Linear convergence of an algorithm for largest singular value of a nonnegative rectangular tensor

Abstract

An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we establish a linear convergence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.

Keywords

Singular value / nonnegative tensor / rectangular tensor / algorithm / convergence

References

[1] Chang K C, Pearson K, Zhang T. Perron-Frobenius theorem for nonnegative tensors. Commun Math Sci, 2008, 6: 507-520

[2] Chang K C, Pearson K, Zhang T. On eigenvalue problems of real symmetric tensors. J Math Anal Appl, 2009, 350: 416-422

[3] Chang K C, Qi L, Zhou G. Singular values of a real rectangular tensor. J Math Anal Appl, 2010, 370: 284-294

[4] Dahl D, Leinass J M, Myrheim J, Ovrum E. A tensor product matrix approximation problem in quantum physics. Linear Algebra Appl, 2007, 420: 711-725

[5] Knowles J K, Sternberg E. On the ellipticity of the equations of non-linear elastostatics for a special material. J Elasticity, 1975, 5: 341-361

[6] Lim L-H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi- Sensor Adaptive Processing (CAMSAP ’05), Vol 1. Piscataway: IEEE Computer Society Press, 2005, 129-132

[7] Ng M, Qi L, Zhou G. Finding the largest eigenvalue of a nonnegative tensor. SIAM J Matrix Anal Appl, 2009, 31: 1090-1099

[8] Ni Q, Qi L, Wang F. An eigenvalue method for the positive definiteness identification problem. IEEE Trans Automat Control, 2008, 53: 1096-1107

[9] Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302-1324

[10] Qi L. Eigenvalues and invariants of tensor. J Math Anal Appl, 2007, 325: 1363-1377

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

[12] Qi L, Sun W, Wang Y. Numerical multilinear algebra and its applications. Front Math China, 2007, 2(4): 501-526

[13] Qi L, Wang F, Wang Y. Z-eigenvalue methods for a global polynomial optimization problem. Math Program, 2009, 118: 301-316

[14] Qi L, Wang Y, Wu E X. D-eigenvalues of diffusion kurtosis tensor. J Comput Appl Math, 2008, 221: 150-157

[15] Rosakis P. Ellipticity and deformation with discontinuous deformation gradients in finite elastostatics. Arch Ration Mech Anal, 1990, 109: 1-37

[16] Wang Y, Aron M. A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J Elasticity, 1996, 44: 89-96

[17] 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, 2009, 16: 589-601

[18] Yang Q, Yang Y. Further results for Perron-Frobenius Theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32: 1236-1250

[19] Yang Y, Yang Q. Further results for Perron-Frobenius Theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517-2530

[20] Yang Y, Yang Q. Singular values of nonnegative rectangular tensors. Front Math China, 2011, 6(2): 363-378

[21] Zhou G, Caccetta L, Qi L. Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor. Linear Algebra Appl, 2013, 438(2): 959-968

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg