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How to compute the minimum norm least squares solution of singular linear system by using the preconditioned HSS method?

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Abstract

For the singular, non-Hermitian, and positive semi-definite system of linear equations Ax = b, we introduce a kind of preconditioners for the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method. Then, the iteration sequence generated by the corresponding PHSS iteration method converges to the minimum norm least squares solution A‡b for any initial guess no matter the singular system of linear equations is consistent or inconsistent. In addition, a kind of PHSS preconditioners are derived from the PHSS iteration method. The PHSS preconditioned generalized minimum residual (PGMRES) methods also determine the minimum norm least squares solution of the consistent singular linear system at breakdown. Numerical experiments are used to verify the effectiveness and the robustness of the PHSS iteration method and the PHSS preconditioner.

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Funding

This work is partially supported by the National Natural Science Foundation of China (Grant nos. 11471150, 11401281) and the National Key Research and Development Program of China (Grant no. 2018YFC0406600).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Yue Hao, Ai-Li Yang & Yu-Jiang Wu

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  1. Yue Hao
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  2. Ai-Li Yang
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  3. Yu-Jiang Wu
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Correspondence to Ai-Li Yang or Yu-Jiang Wu.

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Hao, Y., Yang, AL. & Wu, YJ. How to compute the minimum norm least squares solution of singular linear system by using the preconditioned HSS method?. Numer Algor 83, 1205–1221 (2020). https://doi.org/10.1007/s11075-019-00721-x

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  • DOI: https://doi.org/10.1007/s11075-019-00721-x

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