Skip to main content
Springer Nature Link
Log in

Shifted Inverse Power Method for Computing the Smallest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor

  • Regular Paper
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The strong ellipticity condition (abbr. SE-condition) of the displacement equations of equilibrium for general nonlinearly elastic materials plays an important role in nonlinear elasticity and materials. Qi et al. (Front Math China 4(2):349–364, 2009) pointed out that the SE-condition of the displacement equations of equilibrium can be equivalently transformed into the SE-condition of a fourth-order real partially symmetric tensor \({\mathcal {A}}\), and that the SE-condition of \({\mathcal {A}}\) holds if and only if the smallest M-eigenvalue of \({\mathcal {A}}\) is positive. In order to judge the strong ellipticity of \({\mathcal {A}}\), we propose a shifted inverse power method for computing the smallest M-eigenvalue of \({\mathcal {A}}\) and give its convergence analysis. And then, we borrow and fine-tune an existing initialization strategy to make the sequence generated by the shifted inverse power method rapidly converge to a good approximation of the smallest M-eigenvalue of \({\mathcal {A}}\). Finally, we by numerical examples illustrate the effectiveness of the proposed method in computing the smallest M-eigenvalue of \({\mathcal {A}}\) and judging the SE-condition of the displacement equations of equilibrium.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Algorithm 2
Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data Availability

We do not analyze or generate any datasets, because our work proceeds within a theoretical and mathematical approach.

References

  1. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont, Massachusetts (1999)

    Google Scholar 

  2. Che, H., Chen, H., Wang, Y.: M-positive semi-definiteness and M-positive definiteness of fourth-order partially symmetric Cauchy tensors. J. Inequal. Appl. 2019, 32 (2019)

    Article  MathSciNet  Google Scholar 

  3. Che, H., Chen, H., Wang, Y.: On the M-eigenvalue estimation of fourth-order partially symmetric tensors. J. Ind. Manag. Optim. 16(1), 309–324 (2020)

    Article  MathSciNet  Google Scholar 

  4. Che, H., Chen, H., Xu, N., Zhu, Q.: New lower bounds for the minimum M-eigenvalue of elasticity M-tensors and applications. J. Inequal. Appl. 2020, 188 (2020)

    Article  MathSciNet  Google Scholar 

  5. Che, H., Chen, H., Zhou, G.: New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors. J. Ind. Manag. Optim. 17(6), 3685–3694 (2021)

    Article  MathSciNet  Google Scholar 

  6. Chen, H., He, H., Wang, Y., Zhou, G.: An efficient alternating minimization method for fourth degree polynomial optimization. J. Global Optim. 82, 83–103 (2022)

    Article  MathSciNet  Google Scholar 

  7. Chirita, S., Danescu, A., Ciarletta, M.: On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87, 1–27 (2007)

    Article  MathSciNet  Google Scholar 

  8. Ciarlet, P.G.: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity. Society for Industrial and Applied Mathematics, Philadelphia (2022)

  9. Ding, W., Liu, J., Qi, L., Yan, H.: Elasticity M-tensors and the strong ellipticity condition. Appl. Math. Comput. 373, 124982 (2020)

    Article  MathSciNet  Google Scholar 

  10. Fichera, G.: Existence theorems in elasticity. In: Truesdell, C. (eds.) Linear theories of elasticity and thermoelasticity. Springer, Berlin (1973). https://doi.org/10.1007/978-3-662-39776-3_3

  11. Han, D., Dai, H.H., Qi, L.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97, 1–13 (2009)

    Article  MathSciNet  Google Scholar 

  12. Han, D., Qi, L.: A successive approximation method for quantum separability. Front. Math. China 8(6), 1275–1293 (2013)

    Article  MathSciNet  Google Scholar 

  13. He, J., Li, C., Wei, Y.: M-eigenvalue intervals and checkable sufficient conditions for the strong ellipticity. Appl. Math. Lett. 102, 106137 (2020)

