Fourth-order tensor Riccati equations with the Einstein product

In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the -tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati...

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Bibliographic Details
Published inLinear & multilinear algebra Vol. 70; no. 10; pp. 1831 - 1853
Main Authors Miao, Yun, Wei, Yimin, Chen, Zhen
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 03.07.2022
Taylor & Francis Ltd
Subjects
Algorithms
condition number
Einstein product
Mathematical analysis
Multiplication
Permutations
Perturbation methods
Riccati equation
spectrum
tensor
tensor blocks
tensor Riccati equation
Tensors
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ISSN0308-1087
1563-5139
DOI10.1080/03081087.2020.1777248

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Summary:In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the -tensor, the spectrum and the block tensor multiplication. We also present an expression of the fourth-order permutation tensors and generalize the Riccati equation from the matrix to the tensor case. The existence and uniqueness of the solution for the fourth-order -tensor Riccati equation are studied. Furthermore, we provide a perturbation analysis of the mixed and componentwise condition numbers for the -tensor Riccati equation. An algorithm for solving the equation is proposed. Numerical examples are presented to show the effectiveness of our algorithm.
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ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2020.1777248