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On MrR (Mister R) Method for Solving Linear Equations with Symmetric Matrices

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Modeling, Design and Simulation of Systems (AsiaSim 2017)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 752))

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Abstract

Krylov subspace methods, such as the Conjugate Gradient (CG) and Conjugate Residual (CR) methods, are treated for efficiently solving a linear system of equations with symmetric matrices. AZMJ variant of Orthomin(2) (abbreviated as AZMJ) [1] has recently been proposed for solving the linear equations. In this paper, an alternative AZMJ variant is redesigned, i.e., an alternative minimum residual method for symmetric matrices is proposed by using the coupled two-term recurrences formulated by Rutishauser. The recurrence coefficients are determined by imposing the A-orthogonality on the residuals as well as CR. Our proposed variant is referred to as MrR. It is mathematically equivalent to CR and AZMJ, but the implementations are different; the recurrence formulae contain alternative expressions for the auxiliary vectors and the recurrence coefficients. Through numerical experiments on the linear equations with real symmetric matrices, it is demonstrated that the residual norms of MrR converge faster than those of CG and AZMJ.

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Acknowledgment

We appreciate Mr. K. Iwasato of Kyushu University for executing numerical experiments. We would like to appreciate Professor G.L.G. Sleijpen and the reviewers for their insightful and helpful suggestions. This research was partly supported by JSPS KAKENHI Grant Number 26390136, 2016.

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Authors and Affiliations

  1. Faculty of Economics and Information, Gifu Shotoku University, Nakauzura, Gifu, 5008288, Japan

    Kuniyoshi Abe

  2. Professor Emeritus, Kyushu University, Fukuoka, 8128581, Japan

    Seiji Fujino

Authors
  1. Kuniyoshi Abe
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  2. Seiji Fujino
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Correspondence to Kuniyoshi Abe .

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Editors and Affiliations

  1. Universiti Teknologi Malaysia, Skudai, Malaysia

    Mohamed Sultan Mohamed Ali

  2. Universiti Teknologi Malaysia, Skudai, Malaysia

    Herman Wahid

  3. Universiti Teknologi Malaysia, Skudai, Malaysia

    Nurul Adilla Mohd Subha

  4. Universiti Teknologi Malaysia, Skudai, Malaysia

    Shafishuhaza Sahlan

  5. Universiti Teknologi Malaysia, Skudai, Malaysia

    Mohd Amri Md. Yunus

  6. Universiti Teknologi Malaysia, Skudai, Malaysia

    Ahmad Ridhwan Wahap

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Abe, K., Fujino, S. (2017). On MrR (Mister R) Method for Solving Linear Equations with Symmetric Matrices. In: Mohamed Ali, M., Wahid, H., Mohd Subha, N., Sahlan, S., Md. Yunus, M., Wahap, A. (eds) Modeling, Design and Simulation of Systems. AsiaSim 2017. Communications in Computer and Information Science, vol 752. Springer, Singapore. https://doi.org/10.1007/978-981-10-6502-6_45

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  • DOI: https://doi.org/10.1007/978-981-10-6502-6_45

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-6501-9

  • Online ISBN: 978-981-10-6502-6

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