PSLQ: An Algorithm to Discover Integer Relations

By an integer relation algorithm, we mean a practical computational scheme that can recover the vector of integers ai, if it exists, or can produce bounds within which no integer relation exists. As we will see in the examples below, an integer relation algorithm can be used to recognize a computed constant in terms of a formula involving known constants, or to discover an underlying relation between quantities that can be computed to high precision. At the present time, the most widely used algorithm for integer relation detection is the “PSLQ” algorithm of mathematician-sculptor Helaman Ferguson [11, 4], although the “LLL” algorithm is also used for this purpose. One detailed comparison of these two methods found that PSLQ appears to be more numerically stable than LLL, in the sense that if PSLQ reliably finds a relation beginning at a nearly minimal precision level, whereas LLL sometimes finds a relation at one level but fails at a somewhat higher level [10]. This study also found that tuned implementations of PSLQ (which select multiple pairs of indices, and which employ two or three levels of precision [4]) are significantly more efficient than typical implementations of LLL. Additional research may further cast light on the relative merits of these two schemes. In the following, though, we will focus on PSLQ. PSLQ operates by constructing a sequence of integer-valued matrices Bn that reduces the vector y = xBn, until either the relation is found (as one of the columns of Bn), or else precision is exhausted. At the same time, PSLQ generates a steadily growing bound on the size of any possible relation. When a relation is found, the size of smallest entry of the vector y abruptly drops to roughly “epsilon” (i.e. 10, where p is the number of digits of precision). The size of this drop can be viewed as a “confidence level” that