An introspective algorithm for the integer determinant

TL;DR

This paper introduces an introspective algorithm for computing the integer matrix determinant that combines multiple methods for efficiency and speed, outperforming existing implementations.

Contribution

The paper presents a novel introspective algorithm that adaptively combines various determinant computation methods for improved performance.

Findings

Order of magnitude faster than existing implementations

Expected complexity is O(n^3 log^{2.5}(n ||A||)) for dense matrices

Efficient for both dense and sparse matrices

Abstract

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.