Matroid theory: Efficient algorithms for finding all minimal sets of linearly dependent variables

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Are there any algorithms in the literature of matroid theory (or elsewhere) that find all minimal sets of linearly dependent variables (also termed as circuits in matroid theory) from the given global set of variables? The only paper that I could find so far is this, but that requires one to have the set of all bases.

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Algorithms for computing the circuits of a matroid represented by a matrix have been implemented in Sage and in the Matroids package for Macaulay2. Both are open source software, so the algorithms are readily available.

Moreover, a quick search on the arxiv returns a number of results including this paper that gives an algorithm for computing the circuits of algebraic matroids. Since every linear matroid (i.e., a matroid defined as the linear dependencies among the columns of a matrix) is algebraic, this may also be worth looking into.