Parallel computation of the rank of large sparse matrices from algebraic K-theory

TL;DR

This paper presents parallel algorithms for computing the rank and Smith forms of extremely large sparse matrices from algebraic K-theory, enabling the first cohomology computation of GL_7(Z).

Contribution

It introduces new parallel linear algebra methods tailored for massive sparse matrices in algebraic K-theory, linking matrix computations to motivic cohomology.

Findings

Matrices with up to 37 million non-zero entries processed

Largest rank computation took over 35 days on 50 processors

First cohomology computation of GL_7(Z) achieved

Abstract

This paper deals with the computation of the rank and of some integer Smith forms of a series of sparse matrices arising in algebraic K-theory. The number of non zero entries in the considered matrices ranges from 8 to 37 millions. The largest rank computation took more than 35 days on 50 processors. We report on the actual algorithms we used to build the matrices, their link to the motivic cohomology and the linear algebra and parallelizations required to perform such huge computations. In particular, these results are part of the first computation of the cohomology of the linear group GL_7(Z).