# Cache-oblivious Matrix Multiplication for Exact Factorisation

## TL;DR

This paper introduces a cache-oblivious matrix multiplication method using Morton-hybrid space-filling curves, significantly improving runtime for exact matrix factorization over finite fields.

## Contribution

It develops a novel cache-oblivious approach for matrix multiplication tailored for parallel TU decomposition with Morton-hybrid layout, enhancing efficiency.

## Key findings

- Orders of magnitude faster sequential evaluation
- Low span in recursive matrix multiplication
- Effective incorporation into parallel decomposition

## Abstract

We present a cache-oblivious adaptation of matrix multiplication to be incorporated in the parallel TU decomposition for rectangular matrices over finite fields, based on the Morton-hybrid space-filling curve representation. To realise this, we introduce the concepts of alignment and containment of sub-matrices under the Morton-hybrid layout. We redesign the decompositions within the recursive matrix multiplication to force the base case to avoid all jumps in address space, at the expense of extra recursive matrix multiplication (MM) calls. We show that the resulting cache oblivious adaptation has low span, and our experiments demonstrate that its sequential evaluation order demonstrates orders of magnitude improvement in run-time, despite the recursion overhead.