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<title>关于高相对精度特征值计算的评述</title>
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<b>下一节：</b><a name="tex2html3053" href="node155.html">广义厄米特征值问题</a>
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关于高相对精度特征值计算的评述</a>
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过去10年左右，高精度特征值计算一直备受关注。在理论理解和数值算法方面都取得了巨大进展。但详细介绍超出了本书的范围。感兴趣的读者可参考相关文献。在算法方面，有Demmel-Kahan QR方法用于双对角奇异值计算[<a href="node421.html#deka:90">123</a>]，以及（双侧）Jacobi方法用于正定矩阵的特征值问题。对于奇异值计算[<a href="node421.html#deve92">124</a>,<a href="node421.html#math:95a">317</a>,<a href="node421.html#slap92">406</a>]，有针对缩放对角占优矩阵的二分法[<a href="node421.html#bade90">40</a>]，以及针对无环图矩阵的新qd方法实现[<a href="node421.html#degr:93">117</a>,<a href="node421.html#kaha:66c">255</a>]和Demmel的结构化矩阵算法[<a href="node421.html#demm:00">115</a>]。最近，[<a href="node421.html#dgesvd:99">118</a>]展示了如何为可精确分解为<span class="math-inline">B=X\Gamma Y^*</span>的矩阵计算高相对精度的SVD，其中<span class="math-inline">\Gamma</span>是对角矩阵，<span class="math-inline">X</span>和<span class="math-inline">Y</span>是任意良态矩阵。在理论方面，对于乘性扰动<span class="math-inline">A\to\widetilde A=D^*AE</span>（当<span class="math-inline">A</span>是厄米矩阵时<span class="math-inline">E=D</span>），得到了许多关于绝对扰动的著名定理的类似结果[<a href="node421.html#eiip:95">157</a>,<a href="node421.html#li:97a">300</a>,<a href="node421.html#rcli98">301</a>,<a href="node421.html#rcli99">302</a>,<a href="node421.html#li:99a">303</a>,<a href="node421.html#lima:99">297</a>]。
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<address>
Susan Blackford
2000-11-20
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