Towards an exact adaptive algorithm for the determinant of a rational matrix

TL;DR

This paper develops and compares strategies for the exact computation of rational matrix determinants, introducing an adaptive algorithm that selects the most efficient method based on matrix properties, implemented in the LinBox library.

Contribution

It proposes new strategies for rational determinant computation, including an adaptive algorithm that optimally chooses methods based on matrix characteristics.

Findings

The adaptive algorithm outperforms individual strategies in various cases.

Preconditioning techniques can simplify rational determinant calculations.

Experimental results validate the efficiency of the proposed methods.

Abstract

In this paper we propose several strategies for the exact computation of the determinant of a rational matrix. First, we use the Chinese Remaindering Theorem and the rational reconstruction to recover the rational determinant from its modular images. Then we show a preconditioning for the determinant which allows us to skip the rational reconstruction process and reconstruct an integer result. We compare those approaches with matrix preconditioning which allow us to treat integer instead of rational matrices. This allows us to introduce integer determinant algorithms to the rational determinant problem. In particular, we discuss the applicability of the adaptive determinant algorithm of [9] and compare it with the integer Chinese Remaindering scheme. We present an analysis of the complexity of the strategies and evaluate their experimental performance on numerous examples. This experience allows us to develop an adaptive strategy which would choose the best solution at the run time, depending on matrix properties. All strategies have been implemented in LinBox linear algebra library.