A fast algorithm for the spectral radii of weakly reducible nonnegative tensors

In this paper, we propose a fast algorithm for computing the spectral radii of symmetric nonnegative tensors. In particular, by this proposed algorithm, we are able to obtain the spectral radii of weakly reducible symmetric nonnegative tensors without requiring the partition of the tensors. As we know, it is very costly to determine the partition for large‐sized weakly reducible tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radii of large‐sized tensors. As an application, we present an algorithm for testing the positive definiteness of Z‐tensors. By this algorithm, it is guaranteed to determine the positive definiteness for any Z‐tensor.

[1]  Liping Chen,et al.  A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor , 2017, J. Comput. Appl. Math..

[2]  Jianyu Pan,et al.  Shifted power method for computing tensor H‐eigenpairs , 2016, Numer. Linear Algebra Appl..

[3]  Liping Chen,et al.  Computing Tensor Eigenvalues via Homotopy Methods , 2015, SIAM J. Matrix Anal. Appl..

[4]  Qin Ni,et al.  A quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map , 2014, Journal of Global Optimization.

[5]  Liqun Qi,et al.  M-Tensors and Some Applications , 2014, SIAM J. Matrix Anal. Appl..

[6]  L. Qi,et al.  Strictly nonnegative tensors and nonnegative tensor partition , 2011, Science China Mathematics.

[7]  Liqun Qi,et al.  On the largest eigenvalue of a symmetric nonnegative tensor , 2013, Numer. Linear Algebra Appl..

[8]  Tan Zhang,et al.  A survey on the spectral theory of nonnegative tensors , 2013, Numer. Linear Algebra Appl..

[9]  S. Gaubert,et al.  Perron–Frobenius theorem for nonnegative multilinear forms and extensions , 2009, 0905.1626.

[10]  L. Qi.  Symmetric nonnegative tensors and copositive tensors , 2012, 1211.5642.

[11]  L. Qi,et al.  Finding the Spectral Radius of a Nonnegative Tensor , 2011, 1111.2138.

[12]  Tan Zhang,et al.  Primitivity, the Convergence of the NQZ Method, and the Largest Eigenvalue for Nonnegative Tensors , 2011, SIAM Journal on Matrix Analysis and Applications.

[13]  Tamara G. Kolda,et al.  Shifted Power Method for Computing Tensor Eigenpairs , 2010, SIAM J. Matrix Anal. Appl..

[14]  Yongjun Liu,et al.  An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor , 2010, J. Comput. Appl. Math..

[15]  Michael K. Ng,et al.  Finding the Largest Eigenvalue of a Nonnegative Tensor , 2009, SIAM J. Matrix Anal. Appl..

[16]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[17]  Kung-Ching Chang,et al.  Perron-Frobenius theorem for nonnegative tensors , 2008 .

[18]  Fei Wang,et al.  An Eigenvalue Method for Testing Positive Definiteness of a Multivariate Form , 2008, IEEE Transactions on Automatic Control.

[19]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[20]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[21]  Joos Vandewalle,et al.  On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors , 2000, SIAM J. Matrix Anal. Appl..

[22]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .