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 = (x 1 ,...,x n ) be a vector in K n . The vector x has an integer relation if there exists a vector m = (m 1 ,...,m n ) ∈ O(K) n , m ¬= 0, such that m 1 x 1 + m 2 x 2 +... + m n x n = 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 = (x 1 ,...,x n ) ∈ K n , 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 M x 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(n 3 + n 2 logM x ) iterations.