GBLA -- Gr\"obner Basis Linear Algebra Package

TL;DR

GBLA is an open-source C library that optimizes matrix reduction in Gr"obner basis computations, achieving significant speedups and better scalability through novel data structures, operation ordering, and storage formats.

Contribution

It introduces a new data structure, operation order, and storage format for Gr"obner basis matrix reduction, improving performance and scalability over existing methods.

Findings

Up to 40% faster than GB reduction in sequential runs

Scales up to 26 times better on 32 cores

Reduces storage size by up to 4 times

Abstract

This is a system paper about a new GPLv2 open source C library GBLA implementing and improving the idea of Faug\`ere and Lachartre (GB reduction). We further exploit underlying structures in matrices generated during Gr\"obner basis computations in algorithms like F4 or F5 taking advantage of block patterns by using a special data structure called multilines. Moreover, we discuss a new order of operations for the reduction process. In various different experimental results we show that GBLA performs better than GB reduction or Magma in sequential computations (up to 40% faster) and scales much better than GB reduction for a higher number of cores: On 32 cores we reach a scaling of up to 26. GBLA is up to 7 times faster than GB reduction. Further, we compare different parallel schedulers GBLA can be used with. We also developed a new advanced storage format that exploits the fact that our matrices are coming from Gr\"obner basis computations, shrinking storage by a factor of up to 4. A huge database of our matrices is freely available with GBLA.