Simple, Fast and Practicable Algorithms for Cholesky, LU and QR Decomposition Using Fast Rectangular Matrix Multiplication

TL;DR

This paper introduces simple and efficient algorithms for Cholesky, LU, and QR decompositions that leverage fast rectangular matrix multiplication, achieving near-optimal time complexity and outperforming existing implementations in some cases.

Contribution

The paper proposes novel, straightforward algorithms for matrix decompositions that utilize the fastest known matrix multiplication methods, filling a gap in existing literature.

Findings

Algorithms achieve $O(n^{2.529})$ time complexity.

Implementation with Strassen multiplication outperforms GNU Scientific Library in some cases.

Methods are simple and potentially broadly applicable.

Abstract

This note presents fast Cholesky/LU/QR decomposition algorithms with $O(n^{2.529})$ time complexity when using the fastest known matrix multiplication. The algorithms have potential application, since a quickly made implementation using Strassen multiplication has lesser execution time than the employed by the GNU Scientific Library for the same task in at least a few examples. The underlaying ideas are very simple. Despite this, I have been unable to find these methods in the literature.