A homotopy method for solving multilinear systems with M-tensors. (English) Zbl 1375.65060
Summary: Multilinear systems of equations arise in various applications, such as numerical partial differential equations, data mining, and tensor complementarity problems. In this paper, we propose a homotopy method for finding the unique positive solution to a multilinear system with a nonsingular M-tensor and a positive right side vector. We analyze the method and prove its convergence to the desired solution. We report some numerical results based on an implementation of the proposed method using a prediction-correction approach for path following.
MSC:
| 65F30 | Other matrix algorithms (MSC2010) |
| 65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |
| 15A69 | Multilinear algebra, tensor calculus |
Keywords:
M-tensor; multilinear system; homotopy method; positive solution; convergence; numerical result; prediction-correction approach; path followingSoftware:
TensorToolboxReferences:
| [1] | Ding, W.; Wei, Y., Solving multi-linear systems with M-tensors, J. Sci. Comput., 68, 689-715 (2016) · Zbl 1371.65032 |
| [2] | Li, X.; Ng, M. K., Solving sparse non-negative tensor equations: algorithms and applications, Front. Math. China, 10, 3, 649-680 (2015) · Zbl 1323.65027 |
| [4] | Qi, L., Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40, 1302-1324 (2005) · Zbl 1125.15014 |
| [5] | 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)), 129-132 |
| [6] | Ding, W.; Qi, L.; Wei, Y., M-tensors and nonsingular M-tensors, Linear Algebra Appl., 439(10), 3264-3278 (2013) · Zbl 1283.15074 |
| [7] | Zhang, L.; Qi, L.; Zhou, G., M-tensors and some applications, SIAM J. Matrix Anal. Appl., 35(2), 437-452 (2014) · Zbl 1307.15034 |
| [8] | Berman, A.; Plemmons, R., Nonnegative matrices in the mathematical sciences, (Classics in Applied Mathematics (1994), SIAM: SIAM Philadelphia) · Zbl 0815.15016 |
| [9] | Li, T., Homotopy methods, (Engquist, B., Encyclopedia of Applied and Computational Mathematics (2015), Springer: Springer Berlin), 653-656 · Zbl 1263.65001 |
| [10] | Allgower, E. L.; Georg, K., Numerical continuation methods, an introduction, (Springer Series in Computational Mathematics (1990), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0717.65030 |
| [11] | Sommese, A. J.; Wampler, W. W., The Numerical Solution of Systems of Polynomials Arising in Engineering and Science (2005), World Scientific: World Scientific Singapore · Zbl 1091.65049 |
| [13] | Chang, K. C.; Qi, L.; Zhang, T., A survey of the spectral theory of nonnegative tensors, Numer. Linear Algebra Appl., 20, 891-912 (2013) · Zbl 1313.15015 |
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