Algorithmic Solution for Systems of Linear Equations, in $\mathcal{O}(mn)$ time

TL;DR

This paper introduces a fast, memory-efficient algorithm for solving linear systems of equations with a proven convergence, significantly outperforming existing methods in speed and resource usage, and extends its application to feature selection.

Contribution

A novel, vectorized algorithm for linear systems solving with theoretical convergence proof and practical extension to feature selection tasks.

Findings

> 100x speed-up over state-of-the-art methods

Low memory demands for various system types

High solution accuracy and straightforward control

Abstract

We present a novel algorithm attaining excessively fast, the sought solution of linear systems of equations. The algorithm is short in its basic formulation and, by definition, vectorized, while the memory allocation demands are trivial, because, for each iteration, only one dimension of the given input matrix $\mathbf X$ is utilized. The execution time is very short compared with state-of-the-art methods, exhibiting $> \times 10^2$ speed-up and low memory allocation demands, especially for non-square Systems of Linear Equations, with ratio of equations versus features high (tall systems), or low (wide systems) accordingly. The accuracy is high and straightforwardly controlled, and the numerical results highlight the efficiency of the proposed algorithm, in terms of computation time, solution accuracy and memory demands. The paper also comprises a theoretical proof for the algorithmic convergence, and we extend the implementation of the proposed algorithmic rationale to feature selection tasks.