    Article  MathSciNet  Google Scholar 

  14. He, J., Liu, Y., Xu, G.: New S-type inclusion theorems for the M-eigenvalues of a 4th-order partially symmetric tensor with applications. Appl. Math. Comput. 398, 125992 (2021)

    Article  MathSciNet  Google Scholar 

  15. He, J., Liu, Y., Xu, G.: New M-eigenvalue inclusion sets for fourth-order partially symmetric tensors with applications. Bull. Malays. Math. Sci. Soc. 44(6), 3929–3947 (2021)

    Article  MathSciNet  Google Scholar 

  16. He, J., Xu, G., Liu, Y.: Some inequalities for the minimum M-eigenvalue of elasticity M-tensors. J. Ind. Manag. Optim. 16(6), 3035–3045 (2020)

    Article  MathSciNet  Google Scholar 

  17. Huang, Z.H., Li, X., Wang, Y.: Bi-block positive semidefiniteness of bi-block symmetric tensors. Front. Math. China 16(1), 141–169 (2021)

    Article  MathSciNet  Google Scholar 

  18. Knowles, J.K., Sternberg, E.: On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elast. 5(3–4), 341–361 (1975)

    Article  MathSciNet  Google Scholar 

  19. Knowles, J.K., Sternberg, E.: On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Ration. Mech. Anal. 63(4), 321–336 (1976)

    Article  MathSciNet  Google Scholar 

  20. Li, M., Chen, H., Zhou, G.: Fourth-order partially symmetric tensors: theory and algorithm. Pac. J. Optim. 18(1), 233–264 (2022)

    MathSciNet  CAS  Google Scholar 

  21. Li, S., Chen, Z., Liu, Q., Lu, L.: Bounds of M-eigenvalues and strong ellipticity conditions for elasticity tensors. Linear Multilinear Algebra 70(19), 4544–4557 (2022)

    Article  MathSciNet  Google Scholar 

  22. Li, S., Li, C., Li, Y.: M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor. J. Comput. Appl. Math. 356, 391–401 (2019)

    Article  MathSciNet  Google Scholar 

  23. Li, S., Li, Y.: Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors. Appl. Math. Comput. 403, 126134 (2021)

    Article  MathSciNet  Google Scholar 

  24. Li, S., Li, Y.: Bounds for the M-spectral radius of a fourth-order partially symmetric tensor. J. Inequal. Appl. 2018, 18 (2018)

    Article  MathSciNet  PubMed  PubMed Central  Google Scholar 

  25. Li, S., Li, Y.: Checkable criteria for the M-positive definiteness of fourth-order partially symmetric tensors. Bull. Iran. Math. Soc. 46, 1455–1463 (2020)

    Article  MathSciNet  Google Scholar 

  26. Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20(3), 1286–1310 (2009)

    Article  MathSciNet  Google Scholar 

  27. Liu, K., Che, H., Chen, H., Li, M.: Parameterized S-type M-eigenvalue inclusion intervals for fourth-order partially symmetric tensors and its applications. J. Ind. Manag. Optim. 19(5), 3060–3074 (2023)

    Article  MathSciNet  Google Scholar 

  28. Liu, L., Wang, G., Zhang, Z., Liu, S.: Bound estimations of bi-block M-eigenvalues for bi-block symmetric tensors. J. Ind. Manag. Optim. 19(8), 6244–6263 (2023)

    Article  MathSciNet  Google Scholar 

  29. Qi, L., Chen, H., Chen, Y.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)

    Book  Google Scholar 

  30. Qi, L., Dai, H.H., Han, D.: Conditions for strong ellipticity and M-eigenvalues. Front. Math. China 4(2), 349–364 (2009)

    Article  MathSciNet  Google Scholar 

  31. Rosakis, P.: Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Rational Mech. Anal. 109, 1–37 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  32. Sang, C., Zhao, J.: Criteria for the strong ellipticity condition of a partially symmetric tensor. Acta Appl. Math. 180, 10 (2022)

    Article  MathSciNet  Google Scholar 

  33. Sheng, Z., Ni, Q.: Computing tensor Z-eigenvalues via shifted inverse power method. J. Comput. Appl. Math. 398, 113717 (2021)

