A Randomized Parallel Algorithm with Run Time $O(n^2)$ for Solving an $n \times n$ System of Linear Equations

TL;DR

This paper presents a parallel randomized algorithm for solving real-valued linear systems with an $O(n^2)$ runtime, extending previous algorithms over prime fields and removing symmetry constraints.

Contribution

It introduces a parallelized randomized algorithm for real linear systems with quadratic runtime, generalizing prior methods to non-symmetric matrices.

Findings

Algorithm solves $A x = b$ in $O(n^2)$ time with probability one.

Extends previous algorithms from prime fields to reals.

Does not require matrix symmetry.

Abstract

In this note, following suggestions by Tao, we extend the randomized algorithm for linear equations over prime fields by Raghavendra to a randomized algorithm for linear equations over the reals. We also show that the algorithm can be parallelized to solve a system of linear equations $A x = b$ with a regular $n \times n$ matrix $A$ in time $O(n^2)$, with probability one. Note that we do not assume that $A$ is symmetric.