On the Multilinear Systems, Part I: Comparison of Splittings of Tensors

doi:10.1007/s42967-025-00511-4

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (4): 1587-1603.doi: 10.1007/s42967-025-00511-4

On the Multilinear Systems, Part I: Comparison of Splittings of Tensors

Eisa Khosravi Dehdezi

Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran

收稿日期:2024-06-01 修回日期:2025-06-02 接受日期:2025-06-05 出版日期:2026-08-20 发布日期:2026-09-17
通讯作者: Eisa Khosravi Dehdezi, Email: esakhosravidehdezi@gmail.com E-mail:esakhosravidehdezi@gmail.com
基金资助:
The author received no financial support for the research, authorship, and/or publication of this article.

On the Multilinear Systems, Part I: Comparison of Splittings of Tensors

Eisa Khosravi Dehdezi

Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran

Received:2024-06-01 Revised:2025-06-02 Accepted:2025-06-05 Online:2026-08-20 Published:2026-09-17
Supported by:
The author received no financial support for the research, authorship, and/or publication of this article.

摘要/Abstract

摘要： Tensors (hypermatrices) are multidimensional analogs of matrices. In the last few years, the tensor splitting problem has attracted much attention and has been studied extensively, from theory to solution methods and applications. This work, with its two parts, aims to contribute to reviewing the state-of-the-art studies for the tensor splittings and iterative methods for solving multilinear systems (tensor equations). In the first part of this paper, some new comparison theorems for two regular and weak regular splittings of real tensors are derived. These theorems are extensions of the classical comparison theorems of splittings of matrices. We show that, under certain conditions, the iterates implied for solving the multilinear system Ax^m-1 = b are monotone. We generalize theorems about H-splittings from matrices to tensors. These and comparison theorems for two regular and weak regular splittings can be easily extended for complex tensors. In addition, we discuss triangular splitting, M-splitting, H-splitting, H-compatible splitting, and direct splitting of tensors. Some iterative methods and associated comparison theorems for solving the multilinear system Ax^m-1 = b, when A is an H-tensor, and a fast and smooth iterative method to solve Ax^m-1 = b with the symmetric coefficient tensor A are given in the second part.

关键词: Tensor splitting, Multilinear system, H-tensor

Abstract: Tensors (hypermatrices) are multidimensional analogs of matrices. In the last few years, the tensor splitting problem has attracted much attention and has been studied extensively, from theory to solution methods and applications. This work, with its two parts, aims to contribute to reviewing the state-of-the-art studies for the tensor splittings and iterative methods for solving multilinear systems (tensor equations). In the first part of this paper, some new comparison theorems for two regular and weak regular splittings of real tensors are derived. These theorems are extensions of the classical comparison theorems of splittings of matrices. We show that, under certain conditions, the iterates implied for solving the multilinear system Ax^m-1 = b are monotone. We generalize theorems about H-splittings from matrices to tensors. These and comparison theorems for two regular and weak regular splittings can be easily extended for complex tensors. In addition, we discuss triangular splitting, M-splitting, H-splitting, H-compatible splitting, and direct splitting of tensors. Some iterative methods and associated comparison theorems for solving the multilinear system Ax^m-1 = b, when A is an H-tensor, and a fast and smooth iterative method to solve Ax^m-1 = b with the symmetric coefficient tensor A are given in the second part.

Key words: Tensor splitting, Multilinear system, H-tensor

Eisa Khosravi Dehdezi. On the Multilinear Systems, Part I: Comparison of Splittings of Tensors[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1587-1603.

