Online order basis algorithm and its impact on the block Wiedemann algorithm

Order bases are a fundamental tool for linear algebra with polynomial coefficients. In particular, block Wiedemann methods are nowadays able to tackle large sparse matrix problems because they benefit from fast order basis algorithms. However, such fast algorithms suffer from two practical drawbacks: they are not designed for early termination and often require more knowledge on the input than necessary. In this paper, we propose an online algorithm for order basis which allows for both early termination and minimal input requirement while keeping quasi-optimal complexity in the order. Using this algorithm inside block Wiedemann methods leads to an improvement of their practical performance by a constant factor.