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Two New Variants of the Simpler Block GMRES Method with Vector Deflation and Eigenvalue Deflation for Multiple Linear Systems

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Abstract

In this paper, two effective methods based on the simpler block GMRES method are established in order to solve the linear systems of equations with multiple right-hand sides. The first method is derived from the simpler block GMRES method with vector deflation restarting (SBGMRES-DR). The second method is constructed from a combination of SBGMRES-DR with the eigenvalue deflation technique, which is called the deflated simpler block GMRES method with vector deflation restarting (D-SBGMRES-DR). To be more specific, SBGMRES-DR is capable of removing linearly or almost linearly dependent vectors created by the block Arnoldi process. On the other hand, D-SBGMRES-DR not only deletes linearly or almost linearly dependent vectors but also retains harmonic Ritz vectors associated with the smallest harmonic Ritz values in magnitude, and adds them to the new search subspace at the time of restart. Finally, a wide range of practical experiments are carried out to assess the efficiency of the proposed methods. The numerical results indicate that the D-SBGMRES-DR method outperforms the compared methods with respect to the number of matrix–vector products and the computational time.

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Data Availibility

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

The authors would like to express their gratitude towards the anonymous reviewers for the valuable comments and suggestions that helped us to enhance the quality of this work. We are also grateful to Dr. Kathryn Lund for providing us with the codes of the loop-interchange block GMRES method.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

    Azita Tajaddini & Najmeh Azizizadeh

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Gang Wu

  3. Department of Applied Mathematics, Faculty of Sciences and Modern Technologies, Graduate University of Advanced Technology, Kerman, Iran

    Farid Saberi-Movahed

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  1. Azita Tajaddini
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  2. Gang Wu
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  4. Najmeh Azizizadeh
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Correspondence to Azita Tajaddini.

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Tajaddini, A., Wu, G., Saberi-Movahed, F. et al. Two New Variants of the Simpler Block GMRES Method with Vector Deflation and Eigenvalue Deflation for Multiple Linear Systems. J Sci Comput 86, 9 (2021). https://doi.org/10.1007/s10915-020-01376-w

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  • DOI: https://doi.org/10.1007/s10915-020-01376-w

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