A Novel Randomized XR-Based Preconditioned CholeskyQR Algorithm

TL;DR

This paper introduces a randomized preconditioning framework for CholeskyQR, enhancing its stability and speed, and proposes two new methods with proven effectiveness and scalability.

Contribution

It presents a novel randomized preconditioner framework for CholeskyQR, including two new methods that improve stability and efficiency.

Findings

Preconditioners effectively reduce the condition number.

Methods are more stable and faster than existing algorithms.

Numerical tests demonstrate good scalability.

Abstract

CholeskyQR is a simple and fast QR decomposition via Cholesky decomposition, while it has been considered highly sensitive to the condition number. In this paper, we provide a randomized preconditioner framework for CholeskyQR algorithm. Under this framework, two methods (randomized LU-CholeskyQR and randomized QR-CholeskyQR) are proposed and discussed. We prove the proposed preconditioners can effectively reduce the condition number, which is also demonstrated by numerical tests. Abundant numerical tests indicate our methods are more stable and faster than all the existing algorithms and have good scalability.