An iterative scheme for testing the positive definiteness of multivariate homogeneous forms^*

* The authors confirm contribution to the paper as follows: study conception and design: Kaili Zhang, Haibin Zhang; analysis and interpretation of results: Kaili Zhang, Haibin Zhang and Pengfei Zhao; numerical experiment: Pengfei Zhao, Kaili Zhang; draft manuscript preparation: Kaili Zhang, Xueyong Wang. All authors reviewed the results and approved the final version of the manuscript.

ABSTRACT

A positive-definite homogeneous multivariate form plays a critical role in the medical imaging and automatic control, 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 H-tensor. Then based on weak reducibility, some criteria were provided to identify strong H-tensors. Furthermore, with these relations, we establish an iterative scheme to identify the positive-definite multivariate homogeneous form and prove it is theoretically valid. Numerical experiments were given to illustrate the practicality of the scheme.

KEYWORDS:

• Positive definiteness
• strong H-tensor
• homogeneous multivariate form
• weakly reducible
• iterative scheme

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A18
• 15A69
• 12E05
• 65F10

Acknowledgments

The authors are very grateful to Beijing Innovation Center for Engineering Science and Advanced Technology, Peking University and Beijing University of Technology for their joint project support. And we would like to thank the anonymous reviewers and the editors for their very careful and helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

* The authors confirm contribution to the paper as follows: study conception and design: Kaili Zhang, Haibin Zhang; analysis and interpretation of results: Kaili Zhang, Haibin Zhang and Pengfei Zhao; numerical experiment: Pengfei Zhao, Kaili Zhang; draft manuscript preparation: Kaili Zhang, Xueyong Wang. All authors reviewed the results and approved the final version of the manuscript.

Additional information

Funding

This work was supported by Natural Science Foundation of China [grant number 11771003, 11801309, 11671228].