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<p>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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-tensor. Then, based on diagonal dominance, some criteria are presented to test strong <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>-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.</p>

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New Iterative Scheme-Based Strong \(\mathcal {H}\)-Tensors for Identifying the Positive Definiteness of Multivariate Homogeneous Forms

  • 期刊论文

  • 发表日期：

    2025年1月4日

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  • Pengcheng Zhao,
  • Deshu Sun,
  • Lanlan Liu

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 \(\mathcal {H}\) H -tensor. Then, based on diagonal dominance, some criteria are presented to test strong \(\mathcal {H}\) H -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.