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Expand Up @@ -84,22 +84,22 @@ <h1><a name="SECTION002000000000000000000"></a><a name="chap:precond"></a>

<ul>
<li><a name="tex2html6831" href="node397.html">引言</a>
<li><a name="tex2html6832" href="node398.html">不精确方法
<li><a name="tex2html6832" href="node398.html">不精确方法
<br>&nbsp; <em>K.&nbsp;Meerbergen and R.&nbsp;Morgan</em></a>
<ul>
<li><a name="tex2html6833" href="node399.html">矩阵变换</a>
<li><a name="tex2html6834" href="node400.html">不精确矩阵变换</a>
<li><a name="tex2html6835" href="node401.html">Arnoldi方法与不完全Cayley变换</a>
<li><a name="tex2html6835" href="node401.html">带非精确Cayley变换的Arnoldi方法</a>
<li><a name="tex2html6836" href="node402.html">Davidson方法</a>
<ul>
<li><a name="tex2html6837" href="node403.html"><b>Example 11.2.1.</b></a>
<li><a name="tex2html6838" href="node404.html">Example 11.2.2.</a>
<li><a name="tex2html6837" href="node403.html">例11.2.1.</a>
<li><a name="tex2html6838" href="node404.html">例11.2.2.</a>
</ul>
<li><a name="tex2html6839" href="node405.html">Jacobi-Davidson方法与Cayley变换</a>
<li><a name="tex2html6839" href="node405.html">带Cayley变换的Jacobi-Davidson方法</a>
<li><a name="tex2html6840" href="node406.html">预处理Lanczos方法</a>
<li><a name="tex2html6841" href="node407.html">非精确有理Krylov方法</a>
<li><a name="tex2html6841" href="node407.html">不精确有理Krylov方法</a>
<ul>
<li><a name="tex2html6842" href="node408.html">Example 11.2.3.</a>
<li><a name="tex2html6842" href="node408.html">例11.2.3.</a>
</ul>
<li><a name="tex2html6843" href="node409.html">不精确位移反演法</a>
</ul>
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<ul>
<li><a name="tex2html6845" href="node411.html">引言</a>
<li><a name="tex2html6846" href="node412.html">预处理的一般框架</a>
<li><a name="tex2html6847" href="node413.html">预条件转移幂法</a>
<li><a name="tex2html6847" href="node413.html">预处理位移幂法</a>
<li><a name="tex2html6848" href="node414.html">预处理最速上升/下降法</a>
<li><a name="tex2html6849" href="node415.html">预处理Lanczos方法</a>
<li><a name="tex2html6850" href="node416.html">Davidson方法</a>
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<button class="navigate">索引</button></a>
<br>
<b>下一节：</b><a name="tex2html6868" href="node398.html">Inexact Methods &nbsp; K.&nbsp;Meerbergen</a>
<b>下一节：</b><a name="tex2html6868" href="node398.html">不精确方法</a>
<b>上一级：</b><a name="tex2html6862" href="node396.html">预处理技术</a>
<b>上一节：</b><a name="tex2html6856" href="node396.html">预处理技术</a>


<br>
<br>
<!--End of Navigation Panel--><h1><a name="SECTION002010000000000000000"></a> <a name="sec:introduction"></a>
<br>
引言
</h1>

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<p>
我们再看一个特征值预处理方法之间的相似之处。为便于阐述，我们假设存在一个标准的对称特征值问题。似乎所有方法的共同点是算子
<div class="math-display" id="eq:keyop">M^{-1}(A-\nu I), \tag{272}</div>
<div class="math-display" id="eq:keyop">M^{-1}(A-\nu I), \tag{11.1}</div>

其中<span class="math-inline">\nu</span>是近似特征值，<span class="math-inline">M</span>是<span class="math-inline">A-\nu I</span>的（可能变化的）近似。我们将探讨一些不同方法如何使用此算子的例子；有关一般讨论，请参见§<a href="node412.html#sec:prec_preconditioning">11.3.2</a>。讨论的四种方法包括预处理的幂法、Rayleigh商迭代（RQI）、Davidson方法和Jacobi-Davidson方法。预处理的幂法是§<a href="node410.html#sec:prec">11.3</a>中的算法<a href="node413.html#fig:prec_powerBA">11.5</a>。当<span class="math-inline">B=I</span>时，该算法的步骤4和5执行
<div class="math-display">w =T(x^{(i)}-\mu^{(i)}Ax^{(i)}) =-\mu^{(i)} T(A - {1/\mu^{(i)}} I) x^{(i)}. </div>
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<button class="navigate">索引</button></a>
<br>
<b>下一节：</b><a name="tex2html6868" href="node398.html">Inexact Methods &nbsp; K.&nbsp;Meerbergen</a>
<b>下一节：</b><a name="tex2html6868" href="node398.html">不精确方法</a>
<b>上一级：</b><a name="tex2html6862" href="node396.html">预处理技术</a>
<b>上一节：</b><a name="tex2html6856" href="node396.html">预处理技术</a>


