For the equation:
$$ \sum_{i=1}^na_ix_i=0 $$ where all $x_i$ are real numbers and all $a_i$ are integers, the PSLQ algorithm can either find an integer relation or give lower bounds on the norm of vector $a$ of any possible integer relation for a given precision approximation of all $x_i$.
See Integer Relation Detection by Bailey and Borwein for more information.
I was testing Bailey's arbitrary precision software implementation of PSLQ and comparing it to Maple's implementation by giving them a list of $n=16$ linearly independent real numbers and seeing what they came up with for different precision approximations of $x_1\dots x_{16}$.
Since all $x_i$ are linearly independent real numbers, I know no integer relation exists, but what was interesting was the Bailey's software exhausts precision and gives up when the norm grows to some size (I'm not sure exactly how it determines the size) and reports it has not found a relation, while the Maple implementation keeps increasing the bound until it finds a relation, even if it is erroneous. I would have tried Pari/GP and Sage as well, but they do not seem to have a PSLQ algorithm available and their LLL implementations do not seem to give this lower bound on the norm of any possible integer relation.
So my question is this:
How can I use this lower bound to determine the maximum size $M$ of coefficients $a_i$ for which a given precision will not give a (wrong) relation?
I'd like to be able to judge that Maple is giving me an actual integer relation rather than garbage. I'm considering writing my own PSLQ implementation based on Bailey's and linking it to Maple for ease of use, but I want to understand how to use the bound to control the stopping criteria for the algorithm. In particular, I want to understand how the bound relates to the size of the $a_i$'s. Maple's implementation doesn't give you this level of control.
