New Iterative Scheme-Based Strong ��-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms

doi:10.1007/s42967-024-00451-5

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (2): 472-489.doi: 10.1007/s42967-024-00451-5

New Iterative Scheme-Based Strong ��-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms

Pengcheng Zhao, Deshu Sun, Lanlan Liu

College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, Guizhou, China

收稿日期:2024-05-04 修回日期:2024-07-29 出版日期:2026-04-07 发布日期:2026-04-07
通讯作者: Pengcheng Zhao,E-mail:zpc13142022@163.com E-mail:zpc13142022@163.com
基金资助:
This research is supported by Guizhou Provincial Science and Technology Projects, China (20191161), the Natural Science Research Project of Department of Education of Guizhou Province, China (QJJ2023012), and the Research Foundation of Guizhou Minzu University, China (GZMUZK[2023]YB10).

New Iterative Scheme-Based Strong ��-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms

Pengcheng Zhao, Deshu Sun, Lanlan Liu

College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, Guizhou, China

Received:2024-05-04 Revised:2024-07-29 Online:2026-04-07 Published:2026-04-07
Contact: Pengcheng Zhao,E-mail:zpc13142022@163.com E-mail:zpc13142022@163.com
Supported by:
This research is supported by Guizhou Provincial Science and Technology Projects, China (20191161), the Natural Science Research Project of Department of Education of Guizhou Province, China (QJJ2023012), and the Research Foundation of Guizhou Minzu University, China (GZMUZK[2023]YB10).

摘要/Abstract

摘要： The positive definite homogeneous multivariate form plays an important role in the automatic control and medical imaging, and the definiteness of this form can be identified by a special structure tensor. In this paper, we first state the equivalence between the positive definite multivariate form and the corresponding tensor and account for the links between the positive definite tensor with a strong \begin{document}$ \mathcal {H} $\end{document}-tensor. Then, based on diagonal dominance, some criteria are presented to test strong \begin{document}$ \mathcal {H} $\end{document}-tensors. Furthermore, with these relations, we provide an iterative scheme to identify the positive definite multivariate homogeneous form and prove its theoretically valid. The advantages of the results obtained are illustrated by numerical examples.

关键词: Homogeneous multivariate form, Positive definiteness, \begin{document}$ \mathcal {H} $\end{document}-tensor, Diagonal dominance, Irreducible, Non-zero element chain, Iterate scheme

Abstract: The positive definite homogeneous multivariate form plays an important role in the automatic control and medical imaging, and the definiteness of this form can be identified by a special structure tensor. In this paper, we first state the equivalence between the positive definite multivariate form and the corresponding tensor and account for the links between the positive definite tensor with a strong \begin{document}$ \mathcal {H} $\end{document}-tensor. Then, based on diagonal dominance, some criteria are presented to test strong \begin{document}$ \mathcal {H} $\end{document}-tensors. Furthermore, with these relations, we provide an iterative scheme to identify the positive definite multivariate homogeneous form and prove its theoretically valid. The advantages of the results obtained are illustrated by numerical examples.

Key words: Homogeneous multivariate form, Positive definiteness, \begin{document}$ \mathcal {H} $\end{document}-tensor, Diagonal dominance, Irreducible, Non-zero element chain, Iterate scheme

中图分类号:

Pengcheng Zhao, Deshu Sun, Lanlan Liu. New Iterative Scheme-Based Strong ��-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 472-489.

参考文献

[1] Ding, W.Y., Qi, L.Q., Wei, Y.M.:

\begin{document}$ \cal{M} $\end{document}

-tensors and nonsingular

\begin{document}$ \cal{M} $\end{document}

-tensors. Linear Algebra Appl. 439, 3264–3278 (2013)
[2] Huang, B.H., Ma, C.F.: Iterative criteria for identifying strong

\begin{document}$ \cal{H} $\end{document}

-tensors. J. Comput. Appl. Math. 352, 93–109 (2019)
[3] Kannan, M.R., Monderer, N.S., Berman, A.: Some properties of strong

\begin{document}$ \cal{H} $\end{document}

-tensors and general

\begin{document}$ \cal{H} $\end{document}

-tensors. Linear Algebra Appl. 476, 42–55 (2015)
[4] Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23(3), 863–884 (2002)
[5] Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32(4), 1095–1124 (2011)
[6] Li, C.Q., Li, Y.T.: Double

\begin{document}$ B $\end{document}

-tensors and quasi-double

\begin{document}$ B $\end{document}

-tensors. Linear Algebra Appl. 466, 343–356 (2015)
[7] Li, C.Q., Qi, L.Q., Li, Y.T.:

