经验数据的PSLQ算法
The PSLQ Algorithm for Empirical Data
摘要：
著名的整数关系发现算法PSLQ已成功地应用于许多应用中。PSLQ只对精确的输入数据进行了理论分析，但当输入数据是无理数时，由于计算机的精度有限，它们必须是近似的。当算法以经验数据（误差有界的不精确数据）而不是精确的实数作为输入时，我们如何在理论上确保算法的输出是精确的整数关系？在本文中，我们研究的PSLQ算法的经验数据作为其输入。首先，我们给出了这种情况的终止条件。其次，我们分析了从输入数据构造的超平面矩阵上的扰动，从而揭示了输入数据的精度和输出质量之间的关系（精确数据和计算的整数关系的内积的绝对值的上界），这自然会导致PSLQ的错误控制策略。此外，我们分析了经验数据的PSLQ算法的复杂性界。超越数和代数数的例子表明了我们的错误控制策略的意义。
英文摘要：
The celebrated integer relation finding algorithm PSLQ has been successfully used in many applications. PSLQ was only analyzed theoretically for exact input data, however, when the input data are irrational numbers, they must be approximate ones due to the finite precision of the computer. When the algorithm takes empirical data (inexact data with error bounded) instead of exact real numbers as its input, how do we theoretically ensure the output of the algorithm to be an exact integer relation? In this paper, we investigate the PSLQ algorithm for empirical data as its input. Firstly, we give a termination condition for this case. Secondly, we analyze a perturbation on the hyperplane matrix constructed from the input data and hence disclose a relationship between the accuracy of the input data and the output quality (an upper bound on the absolute value of the inner product of the exact data and the computed integer relation), which naturally leads to an error control strategy for PSLQ. Further, we analyze the complexity bound of the PSLQ algorithm for empirical data. Examples on transcendental numbers and algebraic numbers show the meaningfulness of our error control strategy.
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