An Improved Iterative Algorithm for Identifying Strong H-Tensors

doi:10.1007/s42967-023-00362-x

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1598-1614.doi: 10.1007/s42967-023-00362-x

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An Improved Iterative Algorithm for Identifying Strong H-Tensors

Wenbin Gong, Yan Li, Yaqiang Wang

School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, 721013, Shaanxi, China

Received:2023-07-08 Revised:2023-11-02 Accepted:2023-12-03 Online:2024-04-02 Published:2024-04-02
Supported by:
The authors are grateful to the referee for carefully reading of the paper and valuable suggestions and comments.

Abstract

Abstract: Strong H-tensors play a significant role in identifying the positive definiteness of an even-order real symmetric tensor. In this paper, first, an improved iterative algorithm is proposed to determine whether a given tensor is a strong H-tensor, and the validity of the iterative algorithm is proved theoretically. Second, the iterative algorithm is employed to identify the positive definiteness of an even-order real symmetric tensor. Finally, numerical examples are presented to illustrate the advantages of the proposed algorithm.

Key words: Strong H-tensors, Iterative algorithm, Positive diagonal matrix, Symmetric tensors

CLC Number:

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.

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References

[1] Anderson, B.D., Bose, N.K.: Output feedback stabilization and related problems solution via decision methods. IEEE Trans. Automat. Control. 20(1), 53-66 (1975)
[2] Bose, N.K., Kamat, P.S.: Algorithm for stability test of multidimensional filters. IEEE Trans. Acoust. Speech Signal Process. 22(5), 307-314 (1974)
[3] Bose, N.K., Newcomb, R.W.: Tellegen’s theorem and multivariable realizability theory. Int. J. Electron. 36(3), 417-425 (1974)
[4] Chang, K.C., Pearson, K., Zhang, T.: Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6, 507-520 (2008)
[5] Cui, J.J., Peng, G.H., Lu, Q., Huang, Z.G.: New iterative criteria for strong H-tensors and an application. Jinequal Appl. 49, 2-14 (2017)
[6] Ding, W.Y., Qi, L.Q., Wei, Y.M.: M-tensors and nonsingular M-tensors. Linear Algebra Appl. 439, 3264-3278 (2013)
[7] Ding, W.Y., Wei, Y.M.: Solving multilinear systems with M-tensors. J. Sci. Comput. 68(2), 689-715 (2016)
[8] Gong, W.B., Wang, Y.Q.: Some new criteria for judging H-tensor and their applications. AIMS Mathematics. 8, 7606-7617 (2023)
[9] Kannan, M.R., Shaked-Monderer, N., Berman, A.: Some properties of strong H-tensors and general H-tensors. Linear Algebra Appl. 476, 42-55 (2015)
[10] Li, C.Q., Li, Y.T.: Double B-tensors and quasi-double B-tensors. Linear Algebra Appl. 466, 343-356 (2015)
[11] Li, C.Q., Li, Y.T., Xu, K.: New eigenvalue inclusion sets for tensor. Numer. Algebra App. 21, 39-50 (2014)
[12] Li, C.Q., Wang, F., Zhao, J.X., Li, Y.T.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1-14 (2014)
[13] Li, G., Zhang, Y.C., Feng, Y.: Criteria for nonsingular H-tensors. Adv. Appl. Math. 3, 66-71 (2018)
[14] Li, Y.T., Liu, Q.L., Qi, L.Q.: Programmable criteria for strong H-tensors. Numer. Algorithms 74, 199-211 (2017)
[15] Liu, Q.L., Li, C.Q., Li, Y.T.: On the iterative criterion for strong H-tensors. Comput. Appl. Math. 36, 1623-1635 (2017)
[16] Liu, Q.L., Zhao, J.X., Li, C.Q., Li, Y.T.: An iterative algorithm based on strong H-tensors for identifying positive definiteness of irreducible homogeneous polynomial forms. J. Comput. Appl. Math. 369, 112581 (2020)
[17] Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302-1324 (2005)
[18] Qi, L.Q., Song, Y.S.: An even order symmetric B tensor is positive definite. Linear Algebra Appl. 457, 303-312 (2014)
[19] Qi, L.Q., Yu, G.H., Xu, Y.: Nonnegative diffusion orientation distribution function. J. Math. Imaging Vis. 45, 103-113 (2013)
[20] Shao, J.Y.: A general product of tensors with applications. Linear Algebra Appl. 439, 2350-2366 (2013)
[21] Sigmund, K., Hofbauer, J.: Evolutionary game dynamics. B. Am. Math. Soc. 40, 479-519 (2003)
[22] Sun, D.S., Bai, D.J.: New criteria-based H-tensors for identifying the positive definiteness of multivariate homogeneous forms. Open Math. 19, 551-561 (2021)
[23] Vidyasagar, M., Desoer, C.A.: Nonlinear systems analysis. In: IEEE Transactions on Systems, Man, and Cybernetics, vol. 10, no. 8, pp. 537-538. IEEE (1980)
[24] Wang, F., Sun, D.S.: New criteria forH-tensors and an application. J. Inequal. Appl. 96, 1-11 (2016)
[25] Wang, F., Sun, D.S., Xu, Y.M.: Some criteria for identifying H-tensors and its applications. Calcolo. 56, 2-18 (2019)
[26] Wang, F., Sun, D.S., Zhao, J.X., Li, C.Q.: New practical criteria for H-tensors and its application. Linear Multilinear A. 65, 269-283 (2017)
[27] Wang, G., Tan, F.: Some criteria for H-tensors. Commun. Appl. Math. Comput. 2, 641-651 (2020)
[28] Wang, X.Z., Che, M.L., Wei, Y.M.: Existence and uniqueness of positive solution for H-tensor equations. Appl. Math. Lett. 98, 191-198 (2019)
[29] Wang, Y.J., Zhou, G.L., Caccetta, L.: Nonsingular H-tensor and its criteria. J. Ind. Manag. Optim. 2016, 1173-1186
[30] Wei, Y.M., Ding, W.Y.: Theory and Computation of Tensors: Multi-dimensional Arrays. Elsevier Science, San Diego (2016)
[31] Xu, Y.Y., Zhao, R.J., Zheng, B.: Some criteria for identifying strong H-tensors. Numer. Algorithms 80, 1121-1141 (2019)
[32] Yang, Y.N., Yang, Q.Z.: Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31, 2517-2530 (2010)
[33] Zhang, K.L., Wang, Y.J.: An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. J. Comput. Appl. Math. 305, 1-10 (2016)
[34] Zhao, R.J., Gao, L., Liu, Q.L., Li, Y.T.: Criterions for identifying H-tensors. Front. Math. China 11(11), 660-678 (2016)

An Improved Iterative Algorithm for Identifying Strong H-Tensors

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