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<title>Rayleigh商迭代</title>
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<h4><a name="SECTION001440030000000000000">
Rayleigh商迭代</a>
</h4>
类似于逆迭代法，第§<a href="node94.html#sec_power_sym">4.3</a>节中介绍的瑞利商迭代(RQI)方法同样可以推广应用于求解问题 (<a href="node156.html#gsymeig">5.1</a>)。
<a name="17235"></a>
<pre class="algorithm" id="rqi_gsymeig">
算法 5.3 GHEP的Rayleigh商迭代
(1) 选择一个初始向量 <span class="math-inline">v</span>，要求 <span class="math-inline">v^*Bv = 1</span>，令 <span class="math-inline">w = Bv</span>，并取 <span class="math-inline">\rho_1</span>
(2) for <span class="math-inline">k = 1,2,\dots</span>
(3) 计算 <span class="math-inline">y = (A - \rho_k B)^{-1}w</span>
(4) 如果奇异，停止
(5) <span class="math-inline">\theta = w^*y</span>
(6) 计算 <span class="math-inline">w = By</span>
(7) <span class="math-inline">\eta = (w^*y)^{1/2}, \; v = y/\eta, \; w = w / \eta</span>
(8) <span class="math-inline">\rho_{k+1} = \rho_k + \theta/\eta^2</span>
(9) 如果 <span class="math-inline">\vert \theta \vert \ge \epsilon_M^{-1/2}</span>，停止
(10) end for
(11) 获得结果 <span class="math-inline">\lambda = \rho_k, \; x = v</span>
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算法<a href="node167.html#invit_gsymeig">5.2</a>与算法<a href="node168.html#rqi_gsymeig">5.3</a>之间的唯一区别在于步骤(8)，即更新位移的部分。这导致在每次迭代中，步骤(3)必须进行稀疏分解。然而，这种额外的工作换来的是三次收敛速率。
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<address>
Susan Blackford
2000-11-20
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