We present an algorithm computing the determinant of an integer matrix A. The
algorithm is introspective in the sense that it uses several distinct
algorithms that run in a concurrent manner. During the course of the algorithm
partial results coming from distinct methods can be combined. Then, depending
on the current running time of each method, the algorithm can emphasize a
particular variant. With the use of very fast modular routines for linear
algebra, our implementation is an order of magnitude faster than other existing
implementations. Moreover, we prove that the expected complexity of our
algorithm is only O(n^3 log^{2.5}(n ||A||)) bit operations in the dense case
and O(Omega n^{1.5} log^2(n ||A||) + n^{2.5}log^3(n||A||)) in the sparse case,
where ||A|| is the largest entry in absolute value of the matrix and Omega is
the cost of matrix-vector multiplication in the case of a sparse matrix.Comment: Published in Transgressive Computing 2006, Grenade : Espagne (2006