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Greville’s method for preconditioning least squares problems

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Abstract

In this paper, we propose a preconditioning algorithm for least squares problems \(\displaystyle{\min_{x\in{{\mathbb{R}}}^n}}\|b-Ax\|_2\), where A can be matrices with any shape or rank. The preconditioner is constructed to be a sparse approximation to the Moore–Penrose inverse of the coefficient matrix A. For this preconditioner, we provide theoretical analysis to show that under our assumption, the least squares problem preconditioned by this preconditioner is equivalent to the original problem, and the GMRES method can determine a solution to the preconditioned problem before breakdown happens. In the end of this paper, we also give some numerical examples showing the performance of the method.

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Author notes

  1. Xiaoke Cui

    Present address: Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8568, Japan

Authors and Affiliations

  1. Department of Informatics, School of Multidisciplinary Sciences, The Graduate University for Advanced Studies (Sokendai), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Xiaoke Cui & Ken Hayami

  2. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Ken Hayami

  3. Department of Mathematics, Tongji University, 1239 Siping Road, Shanghai, 200092, People’s Republic of China

    Jun-Feng Yin

Authors
  1. Xiaoke Cui
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  2. Ken Hayami
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  3. Jun-Feng Yin
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Corresponding author

Correspondence to Xiaoke Cui.

Additional information

Communicated by Rafael Bru.

This research was supported by the Grants-in-Aid for Scientific Research(C) of the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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Cui, X., Hayami, K. & Yin, JF. Greville’s method for preconditioning least squares problems. Adv Comput Math 35, 243–269 (2011). https://doi.org/10.1007/s10444-011-9171-x

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  • DOI: https://doi.org/10.1007/s10444-011-9171-x

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