113
Views
0
CrossRef citations to date
0
Altmetric
Original Articles

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

, , &
Pages 2461-2472 | Received 31 Aug 2018, Accepted 03 Jan 2019, Published online: 22 Jan 2019

References

  • A. Barmpoutis, M.S. Hwang, D. Howland, J.R. Forder, and B.C. Vemuri, Regularized positive-definite fourth order tensor field estimation from DW-MRI, Neuroimage 45(1) (2009), pp. S153–S162.
  • P.J. Basser and D.K. Jones, Diffusion-tensor MRI: theory, experimental design and data analysis-a technical review, NMR Biomed. 15 (2002), pp. 456–467.
  • N.K. Bose and A.R. Modaress, General procedure for multivariable polynomial positivity test with control applications, IEEE Trans. Automat. Control. 21 (1976), pp. 696–701.
  • H.T. Che, H.B. Chen, and Y.J. Wang, C-eigenvalue inclusion theorems for piezoelectric-type tensors, Appl. Math. Lett. 89 (2019), pp. 41–49.
  • H.B. Chen and L.Q. Qi, Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors, J. Ind. Manag. Optim. 11 (2015), pp. 1263–1274.
  • H.B. Chen and Y.J. Wang, On computing minimal H-eigenvalue of sign-structured tensors, Front. Math. China 12(6) (2017), pp. 1289–1302.
  • H.B. Chen and Y.J. Wang, High-order copositive tensors and its applications, J. Appl. Anal. Comput. 6(8) (2018), pp. 1863–1885.
  • H.B. Chen, Y.N. Chen, G.Y. Li, and L.Q. Qi, A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test, Numer. Linear Algebra Appl. 25(1) (2018), p. e2125.
  • H.B. Chen, L.Q. Qi, and Y.S. Song, Column sufficient tensors and tensor complementarity problems, Front. Math. China 13(2) (2018), pp. 255–276.
  • P. Comon, G. Golub, L.H. Lim, and B. Mourrain, Symmetric tensors and symmetric tensor rank, SIAM J. Matrix Anal. Appl. 30(3) (2008), pp. 1254–1279.
  • W.Y. Ding, L.Q. Qi, and Y.M. Wei, M-tensors and nonsingular M-tensors, Linear Algebra Appl. 439(10) (2013), pp. 3264–3278.
  • S. Friedland, S. Gaubert, and L.X. Han, Perron-Frobenius theorem for nonegitive multilinear forms and extensions, Linear Algebra Appl. 438(2) (2013), pp. 738–749.
  • M.A. Hasan and A.A. Hasan, A procedure for the positive definiteness of forms of even order, IEEE Trans. Automat. Control 41 (1996), pp. 615–617.
  • A.Q. He and J.Z. Liu, A new non-parameter criterion for H-matrices, Comput. Math. Appl. 55 (2008), pp. 1148–1153.
  • S.L. Hu, Z.H. Huang, and L.Q. Qi, Strictly nonnegative tensors and nonnegative tensor partition, Sci. China Math. 57(1) (2014), pp. 181–195.
  • M.A. Hulsen, A sufficient condition for a positive definite configuration tensor in differential models, J. Non-Newton. Fluid Mech. 38(1) (1990), pp. 93–100.
  • M.R. Kannan, N. Shaked-Monderer, and A. Berman, Some properties of strong H-tensors and general H-tensors, Linear Algebra Appl. 476 (2015), pp. 42–55.
  • T.G. Kolda and B.W. Bader, Tensor decompositions and applications, SIAM Rev. 51 (2009), pp. 455–500.
  • L.D. Lathauwer, B.D. Moor, and J. Vandewalle, On the best rank-1 and rank-(r1,r2,…,rn) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl. 21 (2000), pp. 1324–1342.
  • Y.T. Li, Q.L. Liu, and L.Q. Qi, Programmable criteria for strong H-tensor, Numer. Algorithms 74(1) (2017), pp. 199–221.
