多重极值特征对的Monte Carlo确定
Monte Carlo Determination of Multiple Extremal Eigenpairs
摘要：
我们提出了一个Monte Carlo算法，允许同时确定一个非常大的矩阵的几个极值特征对，而不需要计算两个向量的内积或存储任何一个向量的所有组件。新的算法，蒙特卡罗实现的一个确定性的，我们最近基准，是一个扩展的功率方法。在所提出的实现中，我们使用了一种称为梳状的基本蒙特卡罗分裂和终止方法，结合了Arnow { \it et al.}的权重抵消方法，提出了一种新的采样方法--缝合法，将一个大的状态空间采样作为一系列小的状态空间采样。我们通过对不同大小的二维零场伊辛模型的转移矩阵的两个最大特征值的确定来说明该算法的有效性。虽然很可能对其他传递矩阵问题有用，但该算法是相当普遍的，应该找到应用程序，以更大的各种各样的问题，需要几个主导特征值的矩阵。
英文摘要：
We present a Monte Carlo algorithm that allows the simultaneous determination of a few extremal eigenpairs of a very large matrix without the need to compute the inner product of two vectors or store all the components of any one vector. The new algorithm, a Monte Carlo implementation of a deterministic one we recently benchmarked, is an extension of the power method. In the implementation presented, we used a basic Monte Carlo splitting and termination method called the comb, incorporated the weight cancellation method of Arnow {\it et al.}, and exploited a new sampling method, the sewing method, that does a large state space sampling as a succession of small state space samplings. We illustrate the effectiveness of the algorithm by its determination of the two largest eigenvalues of the transfer matrices for variously-sized two-dimensional, zero field Ising models. While very likely useful for other transfer matrix problems, the algorithm is however quite general and should find application to a larger variety of problems requiring a few dominant eigenvalues of a matrix.
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