An Improved Iterative Algorithm for Identifying Strong

Abstract

Strong

\mathcal{H}

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-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

\mathcal{H}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {H}}$$\end{document}

-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.

\mathcal{H}

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-tensors')">Strong

\mathcal{H}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {H}}$$\end{document}

-tensors / Iterative algorithm / Positive diagonal matrix / Symmetric tensors

References

[1] AndersonBD, BoseNK. Output feedback stabilization and related problems solution via decision methods. IEEE Trans. Automat. Control., 1975, 20153-66.

[2] BoseNK, KamatPS. Algorithm for stability test of multidimensional filters. IEEE Trans. Acoust. Speech Signal Process., 1974, 225307-314.

[3] BoseNK, NewcombRW. Tellegen’s theorem and multivariable realizability theory. Int. J. Electron., 1974, 363417-425.

[4] ChangKC, PearsonK, ZhangT. Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci., 2008, 6: 507-520.

[5] CuiJJ, PengGH, LuQ, HuangZG. New iterative criteria for strong H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors and an application. Jinequal Appl., 2017, 49: 2-14

[6] DingWY, QiLQ, WeiYM. M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{M} }$$\end{document}-tensors and nonsingular M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{M} }$$\end{document}-tensors. Linear Algebra Appl., 2013, 439: 3264-3278.

[7] DingWY, WeiYM. Solving multilinear systems with M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{M} }$$\end{document}-tensors. J. Sci. Comput., 2016, 682689-715.

[8] GongWB, WangYQ. Some new criteria for judging H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensor and their applications. AIMS Mathematics., 2023, 8: 7606-7617.

[9] KannanMR, Shaked-MondererN, BermanA. Some properties of strong H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors and general H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors. Linear Algebra Appl., 2015, 476: 42-55.

[10] Li, C.Q., Li, Y.T.: Double B-tensors and quasi-double B-tensors. Linear Algebra Appl. 466, 343–356 (2015)

[11] LiCQ, LiYT, XuK. New eigenvalue inclusion sets for tensor. Numer. Algebra App., 2014, 21: 39-50.

[12] LiCQ, WangF, ZhaoJX, LiYT. Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math., 2014, 255: 1-14.

[13] LiG, ZhangYC, FengY. Criteria for nonsingular H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors. Adv. Appl. Math., 2018, 3: 66-71

[14] LiYT, LiuQL, QiLQ. Programmable criteria for strong H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors. Numer. Algorithms., 2017, 74: 199-211.

[15] LiuQL, LiCQ, LiYT, et al. . On the iterative criterion for strong H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors. Comput. Appl. Math., 2017, 36: 1623-1635.

[16] LiuQL, ZhaoJX, LiCQ, et al. . An iterative algorithm based on strong H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors for identifying positive definiteness of irreducible homogeneous polynomial forms. J. Comput. Appl. Math., 2020, 369: 112581.

[17] QiLQ. Eigenvalues of a real supersymmetric tensor. J. Symb. Comput., 2005, 40: 1302-1324.

[18] QiLQ, SongYS. An even order symmetric B-tensor is positive definite. Linear Algebra Appl., 2014, 457: 303-312.

[19] QiLQ, YuGH, XuY. Nonnegative diffusion orientation distribution function. J. Math. Imaging Vis., 2013, 45: 103-113.

[20] ShaoJY. A general product of tensors with applications. Linear Algebra Appl., 2013, 439: 2350-2366.

[21] SigmundK, HofbauerJ. Evolutionary game dynamics. B. Am. Math. Soc., 2003, 40: 479-519.

[22] SunDS, BaiDJ. New criteria-based H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors for identifying the positive definiteness of multivariate homogeneous forms. Open Math., 2021, 19: 551-561.

[23] VidyasagarM, DesoerCA. Nonlinear Systems Analysis. TSMC., 1978, 428537-538

[24] WangF, SunDS. New criteria for H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors and an application. J. Inequal. Appl., 2016, 96: 1-11

[25] WangF, SunDS, XuYM. Some criteria for identifying H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors and its applications. Calcolo., 2019, 56: 2-18.

[26] WangF, SunDS, ZhaoJX, LiCQ. New practical criteria for H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors and its application. Linear Multilinear A., 2017, 65: 269-283.

[27] WangG, TanF. Some Criteria for H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-Tensors. CAMC., 2020, 2: 1-11

[28] WangXZ, CheML, WeiYM. Existence and uniqueness of positive solution for H+\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }^+$$\end{document}-tensor equations. Appl. Math. Lett., 2019, 98: 191-198.

[29] Wang, Y.J., Zhou, G.L., Caccetta, L.: Nonsingular H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-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] XuYY, ZhaoRJ, ZhengB. Some criteria for identifying strong H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors. Numer. Algorithms., 2019, 80: 1121-1141.

[32] YangYN, YangQZ. Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM. J. Matrix Anal. Appl., 2010, 31: 2517-2530.

[33] ZhangKL, WangYJ. An H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. J. Comput. Appl. Math., 2016, 305: 1-10.

[34] ZhaoRJ, GaoL, LiuQL, LiYT. Criterions for identifying H\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal{H} }$$\end{document}-tensors. Front. Math. China, 2016, 1111660-678

Funding

Natural Science Basic Research Program of Shaanxi, China(2020JM-622)

RIGHTS & PERMISSIONS

Shanghai University