[1] Bai, X., He, H., Ling, C., Zhou, G.: A nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors. Numer. Algorithms 87(3), 1301-1320 (2021)
[2] Beauwens, R.: Factorization iterative methods, M-operators and H-operators. Numer. Math. 31, 335-357 (1979)
[3] Beik, F.P.A., Najafi-Kalyani, M.: Preconditioned iterative methods for multi-linear systems based on the majorization matrix. Linear Multilinear Algebra 70(20), 5827-5846 (2022)
[4] Bozorgmanesh, H., Hajarian, M.: Solving tensor E-eigenvalue problem faster. Appl. Math. Lett. 100, 106020 (2020)
[5] Bru, R., Corral, C., Gimenez, I., Mas, J.: Classes of general H-matrices. Linear Algebra Appl. 429(10), 2358-2366 (2008)
[6] Bu, C., Zhang, X., Zhou, J., Wang, W., Wei, W.: The inverse, rank, and product of tensors. Linear Algebra Appl. 446, 269-280 (2014)
[7] Che, M., Qi, L., Wei, L.: Positive-definite tensors to nonlinear complementarity problems. J. Optim. Theory Appl. 168, 475-487 (2016)
[8] Che, M., Wei, Y.: Theory and Computation of Complex Tensors and Its Applications. Springer, Singapore (2020)
[9] Chen, Y., Li, C.: A new preconditioned AOR method for solving multi-linear systems. Linear Multilinear Algebra 72(9), 1385-1402 (2024)
[10] Csordas, G., Varga, R.S.: Comparisons of regular splittings of matrices. Numer. Math. 44, 23-35 (1984)
[11] Cui, L.B., Li, M.H., Song, Y.: Preconditioned tensor splitting iterations method for solving multi-linear systems. Appl. Math. Lett. 96, 89-94 (2019)
[12] Dehdezi, E.K.: Iterative methods for sparse symmetric multilinear systems. Bull. Iran Math. Soc. 50(3), 1-20 (2024)
[13] Dehdezi, E.K., Karimi, S.: A gradient based iterative method and associated preconditioning technique for solving the large multilinear systems. Calcolo 58(4), 1-19 (2021)
[14] Ding, W., Luo, Z., Qi, L.: P-tensors, {\rm P}_0-tensors, and their applications. Linear Algebra Appl. 555, 336-354 (2018)
[15] Ding, W., Qi, L., Wei, Y.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439(10), 3264-3278 (2013)
[16] Ding, W., Wei, Y.: Solving multi-linear system with M-tensors. J. Sci. Comput. 68, 689-715 (2016)
[17] Du, S., Zhang, L., Chen, C., Qi, L.: Tensor absolute value equations. Sci. Chin. Math. 61, 1695-1710 (2018)
[18] Einstein, A.: The foundation of the general theory of relativity. In: Kox, A.J., Klein, M.J., Schulmann, R. (eds.) The Collected Papers of Albert Einstein, vol. 6, pp. 146-200. Princeton University Press, Princeton (2007)
[19] Frommer, A., Szyld, D.B.: H-splittings and two-stage iterative methods. Numer. Math. 63, 345-356 (1992)
[20] Han, L.: A homotopy method for solving multilinear systems with M-tensors. Appl. Math. Lett. 69, 49-54 (2017)
[21] He, H., Ling, C., Qi, L., Zhou, G.: A globally and quadratically convergent algorithm for solving multilinear systems with M-tensors. J. Sci. Comput. 76(3), 1718-1741 (2018)
[22] Hofbauer, J., Sigmund, K.: Evolutionary game dynamics. Bull. Am. Math. Soc. 40, 479-519 (2003)
[23] Jiang, Z., Li, J.: Solving tensor absolute value equation. Appl. Numer. Math. 170, 255-268 (2021)
[24] Jiang, Z., Li, J.: Slice tensor splitting method for solving tensor equation. Appl. Math. Comput. 463, 128367 (2024). https://doi.org/10.1016/j.amc.2023.128367
[25] Karimi, S., Dehdezi, E.K.: Tensor splitting preconditioners for multilinear systems. J. Math. Model. 12(3), 481-499 (2024)
[26] Kofidis, E., Regalia, P.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863-884 (2002)
[27] Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455-500 (2009)
[28] Li, D., Guan, H.B., Wang, X.Z.: Finding a nonnegative solution to an {{\cal{M}}}-tensor equation (2018). arXiv:1811.11343
[29] Li, D.H., Xie, S., Xu, H.R.: Splitting methods for tensor equations. Numer. Linear Algebra Appl. 24(5), e2102 (2017)
[30] Li, W., Liu, D., Vong, S.W.: Comparison results for splitting iterations for solving multi-linear systems. Appl. Numer. Math. 134, 105-121 (2018)
[31] Li, X., Luo, Z., Chen, Y.: Sparse least squares solutions of multilinear equations. Linear Multilinear Algebra 72(8), 1279-1291 (2024)
[32] Li, X., Ng, M.K.: Solving sparse non-negative tensor equations: algorithms and applications. Front. Math. Chin. 10, 649-680 (2015)
[33] Li, Z., Dai, Y., Gao, H.: Alternating projection method for a class of tensor equations. J. Comput. Appl. Math. 346, 490-504 (2019)
