Arne Storjohann

Ziwen Wang, Stavros Birmpilis, George Labahn et al.

We describe a C implementation of the Las Vegas algorithm of Birmpilis, Labahn and Storjohann from 2020 for computing the Smith normal form of a nonsingular integer matrix. The algorithm computes a Smith massager for the input matrix using $O(n^ω\, \B(\log n + \log \|A\|)\, (\log n)^2)$ bit operations, which is softly equivalent to the cost of multiplying two matrices of the same dimension and entry size. We describe the key implementation techniques that bridge the gap between the theoretical algorithm and practical performance, including BLAS-accelerated modular arithmetic via the Residue Number System and an adaptive batching scheme that collapses the theoretical $O(\log n)$ iterations to $O(1)$ in practice. Experiments on matrices of dimension up to $n = 10007$ show that the implementation's running time scales proportionally to that of a single BLAS matrix multiplication, with both exhibiting the same effective growth rate on a log-log plot.

George Labahn, Arne Storjohann

Given a full column rank $M \in \Z^{\ell \times m}$ and an $F \in \Z^{n \times m}$ we present an algorithm to compute the $n \times n$ basis in Hermite form of the integer lattice comprised of all rows $p \in \Z^{1 \times n}$ such that $pF \in \Z^{1 \times m}$ is in the integer lattice generated by the rows of $M$. The algorithm is randomized of the Las Vegas type, that is, it can fail with probability at most $1/2$, but if fail is not returned it guarantees to produce the correct result. When $M$ is square and $F=I_m$, then the computed basis is the Hermite normal form of $M$, and the algorithm uses about the same number of bit operations as required to multiply together two matrices of the same dimension and size of entries as $M$.