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<address>
Susan Blackford
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<title>Inexact Methods &nbsp; K.&nbsp;Meerbergen and R.&nbsp;Morgan </title>
<title>不精确方法</title>
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<b>下一节：</b><a name="tex2html6882" href="node399.html">矩阵变换</a>
<b>上一级：</b><a name="tex2html6876" href="node396.html">预处理技术</a>
<b>上一节：</b><a name="tex2html6870" href="node397.html">引言</a>


<br>
<br>
<!--End of Navigation Panel--><h1><a name="SECTION002020000000000000000"></a> <a name="sec:iemethods"></a>
<br>
非精确方法
<br>&nbsp; <em>凯·梅尔伯根(K. Meerbergen) 和 罗杰·摩根(R. Morgan)</em>
不精确方法
<br>&nbsp; <em>K.&nbsp;Meerbergen and R.&nbsp;Morgan</em>
</h1>

<p>
针对厄米特征值问题(HEP)的谱变换Lanczos方法和针对非厄米特征值问题(NHEP)的移位-逆变换Arnoldi方法都需要求解线性系统。通常，人们采用直接方法，因为可以利用一次分解进行多次回代变换。然而，当直接求解器的使用变得不可行时，必须采用迭代求解方法。直接方法通常能得到较小的残差范数，而迭代方法的残差范数则取决于一个容差。迭代方法在采用较低残差容差时成本更高。因此，在使用迭代线性系统求解器时，不精确求解线性系统是有利的。基于这个原因，我们将在使用直接方法时的谱变换特征值求解器称为精确方法，而在使用迭代方法时称为非精确方法。本节旨在研究在谱变换中使用非精确线性系统求解器的情况。

<p>
我们从移位-逆变换Arnoldi方法开始，逐渐过渡到(Jacobi)Davidson方法和非精确有理Krylov方法。首先，在§<a href="node399.html#sec:matrixtransformations">11.2.1</a>中，我们回顾了所使用的(精确)矩阵变换及其主要性质。在§<a href="node400.html#sec:inexactmatrixtransformations">11.2.2</a>中，我们引入了非精确变换[<a href="node421.html#mero97">323</a>]的概念，并解释了其重要性。在§<a href="node401.html#sec:inexact-Arnoldi">11.2.3</a>和§<a href="node402.html#sec:davidson">11.2.4</a>中，我们给出了一些关于非精确变换的Rayleigh-Ritz过程的结果。这包括非对称问题(Arnoldi[<a href="node421.html#mero97">323</a>]、Davidson[<a href="node421.html#morg92">332</a>])和对称问题(Lanczos[<a href="node421.html#mosc93">336</a>]、Davidson[<a href="node421.html#mosc86">335</a>,<a href="node421.html#crps94">88</a>])以及Jacobi-Davidson方法的结果。我们还可以利用有理Krylov方法的矩阵递推关系[<a href="node421.html#leme98">291</a>]来计算特征值，即使线性系统求解不精确。这在§<a href="node407.html#sec:inexact-RKS">11.2.7</a>中进行了研究。
我们从移位-逆变换Arnoldi方法开始，逐渐过渡到(Jacobi)Davidson方法和不精确有理Krylov方法。首先，在§<a href="node399.html#sec:matrixtransformations">11.2.1</a>中，我们回顾了所使用的(精确)矩阵变换及其主要性质。在§<a href="node400.html#sec:inexactmatrixtransformations">11.2.2</a>中，我们引入了非精确变换[<a href="node421.html#mero97">323</a>]的概念，并解释了其重要性。在§<a href="node401.html#sec:inexact-Arnoldi">11.2.3</a>和§<a href="node402.html#sec:davidson">11.2.4</a>中，我们给出了一些关于非精确变换的Rayleigh-Ritz过程的结果。这包括非对称问题(Arnoldi[<a href="node421.html#mero97">323</a>]、Davidson[<a href="node421.html#morg92">332</a>])和对称问题(Lanczos[<a href="node421.html#mosc93">336</a>]、Davidson[<a href="node421.html#mosc86">335</a>,<a href="node421.html#crps94">88</a>])以及Jacobi-Davidson方法的结果。我们还可以利用有理Krylov方法的矩阵递推关系[<a href="node421.html#leme98">291</a>]来计算特征值，即使线性系统求解不精确。这在§<a href="node407.html#sec:inexact-RKS">11.2.7</a>中进行了研究。

<p>
<br><hr>
<!--子节列表-->
<a name="CHILD_LINKS"><strong>子节</strong></a>
<a name="CHILD_LINKS"><strong>小节</strong></a>