[34] Liang, M., Zheng, B., Zhao, R.: Alternating iterative methods for solving tensor equations with applications. Numer. Algorithms 80(4), 1437-1465 (2019)
[35] Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing. CAMSAP, vol. 05, no. 1, pp. 129-132. IEEE Computer Society Press, Piscataway (2005)
[36] Liu, D., Li, W., Vong, S.W.: The tensor splitting with application to solve multilinear systems. J. Comput. Appl. Math. 330, 75-94 (2018)
[37] Liu, D., Li, W., Vong, S.W.: A new preconditioned SOR method for solving multilinear systems with an M-tensors. Calcolo 57(2), 15 (2020)
[38] Liu, W., Li, W.: On the inverse of a tensor. Linear Algebra Appl. 495, 199-205 (2016)
[39] Lv, C.Q., Ma, C.F.: A Levenberg-Marquardt method for solving semi-symmetric tensor equations. J. Comput. Appl. Math. 332, 13-25 (2018)
[40] Ma, W., Ding, W., Wei, Y.: Noda iteration for computing generalized tensor eigenpairs (2023). arXiv:2303.01327
[41] Marek, I., Szyld, D.B.: Comparison theorems for weak splittings of bounded operators. Numer. Math. 58(1), 387-397 (1990)
[42] Mollahsani, S., Beik, F.P.A.: Absolute value equations with tensor product structure: unique solvability and numerical solution. Appl. Math-Czech. 67(5), 657-674 (2022)
[43] Neumaier, A.: New techniques for the analysis of linear interval equations. Linear Algebra Appl. 58, 273-325 (1984)
[44] Neumaier, A.: On the comparison of H-matrices with M-matrices. Linear Algebra Appl. 83, 135-141 (1986)
[45] Pearson, K.: Essentially positive tensors. Int. J. Algebra 4, 421-427 (2010)
[46] Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302-1324 (2005)
[47] Qi, L., Luo, Z.: Tensor Analysis: Spectral Theory and Special Tensors. SIAM, Philadelphia (2017)
[48] Schröder, J.: Operator Inequalities. Academic Press, New York (1980)
[49] Shao, J.: A general product of tensors with applications. Linear Algebra Appl. 439, 2350-2366 (2013)
[50] Smith, J.M., Price, G.R.: The logic of animal conflict. Nature 246, 15-18 (1973)
[51] Taylor, P.H., Jonker, L.B.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145-156 (1978)
[52] Wang, C., Chen, H., Wang, Y., Yan, H.: An alternating shifted inverse power method for the extremal eigenvalues of fourth-order partially symmetric tensors. Appl. Math. Lett. (2023). https://doi.org/10.1016/j.aml.2023.108601
[53] Wang, X., Che, M., Mo, C., Wei, Y.: Solving the system of nonsingular tensor equations via randomized Kaczmarz-like method. J. Comput. Appl. Math. (2023). https://doi.org/10.1016/j.cam.2022.114856
[54] Wang, X., Che, M., Wei, Y.: Neural networks based approach solving multi-linear systems with M-tensors. Neurocomputing 351, 33-42 (2019)
[55] Wang, X., Che, M., Wei, Y.: Preconditioned tensor splitting AOR iterative methods for H-tensor equations. Numer. Linear Algebra Appl. 27(6), e2329 (2020)
[56] Xie, Z., Jin, X.Q., Wei, Y.: Tensor methods for solving symmetric M-tensor systems. J. Sci. Comput. 74(1), 412-425 (2018)
[57] Yang, Y., Yang, Q.: Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31(5), 2517-2530 (2010)
[58] Zhang, L., Qi, L., Zhou, G.: M-tensors and some applications. SIAM J. Matrix Anal. Appl. 35(2), 437-452 (2014)

On the Multilinear Systems, Part I: Comparison of Splittings of Tensors

On the Multilinear Systems, Part I: Comparison of Splittings of Tensors

PDF

可视化

摘要/Abstract

引用本文

使用本文

相关文章 5

编辑推荐

Metrics

本文评价

[1]	Guangbin Wang, Fuping Tan. Some Notes on Fixed Point Iteration Methods for Solving Tensor Absolute Value Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1420-1426.
[2]	Dongdong Liu, Ting Hu, Xifu Liu. Restarted Nonnegativity Preserving Tensor Splitting Methods via Relaxed Anderson Acceleration for Solving Multilinear Systems[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 2061-2079.
[3]	Wenbin Gong, Yan Li, Yaqiang Wang. An Improved Iterative Algorithm for Identifying Strong H-Tensors[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1598-1614.
[4]	Zhengbang Cao, Pengpeng Xie. Perturbation Analysis for t-Product-Based Tensor Inverse, Moore-Penrose Inverse and Tensor System[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1441-1456.
[5]	Guangbin Wang, Fuping Tan. Some Criteria for H-Tensors[J]. Communications on Applied Mathematics and Computation, 2020, 2(4): 641-651.