Feasible Newton methods for symmetric tensor <i>Z</i>-eigenvalue problems

Xu, Jie-Feng; Li, Donghui; Bai, Xueli

doi:10.1080/10556788.2022.2142586

Cited by 1 publication

(1 citation statement)

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“…2 Since Qi 3 and Lim 4 established the tensor eigenvalues and expanded the definition of matrix eigenvalues in 2005, the study of the eigenproblems of symmetric tensors has attracted a considerable deal of mathematical attention. [5][6][7][8][9] The M-eigenvalue intervals of fourth-order structured partly symmetric tensors (PSTs) were constructed by Wei et al in order to give some necessary checkable criteria for the M-positive definiteness and strong ellipticity. 10 However, considerably less focused research has been done on the generalized eigenproblems since Chang, Pearson, and Zhang 6,11 introduced the tensor generalized eigenvalues problem.…”

Section: Introductionmentioning

confidence: 99%

Normalized Newton method to solve generalized tensor eigenvalue problems

Pakmanesh,

Afshin,

Hajarian

2024

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The problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z‐eigenpairs under the same ‐Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings.

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