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A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems

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Abstract

By utilizing the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration technique, we establish a non-alternating PHSS (NPHSS) iteration method for solving large sparse non-Hermitian positive definite linear systems. The convergence analysis demonstrates that the iterative series produced by the NPHSS method converge to the unique solution of the linear system when the parameters satisfy some moderate conditions. We also give a possible optimal upper bound for the iterative spectral radius. Moreover, to reduce the computational cost, we establish an inexact variant of the NPHSS (INPHSS) iteration method whose convergence property is studied. Both theoretical and numerical results validate that the NPHSS method outperforms the PHSS method when the Hermitian part of the coefficient matrix is dominant.

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Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which greatly improve the presentation of the paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11471150, 11401281, 11271173), and the CAPES and CNPq in Brazil.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Yu-Jiang Wu

  2. Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, 730000, China

    Yu-Jiang Wu

  3. Institute of Industrial Mathematics, IMPA, CP: 19.081, Curitiba, PR, 81531-980, Brazil

    Yu-Jiang Wu

  4. College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, 730050, China

    Xu Li

  5. Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, China

    Xu Li

  6. Department of Mathematics, Centro Politécnico, Federal University of Paraná, CP: 19.081, Curitiba, PR, 81531-980, Brazil

    Jin-Yun Yuan

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  1. Yu-Jiang Wu
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  2. Xu Li
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  3. Jin-Yun Yuan
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Correspondence to Xu Li.

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Communicated by Ya-xiang Yuan.

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Wu, YJ., Li, X. & Yuan, JY. A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems. Comp. Appl. Math. 36, 367–381 (2017). https://doi.org/10.1007/s40314-015-0231-6

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  • DOI: https://doi.org/10.1007/s40314-015-0231-6

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