I'm an academic mathematician. While trying to verify a counterexample to something in my research, the following computational problem has arisen:
Fix a vector space $V = \mathbb{Q}^n$. Here $n$ is something on the order of $20,000$. I have a procedure that spits out a sequence of vectors $\vec{v}_1,\vec{v}_2,\ldots$ in $V$. The vectors $\vec{v}_i$ all have integer entries; these entries are bounded in size (by something on the order of 20 or 30). For each $k \geq 2$, I need to decide if $\vec{v}_k$ is in the $\mathbb{Q}$-span of the set $\{\vec{v}_1,\ldots,\vec{v}_{k-1}\}$.
What algorithm should I use for this? The naive thing to do would be to just use Gauss-Jordan elimination (or, if I want to mostly stick with integers, compute the Hermite normal form). However, experiments I have done show that the size of the numbers that arise when you do this explode in complexity even for much smaller problems. This accords with stuff I have found while searching the literature, though I don't know the computer science literature very well and have trouble determining the state of the art in stuff like this.