\begin{document}$ MB $\end{document}

-tensors and

\begin{document}$ MB_{0} $\end{document}

-tensors. Linear Algebra Appl. 484, 141–153 (2015)
[8] Li, C.Q., Wang, F., Zhao, J.X., Zhu, Y., Li, Y.T.: Criterions for the positive definiteness of real supersymmetric tensors. J. Comput. Appl. Math. 255, 1–14 (2014)
[9] Li, Y.T., Liu, Q.L., Qi, L.Q.: Programmable criteria for strong

\begin{document}$ \cal{H} $\end{document}

-tensors. Numer. Algorithms 74(1), 1–12 (2017)
[10] Lim, L.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP-05: Proceeding of the IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, pp. 129–132 (2005)
[11] Liu, Q.L., Li, C.Q., Li, Y.T.: On the iterative criterion for strong

\begin{document}$ \cal{H} $\end{document}

-tensors. Comput. Appl. Math. 36(4), 1623–1635 (2017)
[12] Liu, Q.L., Zhao, J.X., Li, C.Q., Li, Y.T.: An iterative algorithm based on strong

\begin{document}$ \cal{H} $\end{document}

-tensors for identifying positive definiteness of irreducible homogeneous polynomial forms. J. Comput. Appl. Math. 369, 112581 (2020)
[13] Ni, G.Y., Qi, L.Q., Wang, F., Wang, Y.J.: The degree of the E-characteristic polynomial of an even order tensor. J. Math. Anal. Appl. 329, 1218–1229 (2007)
[14] Ni, Q., Qi, L.Q., Wang, F.: An eigenvalue method for the positive definiteness identification problem. IEEE Trans. Autom. Control. 53, 1096–1107 (2008)
[15] Qi, L.Q.: Eigenvalues of a real supersymetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)
[16] Qi, L.Q., Song, Y.S.: An even order symmetric

\begin{document}$ B $\end{document}

-tensor is positive definite. Linear Algebra Appl. 457, 303–312 (2014)
[17] Qi, L.Q., Wang, F., Wang, Y.J.:

\begin{document}$ Z $\end{document}

-eigenvalue methods for a global polynomial optimization problem. Math. Program. 118, 301–316 (2009)
[18] Wang, F., Sun, D.S.: New criteria for

\begin{document}$ \cal{H} $\end{document}

-tensors and an application. J. Inequal. Appl. 2016(1), 96 (2016)
[19] Wang, F., Sun, D.S., Xu, Y.M.: Some criteria for identifying

\begin{document}$ \cal{H} $\end{document}

-tensors and its applications. Calcolo 56, 2–19 (2019)
[20] Wang, F., Sun, D.S., Zhao, J.X., Li, C.Q.: New practical criteria for

\begin{document}$ \cal{H} $\end{document}

-tensors and its application. Linear Multilinear Appl. 65(2), 269–283 (2017)
[21] Wang, Y., Zhou, G., Caccetta, L.: Nonsingular

\begin{document}$ \cal{H} $\end{document}

-tensor and its criteria. J. Ind. Manag. Optim. 12(4), 1173–1186 (2016)
[22] Wang, Y.J., Zhang, K.L., Sun, H.C.: Criteria for strong

\begin{document}$ \cal{H} $\end{document}

-tensors. Front. Math. China 11(3), 577–592 (2016)
[23] Yang, Y.N., Yang, Q.Z.: Further results for Perron-Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31, 2517–2530 (2010)
[24] Zhang, K.L., Wang, Y.J.: An

\begin{document}$ \cal{H} $\end{document}

-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms. J. Comput. Appl. Math. 305, 1–10 (2016)
[25] Zhang, L.P., Qi, L.Q., Zhou, G.L.:

\begin{document}$ \cal{M} $\end{document}

-tensors and some applications. SIAM J. Matrix Anal. Appl. 32, 437–452 (2014)

New Iterative Scheme-Based Strong ��-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms

New Iterative Scheme-Based Strong ��-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

编辑推荐

Metrics

本文评价

[1]	Liqun Qi, Chunfeng Cui. Dual Markov Chain and Dual Number Matrices with Nonnegative Standard Parts[J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2442-2461.
[2]	Sheng-Zhong Song, Zheng-Da Huang, Bo-Han Zhang. An Improved SSOR-Like Preconditioner for the Non-Hermitian-Positive Definite Linear System with a Dominant Skew-Hermitian Part[J]. Communications on Applied Mathematics and Computation, 2025, 7(5): 2080-2096.