LU-Cholesky QR algorithms for thin QR decomposition

Abstract

This paper aims to propose the LU-Cholesky QR algorithms for thin QR decomposition (also called economy size or reduced QR decomposition). CholeskyQR is known as a fast algorithm employed for thin QR decomposition, and CholeskyQR2 aims to improve the orthogonality of a Q-factor computed by CholeskyQR. Although such Cholesky QR algorithms can efficiently be implemented in high-performance computing environments, they are not applicable for ill-conditioned matrices, as compared to the Householder QR and the Gram–Schmidt algorithms. To address this problem, we apply the concept of LU decomposition to the Cholesky QR algorithms, i.e., the idea is to use LU-factors of a given matrix as preconditioning before applying Cholesky decomposition. Moreover, we present rounding error analysis of the proposed algorithms on the orthogonality and residual of computed QR-factors. Numerical examples provided in this paper illustrate the efficiency of the proposed algorithms in parallel computing on both shared and distributed memory computers.