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A parallel generalized global conjugate gradient squared algorithm for linear systems with multiple right-hand sides

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Abstract

In this paper, based on the generalized global conjugate gradient squared (GGl-CGS) algorithm in Zhang et al. (Appl Math Comput 216:3694–3706, 2010) and the ideas in Gu et al. (Appl Math Comput 186:1243–1253, 2007), we present a parallel generalized Gl-CGS (PGGl-CGS) algorithm for linear systems with multiple right-hand sides. The new algorithm reduces two global synchronization points to one by changing the computation sequence in the generalized Gl-CGS algorithm, and all inner products per iteration are independent and communication time required for inner product can be overlapped with useful computation. Theoretical analysis and numerical experiments show that the PGGl-CGS method has better parallelism and scalability than the generalized Gl-CGS method, and the parallel performance can be improved by a factor of about 3/2.

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Acknowledgments

The authors would like to thank the referees and Editor for their helpful and detailed suggestions for revising this manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou, Henan , 450015, People’s Republic of China

    Li-Tao Zhang

  2. Institute of Data and Knowledge Engineering, School of Computer and Information Engineering, Henan University, Kaifeng, Henan , 475004, People’s Republic of China

    Xian-Yu Zuo

  3. Laboratory of Computationary Physics, P. O. Box 8009, Beijing , 100088, People’s Republic of China

    Li-Tao Zhang, Tong-Xiang Gu & Xing-Ping Liu

Authors
  1. Li-Tao Zhang
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  2. Xian-Yu Zuo
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  3. Tong-Xiang Gu
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  4. Xing-Ping Liu
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Corresponding author

Correspondence to Li-Tao Zhang.

Additional information

Communicated by Ernesto G. Birgin.

This research of this author is supported by NSFC Tianyuan Mathematics Youth Fund (11226337), NSFC (61203179, 61202098, 61170309, 91130024 and 11171039), Aeronautical Science Foundation of China (2013ZD55006), Project of Youth Backbone Teachers of Colleges and Universities in Henan Province (2013GGJS-142), ZZIA Innovation team fund (2014TD02), Major project of development foundation of science and technology of CAEP (2012A0202008), Basic and Advanced Technological Research Project of of Henan Province (132300410373).

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Zhang, LT., Zuo, XY., Gu, TX. et al. A parallel generalized global conjugate gradient squared algorithm for linear systems with multiple right-hand sides. Comp. Appl. Math. 34, 901–916 (2015). https://doi.org/10.1007/s40314-014-0158-3

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  • DOI: https://doi.org/10.1007/s40314-014-0158-3

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