The PSLQ Algorithm for Empirical Data

TL;DR

This paper extends the PSLQ integer relation algorithm to handle empirical data with bounded errors, providing theoretical guarantees, error control strategies, and complexity analysis for inexact inputs.

Contribution

It offers the first theoretical analysis and error control method for PSLQ with empirical data, including termination conditions and complexity bounds.

Findings

Established a termination condition for empirical data input

Derived an error bound relating input accuracy to output quality

Validated the approach with examples on transcendental and algebraic numbers

Abstract

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.