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

Abstract

International audienceOrder 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

A Relaxed Algorithm for Online Matrix Inversion

“…In this section we show how to incorporate matrix multiplication to solve the problem OnlineInverse in O(n ω ) field operations in K. We adopt two ideas used in relaxed [7] and online [6] algorithms.…”

“…Online algorithms for computer algebra were popularized by van der Hoeven with the introduction of online algorithms for formal power series multiplication [7,13]. Recently, an online algorithm for polynomial matrix order basis computation has been proposed [6].…”

A Relaxed Algorithm for Online Matrix Inversion

“…To incorporate fast matrix multiplication we adopt two ideas used in relaxed [3] and online [2] algorithms.…”

Exact Linear Algebra Algorithmic

“…Although these algorithms are most often based on recursive algorithms and geometric progressions, alternative approaches exist, such as arithmetic progression based trade-offs, used e.g. in the computation of the characteristic polynomial [24] or online update schemes in geometric progressions [16].…”