<ul>
<li><a name="tex2html6883" href="node399.html">矩阵变换</a>
<li><a name="tex2html6884" href="node400.html">非精确矩阵变换</a>
<li><a name="tex2html6885" href="node401.html">带非精确Cayley变换的Arnoldi方法</a>
<li><a name="tex2html6886" href="node402.html">Davidson方法</a>
<ul>
<li><a name="tex2html6887" href="node403.html"><b>例11.2.1.</b></a>
<li><a name="tex2html6887" href="node403.html">例11.2.1.</a>
<li><a name="tex2html6888" href="node404.html">例11.2.2.</a>
</ul>
<li><a name="tex2html6889" href="node405.html">带Cayley变换的Jacobi-Davidson方法</a>
<li><a name="tex2html6890" href="node406.html">预处理Lanczos方法</a>
<li><a name="tex2html6891" href="node407.html">非精确有理Krylov方法</a>
<li><a name="tex2html6891" href="node407.html">不精确有理Krylov方法</a>
<ul>
<li><a name="tex2html6892" href="node408.html">例11.2.3.</a>
</ul>
<li><a name="tex2html6893" href="node409.html">非精确移位-逆变换</a>
<li><a name="tex2html6893" href="node409.html">不精确位移反演法</a>
</ul>
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<b>下一节：</b><a name="tex2html6882" href="node399.html">矩阵变换</a>
<b>上一级：</b><a name="tex2html6876" href="node396.html">预处理技术</a>
<b>上一节：</b><a name="tex2html6870" href="node397.html">引言</a>


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<address>
Susan Blackford
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<b>下一节：</b><a name="tex2html6907" href="node400.html">不精确矩阵变换</a>
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<b>上一节：</b><a name="tex2html6895" href="node398.html">Inexact Methods &nbsp; K.&nbsp;Meerbergen</a>


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<h2><a name="SECTION002021000000000000000"></a>
<a name="sec:matrixtransformations"></a>
<br>
Matrix Transformations
矩阵变换
</h2>

<p>
Consider the eigenvalue problem <span class="math-inline">Ax = \lambda Bx</span>.
The spectral transformation or shift-and-invert transformation (SI)
考虑特征值问题 <span class="math-inline">Ax = \lambda Bx</span>。
谱变换或移位-求逆变换（SI）
<a name="47970"></a>
is defined by
定义为

<div class="math-display">T_{\rm SI} = (A - \mu B)^{-1} B,</div>

where <span class="math-inline">\mu</span> is the shift or pole.
If
<span class="math-inline">Ax = \lambda Bx</span> then
<span class="math-inline">T_{\rm SI} x = \gamma x</span> with
其中 <span class="math-inline">\mu</span> 是移位或极点。
如果
<span class="math-inline">Ax = \lambda Bx</span> 则
<span class="math-inline">T_{\rm SI} x = \gamma x</span> 且

<span class="math-inline">\gamma=(\lambda-\mu)^{-1}</span>.
An alternative is the Cayley transform
<span class="math-inline">\gamma=(\lambda-\mu)^{-1}</span>。
另一种选择是Cayley变换
<a name="47975"></a>

<div class="math-display">T_{\mathrm{C}} = (A-\mu B)^{-1} (A - \nu B),</div>

where <span class="math-inline">\mu</span> is the pole and <span class="math-inline">\nu</span> the zero.
If <span class="math-inline">Ax = \lambda Bx</span> then
<span class="math-inline">T_{\mathrm{C}} x=\zeta x</span> with
其中 <span class="math-inline">\mu</span> 是极点，<span class="math-inline">\nu</span> 是零点。
如果 <span class="math-inline">Ax = \lambda Bx</span> 则
<span class="math-inline">T_{\mathrm{C}} x=\zeta x</span> 且

<span class="math-inline">\zeta = (\lambda-\mu)^{-1}(\lambda-\nu)</span>.
Since
<span class="math-inline">T_{\mathrm{C}} = I + (\mu-\nu) T_{\rm SI}</span> and Krylov spaces are
shift-invariant
with respect to the matrix, we have that
<span class="math-inline">\zeta = (\lambda-\mu)^{-1}(\lambda-\nu)</span>。
由于
<span class="math-inline">T_{\mathrm{C}} = I + (\mu-\nu) T_{\rm SI}</span> 且Krylov空间相对于矩阵具有移位不变性，我们有

<div class="math-display">\mathcal{K}^k(T_{\mathrm{C}},v_1) = \mathcal{K}^k(T_{\rm SI},v_1) \ ,</div>

so, the Arnoldi method applied to <span class="math-inline">T_{\rm SI}</span> or
<span class="math-inline">T_{\mathrm{C}}</span> delivers the same
Ritz vectors and after back transformation of <span class="math-inline">\gamma</span>'s and <span class="math-inline">\zeta</span>'s,
respectively, leads to the same <span class="math-inline">\lambda</span>'s.
因此，应用于 <span class="math-inline">T_{\rm SI}</span> 或
<span class="math-inline">T_{\mathrm{C}}</span> 的Arnoldi方法产生的Ritz向量相同，并且在分别对 <span class="math-inline">\gamma</span> 和 <span class="math-inline">\zeta</span> 进行反变换后，得到相同的 <span class="math-inline">\lambda</span>。

<p>
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