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<title>多重特征值</title>
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多重特征值</a>
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如前一段所述，我们只能收敛到起始向量中表示的特征向量。当矩阵束（<a href="node156.html#gsymeig">5.1</a>）具有多重特征值时，这一点尤为重要。在这种情况下，我们只能从相应的多维不变子空间中得到一个向量。
<p>
与标准情况一样，有两种不同的方法可以获得多重特征值的多个线性独立特征向量。第一种方法是重新启动并运行投影算子，其中所有已收敛的特征方向都已被投影掉；这相当于在算法<a href="node170.html#Gen_Herm_Lanczos">5.4</a>或<a href="node171.html#si-Gen_Herm_Lanczos">5.5</a>的第8步中，将向量<span class="math-inline">r</span>对矩阵<span class="math-inline">B</span>乘以所有已收敛的特征向量进行正交化。只要新的向量收敛，这一过程就会重复进行；例如，参见[<a href="node421.html#MatlPDE">318</a>]。
<p>
我们还可以运行块或带状Lanczos的广义变体，从多个起始方向（例如<span class="math-inline">p</span>个）开始，形成一个块<span class="math-inline">V_1</span>，在第<span class="math-inline">j</span>步中让<span class="math-inline">A</span>对<span class="math-inline">V_j</span>的所有方向进行操作，以计算一个新的<span class="math-inline">B</span>-正交块<span class="math-inline">V_{j+1}</span>。矩阵<span class="math-inline">T</span>将是一个块三对角矩阵，或者更确切地说是一个带状矩阵。详细描述参见[<a href="node421.html#grls94">206</a>]。
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<address>
Susan Blackford
2000-11-20
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