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Accelerated dynamical approaches for finding the unique positive solution of \(\mathcal {K}\mathcal {S}\)-tensor equations

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Abstract

A new class of tensors called \(\mathcal {K}\mathcal {S}\)-tensors, which is a subset of non-singular \(\mathcal {P}\)-tensors and generalization of \({\mathscr{H}}^{+}\)-tensors, is proposed. It is proved that the system of \(\mathcal {K}\mathcal {S}\)-tensor equations always has a unique positive solution for any positive right-hand side by proposing a positive increasing map. Two approaches based on dynamical system are presented to find the unique positive solution. The theoretical analysis results show that the convergence of the proposed models is guaranteed, and numerical examples further illustrate that the given models are feasible and effective in finding the positive solution of \(\mathcal {K}\mathcal {S}\)-tensor equations.

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Funding

Maolin Che is supported by the Fundamental Research Funds for the Central Universities under grant JBK1801058 and the National Natural Science Foundation of China under grant 11901471. Yimin Wei is supported by the National Natural Science Foundation of China under grant 11771099 and Innovation Program of Shanghai Municipal Education Commission. Xuezhong Wang is supported by the National Natural Science Foundation of China under grant 11771099, 12061032; Start-Up Fund of Doctoral Research, Hexi University; Innovative Ability Promotion Program of Gansu Province under grant 2019B-146. Changxin Mo is supported by the National Natural Science Foundation of China under grant 11771099.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hexi University, Zhangye, 734000, People’s Republic of China

    Xuezhong Wang

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Changxin Mo

  3. School of Economic Mathematics, Southwest University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Maolin Che

  4. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

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  1. Xuezhong Wang
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Correspondence to Yimin Wei.

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Wang, X., Mo, C., Che, M. et al. Accelerated dynamical approaches for finding the unique positive solution of \(\mathcal {K}\mathcal {S}\)-tensor equations. Numer Algor 88, 1787–1810 (2021). https://doi.org/10.1007/s11075-021-01095-9

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