    Article  MathSciNet  Google Scholar 

  34. Wang, C., Wang, G., Liu, L.: Sharp bounds on the minimum M-eigenvalue and strong ellipticity condition of elasticity Z-tensors. J. Ind. Manag. Optim. 19(1), 760–772 (2023)

    Article  MathSciNet  Google Scholar 

  35. Wang, G., Sun, L., Liu, L.: M-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order partially symmetric tensors. Complexity 2020, 2474278 (2020)

    Google Scholar 

  36. Wang, G., Sun, L., Wang, X.: Sharp bounds on the minimum M-eigenvalue of elasticity Z-tensors and identifying strong ellipticity. J. Appl. Anal. Comput. 11(4), 2114–2130 (2021)

    MathSciNet  CAS  Google Scholar 

  37. Wang, W., Li, M., Che, H.: A tighter M-eigenvalue localization set for fourth-order partially symmetric tensors. Pac. J. Optim. 16(4), 687–698 (2020)

    MathSciNet  Google Scholar 

  38. Wang, X., Che, M., Wei, Y.: Best rank-one approximation of fourth-order partially symmetric tensors by neural network. Numer. Math. Theor. Meth. Appl. 11(4), 673–700 (2018)

    Article  MathSciNet  Google Scholar 

  39. Wang, Y., Qi, L., Zhang, X.: A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor. Numer. Linear Algebra Appl. 16(7), 589–601 (2009)

    Article  MathSciNet  Google Scholar 

  40. Yao, Y., Wang, G.: Sharp upper bounds on the maximum M-eigenvalue of fourth-order partially symmetric nonnegative tensors. Math. Found. Comput. Sci. 5(1), 33–44 (2022)

    Article  Google Scholar 

  41. Zhang, Y., Sun, L., Wang, G.: Sharp bounds on the minimum M-eigenvalue of elasticity M-tensors. Mathematics 8(2), 250 (2020)

    Article  Google Scholar 

  42. Zhao, J.: Conditions of strong ellipticity and calculations of M-eigenvalues for a partially symmetric tensor. Appl. Math. Comput. 458, 128245 (2023)

    Article  MathSciNet  Google Scholar 

  43. Zhao, J., Luo, Y., Sang, C.: A direct method for calculating M-eigenvalues of an elasticity tensor. Japan J. Indust. Appl. Math. 41, 317–357 (2024). https://doi.org/10.1007/s13160-023-00598-3

    Article  MathSciNet  Google Scholar 

  44. Zubov, L.M., Rudev, A.N.: On necessary and sufficient conditions of strong ellipticity of equilibrium equations for certain classes of anisotropic linearly elastic materials. Z. Angew. Math. Mech. 96(9), 1096–1102 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the anonymous reviewers for their insightful comments and constructive suggestions, which considerably improve our manuscript. The authors also thank Associate Professor Qilong Liu (Guizhou Normal University) for his discussions and suggestions during the revision of the manuscript.

Funding

This work of Jianxing Zhao is supported by Guizhou Provincial Science and Technology Projects (Grant No. QKHJC-ZK[2022]YB215). This work of Caili Sang is supported by Guizhou Provincial Science and Technology Projects (Grant No. QKHJC-ZK[2021]YB013) and Natural Science Research Project of Department of Education of Guizhou Province (Grant Nos. QJJ[2023]012; QJJ[2023]061; QJJ[2023]062).

Author information

Authors and Affiliations

  1. School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, People’s Republic of China

    Jianxing Zhao, Pin Liu & Caili Sang

Authors
  1. Jianxing Zhao
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Pin Liu
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Caili Sang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Caili Sang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Liqun Qi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, J., Liu, P. & Sang, C. Shifted Inverse Power Method for Computing the Smallest M-Eigenvalue of a Fourth-Order Partially Symmetric Tensor. J Optim Theory Appl 200, 1131–1159 (2024). https://doi.org/10.1007/s10957-023-02369-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-023-02369-z

Keywords

Mathematics Subject Classification