Analysis of PSLQ, an integer relation finding algorithm

Let K be either the real, complex, or quaternion number system and let O(K) be the corresponding integers. Let x = (x1, . . . , xn) be a vector in Kn. The vector x has an integer relation if there exists a vector m = (m1, . . . ,mn) ∈ O(K)n, m = 0, such that m1x1+m2x2+ . . .+mnxn = 0. In this paper we define the parameterized integer relation construction algorithm PSLQ(τ), where the parameter τ can be freely chosen in a certain interval. Beginning with an arbitrary vector x = (x1, . . . , xn) ∈ Kn, iterations of PSLQ(τ) will produce lower bounds on the norm of any possible relation for x. Thus PSLQ(τ) can be used to prove that there are no relations for x of norm less than a given size. Let Mx be the smallest norm of any relation for x. For the real and complex case and each fixed parameter τ in a certain interval, we prove that PSLQ(τ) constructs a relation in less than O(n3 + n2 logMx) iterations. Ref: Mathematics of Computation, to appear (1999)