Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
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PRECONDITIONED AOR ITERATIVE METHODS FOR SOLVING MULTI-LINEAR SYSTEMS WITH 𝓜-TENSOR

  • QI, MENG (Department of Mathematics, College of Sciences, Northeastern University) ;
  • SHAO, XINHUI (Department of Mathematics, College of Sciences, College of Sciences, Northeastern University)
  • 투고 : 2021.01.26
  • 심사 : 2021.04.22
  • 발행 : 2021.05.30
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초록

Some problems in engineering and science can be equivalently transformed into solving multi-linear systems. In this paper, we propose two preconditioned AOR iteration methods to solve multi-linear systems with -tensor. Based on these methods, the general conditions of preconditioners are given. We give the convergence theorem and comparison theorem of the two methods. The results of numerical examples show that methods we propose are more effective.

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과제정보

This work was supported by by Central University Basic Scientific Research Business Expenses Special Funds(N2005013).

참고문헌

  1. L.Q. Qi and Z.Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, M. Philadelphia: SIAM, 2017.
  2. W. Ding and Y. Wei, Theory and Computation of Tensors, M. Math. London: Academic Press, 2016.
  3. W.Y. Ding and Y.M. Wei, Solving multi-linear systems with M-tensors, J. Sci. Comput. 68 (2016), 689-715. https://doi.org/10.1007/s10915-015-0156-7
  4. Z. Luo, L. Qi and N. Xiu, The sparsest solutions to Z-tensor complementarity problems, J. Optimi. Lett. 11 (2017), 471-482. https://doi.org/10.1007/s11590-016-1013-9
  5. X.-T. Li and K.N. Michael, Solving sparse non-negative tensor equations: algorithms and applications, J. Front. Math. 10 (2015), 649-680. https://doi.org/10.1007/s11464-014-0377-3
  6. S. Du, L. Zhang, C. Chen and L. Qi, Tensor absolute value equations, Sci. China Math. 61 (2018), 1695-1710. https://doi.org/10.1007/s11425-017-9238-6
  7. Z.J. Xie, X.Q. Jin and Y.M. Wei, Tensor methods for solving symmetric-tensor systems, J. Sci. Comput. 74 (2018), 412-425. https://doi.org/10.1007/s10915-017-0444-5
  8. M.L. Liang, B. Zheng and R.J. Zhao, Alternating iterative methods for solving tensor equations with applications, J. Numerical Algorithms 80 (2019), 1437-1465. https://doi.org/10.1007/s11075-018-0601-4
  9. H. He, C. Ling and L. Qi, G. Zhou, A globally and quadratically convergent algorithm for solving ultilinear systems with M-tensors, J. J. Scient. Comput. 76 (2018), 1718-1741. https://doi.org/10.1007/s10915-018-0689-7
  10. L. Han, A homotopy method for solving multilinear systems with m-tensors, J. Appl. Math. Lett. 69 (2017), 49-54. https://doi.org/10.1016/j.aml.2017.01.019
  11. X.Z. Wang, M.L. Che and Y.M. Wei, Neural networks based approach solving multi-linear systems with M-tensors, J. Neurocomputing on Science Direct 351 (2019), 33-42. https://doi.org/10.1016/j.neucom.2019.03.025
  12. D.D. Liu, W. Li, Seak-WengVonga, The tensor splitting with application to solve multi-linear systems, J. of Computational and Applied Mathematics 330 (2018), 75-94. https://doi.org/10.1016/j.cam.2017.08.009
  13. D.D. Liu, W. Li, Seak-WengVonga, Comparison results for splitting iterations for solving multi-linear systems, J. Applied Numerical Mathematics 134 (2018), 105-121. https://doi.org/10.1016/j.apnum.2018.07.009
  14. L.B. Cui, M.H. Li and Y.S. Song, Preconditioned tensor splitting iterations method for solving multi-linear systems, J. Applied Mathematics Letters 96 (2019), 89-94. https://doi.org/10.1016/j.aml.2019.04.019
  15. W.Y. Ding, L.Q. Qi and Y.M. Wei, M-tensors and nonsingular M-tensors, J. Linear Algebra and its Applications 439 (2013), 3264-3278. https://doi.org/10.1016/j.laa.2013.08.038
  16. Kelly J. Pearson, Essentially Positive Tensors, International Journal of Algebra 4 (2010), 421-427.
  17. Liqun Qi, Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation 40 (2005), 1302-1324. https://doi.org/10.1016/j.jsc.2005.05.007
  18. A. Hadjidimos, Accelerated Overrelaxation Method, J. Mathematics of Computation 32 (1978), 149-157. https://doi.org/10.1090/S0025-5718-1978-0483340-6
  19. H. Niki, K. Harada and M. Morimoto, M. Sakakihara, The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel method, J. of Computational and Applied Mathematics 164 (2004), 587-600. https://doi.org/10.1016/j.cam.2003.11.012
  20. H. Kotakemori, H. Niki and N. Okamotob, Accelerated iterative method for Z-matrices, Journal of Computational and Applied Mathematics 75 (1996), 87-97. https://doi.org/10.1016/S0377-0427(96)00061-1
  21. K.C. Chang, K. Pearson and T. Zhang, Perron-Frobenius Theorem for nonnegative tensors, J. Communications in Mathematical Sciences 6 (2008), 507-520. https://doi.org/10.4310/CMS.2008.v6.n2.a12
  22. G.H. Golub, C.F. Van Loan, Matrix Computations, M. Johns Hopkins University Press, Baltimore, 2013.
  23. Y. Zhang, Q. Liu and Z. Chen, Preconditioned Jacobi type method for solving multi-linear systems with M-tensors, J. Applied Mathematics Letters 104 (2020), 104:106287. https://doi.org/10.1016/j.aml.2020.106287
  24. L.B. Cui, X.Q. Zhang and S.L. Wu, A new preconditioner of the tensor splitting iterative method for solving multi-linear systems with M -tensors, J. Computational and Applied Mathematics 39 (2020), 1-16. https://doi.org/10.1016/0377-0427(92)90216-K
  25. D.D. Liu, W. Li and Seak-WengVonga, A new preconditioned SOR method for solving multi-linear systems with an M-tensors, J. Calcolo 57 (2020), Doi:10.1007/s10092-020-00364-8.