  • C.Q. Li, F. Wang, J.X. Zhao, Y. Zhu, and Y.T. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Appl. Math. 255 (2014), pp. 1–14.
  • Q.L. Liu, C.Q. Li, and Y.T. Li, On the iterative criterion for strong H-tensors, Comput. Appl. Math. 36 (2017), pp. 1623–1635.
  • M. Moakher, On the averaging of symmetric positive-definite tensors, J. Elasticity 82(3) (2006), pp. 273–296.
  • Q. Ni, L.Q. Qi, and F. Wang, An eigenvalue method for testing positive definiteness of a multivariate form, IEEE Trans. Automat. Control 53(5) (2008), pp. 1096–1107.
  • L.Q. Qi, Eigenvalues of a real supersymmetric tensor, J. Symb. Comput. 40(6) (2005), pp. 1302–1324.
  • L.Q. Qi and Z.Y. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, Philadelphia, 2017.
  • L.Q. Qi, F. Wang, and Y.J. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Math. Program. 118(2) (2009), pp. 301–316.
  • L.Q. Qi, Y.J. Wang, and Ed X. Wu, D-eigenvalues of diffusion kurtosis tensors, J. Comput. Appl. Math. 221(1) (2008), pp. 150–157.
  • L.Q. Qi, C.Q. Xu, and Y. Xu, Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm, SIAM J. Matrix Anal. Appl. 35(4) (2014), pp. 1227–1241.
  • L.Q. Qi, G.H. Yu, and E.X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imaging Sci. 3(3) (2010), pp. 416–433.
  • J. Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mt. J. Math. 6(3) (1976), pp. 409–434.
  • X.Y. Wang, Alternating proximal penalization algorithm for the modified multiple-sets split feasibility problems, J. Inequal. Appl. 2018 (2018), pp. 48. https://doi.org/10.1186/s13660-018-1641-y.
  • Y.J. Wang, L. Caccetta, and G.L. Zhou, Convergence analysis of a block improvement method for polynomial optimization over unit spheres, Numer. Linear Algebra Appl. 22(6) (2015), pp. 1059–1076.
  • X.Y. Wang, H.B. Chen, and Y.J. Wang, Solution structures of tensor complementarity problem, Front. Math. China 13(4) (2018), pp. 935–945.
  • Y.J. Wang, L.Q. Qi, S.L. Luo, and Y. Xu, An alternative steepest direction method for the optimization in evaluating geometric discord, Pac. J. Optim. 10 (2014), pp. 137–149.
  • Y.J. Wang, L.Q. Qi, and X.Z. Zhang, A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor, Numer. Linear Algebra Appl. 16(7) (2009), pp. 589–601.
  • Y.J. Wang, K.L. Zhang, and H.C. Sun, Some new criteria for strong H-tensors, Front. Math. China 11(3) (2016), pp. 577–592.
  • Y.J. Wang, G.L. Zhou, and L. Caccetta, Nonsingular H-tensor and its criteria, J. Ind. Manag. Optim. 12 (2016), pp. 1173–1186.
  • G. Wang, G.L. Zhou, and L. Caccetta, Z-eigenvalue inclusion theorems for tensors, Discrete Contin. Dyn. Syst. B 22(1) (2017), pp. 187–198.
  • L.P. Zhang, L.Q. Qi, and G.L. Zhou, M-tensors and some applications, SIAM J. Matrix Anal. Appl. 35(2) (2014), pp. 437–452.
  • K.L. Zhang and Y.J. Wang, An H-tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous, J. Comput. Appl. Math. 305 (2016), pp. 1–10.
  • G.L. Zhou, G. Wang, L.Q. Qi, and M. Alqahtani, A fast algorithm for the spectral radii of weakly reducible nonnegative tensors, Numer. Linear Algebra Appl. 25(2) (2018), p. e2134.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.