Numerical Algorithms

, Volume 77, Issue 4, pp 1183–1197

# A cubically convergent method for solving the largest eigenvalue of a nonnegative irreducible tensor

Original Paper

## Abstract

In this paper, we present a cubically convergent method for finding the largest eigenvalue of a nonnegative irreducible tensor. A cubically convergent method is used to solve an equivalent system of nonlinear equations which is transformed by the tensor eigenvalue problem. Due to particular structure of tensor, Chebyshev’s direction is added to the method with a few extra computation. Two rules are designed such that the descendant property of the search directions is ensured. The global convergence is proved by using the line search technique. Numerical results indicate that the proposed method is competitive and efficient on some test problems.

## Keywords

Cubic convergence Tensor eigenvalue Nonnegative irreducible tensor Chebyshev’s method

## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgments

This work was supported by the Jiangsu Innovation Program for Graduate Education KYZZ160163; the National Natural Science Foundation of China under Grants 11471159,11571169,61661136001; and the Natural Science Foundation of Jiangsu Province under Grant BK20141409.

## References

1. 1.
Chen, Z., Qi, L.Q., Yang, Q.Z., Yang, Y.N.: The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis. Linear algebra Appl. 439, 3713–3733 (2013)
2. 2.
Cui, C.F., Dai, Y.H., Nie, J.W.: All real eigenvalues of symmetric tensors. SIAM. J. Matrix Anal. Appl. 35, 1582–1601 (2004)
3. 3.
Chang, K.C., Pearson, K., Zhang, T.: Perron-Frobenius theorem for nonnegative tensors. Commu. Math. Sci. 6, 507–520 (2008)
4. 4.
Chang, K.C., Pearson, K., Zhang, T.: Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. SIAM. J. Matrix Anal. Appl. 32, 806–819 (2011)
5. 5.
Friedland, S., Gauber, S., Han, L.: Perron-Frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebr. Appl. 438, 738–749 (2013)
6. 6.
Hu, S.L., Huang, Z.H., Qi, L.Q.: Strictly nonnegative tensors and nonnegative tensor partition. Sci. China Math. 57, 181–195 (2014)
7. 7.
Lim, L.H.: Singular values and eigenvalues of tensors, a variational approach Proceedings of the 1st IEEE International Workshop on Computational Advances of Multi-tensor Adaptive Processing, vol. 1, pp 129–132 (2005)Google Scholar
8. 8.
Liu, Y., Zhou, G.N., Ibrahim, F.: An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor. J. Comput. Appl. Math. 235, 286–292 (2010)
9. 9.
Ng, M., Qi, L.Q., Zhou, G.: Finding the largest eigenvalue of a nonnegative tensor. SIAM J. Matrix Anal. Appl. 31, 1090–1099 (2009)
10. 10.
Ni, Q., Qi, L.Q., Wang, F.: An eigenvalue method for testing the positive definiteness of a multivariate form. IEEE Trans. Automat. Control. 53, 1096–1107 (2008)
11. 11.
Ni, Q., Qi, L.Q.: A quadratically convergent algorithm for finding the largest eigenvalue of nonnegative homogeneous polynomial map. J. Glob. Optim. 61, 627–641 (2015)
12. 12.
Nocedal, J., Wright, S.J.: Numerical optimization. Science Press, Beijing (2006)
13. 13.
Qi, L.Q., Teo, K.L.: Multivariate polynomial minimization and its application in signal processing. J. Global. Optim. 26, 419–433 (2003)
14. 14.
Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)
15. 15.
Qi, L.Q., Wang, Y., Wu, E.X.: D-eigenvalues of diffusion kurtosis tensors. J. Compu. Appl. Math. 221, 150–157 (2008)
16. 16.
Qi, L.Q., Yu, G., Wu, E.X.: Higher order positive semi-definite diffusion tensor imaging. SIAM J. Imaging Sci. 3, 416–433 (2010)
17. 17.
Schatz, M.D., Low, T.M., Van De Geijn, R.A., Kolda, T.G.: Exploiting symmetry in tensors for high performance: multiplication with symmetric tensors. SIAM J. Sci. Comput. 36(5), 453–479 (2014)Google Scholar
18. 18.
Werner, W.: Iterative solution of systems of nonlinear equations based upon quadratic approximations. Comp. Maths. with Appls. 12A(3), 331–343 (1986)
19. 19.
Yang, W.W., Ni, Q.: A new cubically convergent method for solving a system of nonlinear equations to submit to International Journal of Computer Mathematics (2016)Google Scholar
20. 20.
Yang, Y.N., Yang, Q.Z.: Further results for Perron-Frobenius Theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31, 2517–2530 (2010)
21. 21.
Yang, Y.N., Yang, Q.Z.: Further results for Perron-Frobenius Theorem for nonnegative tensors II. SIAM J. Matrix Anal. Appl. 32, 1236–1250 (2011)
22. 22.
Zhang, L., Qi, L.Q.: Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor. Numer. Linear Algebra Appl. 19, 830–841 (2012)