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

Liping Zhang

Front. Math. China ›› 2012, Vol. 8 ›› Issue (1) : 141-153.

PDF(127 KB)
Front. Math. China All Journals
PDF(127 KB)
Front. Math. China ›› 2012, Vol. 8 ›› Issue (1) : 141-153. DOI: 10.1007/s11464-012-0260-z
Research Article
RESEARCH ARTICLE

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

Author information +
History +

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

Cite this article

Download citation ▾
Liping Zhang. Linear convergence of an algorithm for largest singular value of a nonnegative rectangular tensor. Front. Math. China, 2012, 8(1): 141‒153 https://doi.org/10.1007/s11464-012-0260-z
This is a preview of subscription content, contact us for subscripton.

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
CrossRef Google scholar
[3.]
Chang K. C., Qi L., Zhou G.. Singular values of a real rectangular tensor. J Math Anal Appl, 2010, 370: 284-294
CrossRef Google scholar
[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
CrossRef Google scholar
[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
CrossRef Google scholar
[6.]
Lim L.-H.. Singular values and eigenvalues of tensors: a variational approach. Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP’ 05), Vol 1, 2005, Piscataway: IEEE Computer Society Press 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
CrossRef Google scholar
[8.]
Ni Q., Qi L., Wang F.. An eigenvalue method for the positive definiteness identification problem. IEEE Trans Automat Control, 2008, 53: 1096-1107
CrossRef Google scholar
[9.]
Qi L.. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302-1324
CrossRef Google scholar
[10.]
Qi L.. Eigenvalues and invariants of tensor. J Math Anal Appl, 2007, 325: 1363-1377
CrossRef Google scholar
[11.]
Qi L., Dai H. H., Han D.. Conditions for strong ellipticity and M-eigenvalues. Front Math China, 2009, 4(2): 349-364
CrossRef Google scholar
[12.]
Qi L., Sun W., Wang Y.. Numerical multilinear algebra and its applications. Front Math China, 2007, 2(4): 501-526
CrossRef Google scholar
[13.]
Qi L., Wang F., Wang Y.. Z-eigenvalue methods for a global polynomial optimization problem. Math Program, 2009, 118: 301-316
CrossRef Google scholar
[14.]
Qi L., Wang Y., Wu E. X.. D-eigenvalues of diffusion kurtosis tensor. J Comput Appl Math, 2008, 221: 150-157
CrossRef Google scholar
[15.]
Rosakis P.. Ellipticity and deformation with discontinuous deformation gradients in finite elastostatics. Arch Ration Mech Anal, 1990, 109: 1-37
CrossRef Google scholar
[16.]
Wang Y., Aron M.. A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J Elasticity, 1996, 44: 89-96
CrossRef Google scholar
[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
CrossRef Google scholar
[18.]
Yang Q., Yang Y.. Further results for Perron-Frobenius Theorem for nonnegative tensors II. SIAM J Matrix Anal Appl, 2011, 32: 1236-1250
CrossRef Google scholar
[19.]
Yang Y., Yang Q.. Further results for Perron-Frobenius Theorem for nonnegative tensors. SIAM J Matrix Anal Appl, 2010, 31: 2517-2530
CrossRef Google scholar
[20.]
Yang Y., Yang Q.. Singular values of nonnegative rectangular tensors. Front Math China, 2011, 6(2): 363-378
CrossRef Google scholar
[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
CrossRef Google scholar
AI Summary AI Mindmap
PDF(127 KB)

802

Accesses

6

Citations

Detail

Sections
Recommended

/