Suppose I have an ordered basis $\{b_1, \dots, b_n\}$ of a lattice in $\mathbb{R}^n$, but I do not assume that $b_i \in \mathbb{Z}^n$ for all $1 \leq i \leq n$. I would like to perform lattice reduction: for instance, I would like to get a reduced basis in the sense of the Lenstra–Lenstra–Lovász lattice reduction algorithm (LLL-algorithm), which, as far as I understand, still makes sense if the basis does not have integer coefficients. So far, in examples, I have been applying the LLL-algorithm to such instances quite successfully, but I haven't been able to find any theoretical analyses of lattice reduction algorithms for the case where the basis does not have integer coefficients. The closest thing I found is the floating point extensions of the LLL-algorithm (e.g., Floating-point LLL revisited. PQ Nguên, D Stehlé), but, as far as I can tell, these assume that the input of the algorithm is a floating point approximation of a basis with integer coefficients.
My question is: Have people consider the problem of lattice reduction for bases with non-integer coefficients? Are there adaptations of the LLL-algorithm with complexity guarantees that work in this case?
