Susan Blackford | ||
|
|
||
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