TY - JOUR

T1 - LU-Cholesky QR algorithms for thin QR decomposition

AU - Terao, Takeshi

AU - Ozaki, Katsuhisa

AU - Ogita, Takeshi

N1 - Funding Information:
This work was partially funded by JSPS KAKENHI Grant Numbers 16H03917, JST CREST, and MEXT as “Exploratory Issue on Post-K computer” (Development of verified numerical computations and super high-performance computing environment for extreme researches) using computational resources of the K computer and other computers of the HPCI system provided by RIKEN R-CCS and Nagoya University through the HPCI System Research Project (Project ID: hp180222 , hp190192 ).
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2020/4

Y1 - 2020/4

N2 - 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.

AB - 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.

KW - High-performance computing

KW - Numerical linear algebra

KW - Rounding error analysis

KW - Thin QR decomposition

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U2 - 10.1016/j.parco.2019.102571

DO - 10.1016/j.parco.2019.102571

M3 - Article

AN - SCOPUS:85076601603

SN - 0167-8191

VL - 92

JO - Parallel Computing

JF - Parallel Computing

M1 - 102571

ER -