2014, 4(1): 75-91. doi: 10.3934/naco.2014.4.75

An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials

1. 

Department of Mathematics and Statistics, Curtin University, Bentley, WA, Australia

Received  February 2013 Revised  November 2013 Published  December 2013

In this paper, we propose an iterative method for calculating the largest eigenvalue of nonhomogeneous nonnegative polynomials. This method is a generalization of the method in [19]. We also prove this method is convergent for irreducible nonhomogeneous nonnegative polynomials.
Citation: Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75
References:
[1]

M. Akian, S. Gaubert and A. Guterman, Tropical polyhedra are equivalent to mean payoff games, arXiv:0912.2462.

[2]

F. Baccelli, G. Cohen, G. J. Olsder and J. P. Quadrat, Synchronization and Linearity, Wiley Series in Probability and Mathematical Statistics, John Wiley, 1992.

[3]

L. Baratchart, M. Berthod and L. Pottier, Optimization of positive generalized polynomials under lpconstraints Journal of Convex Analysis, 5 (1998), 353-379.

[4]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Medical Image Computing and Computer-Assisted Intervention MICCAI, (2008), 1-8.

[5]

L. Collatz, Einschliessungssatz für die charakteristischen Zahlen von Matrizen, Math. Zeit., 48 (1942), 221-226.

[6]

K. C. Chang, K. J. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.

[7]

K. C. Chang, K. J. Pearson and T. Zhang, Primitivity, the convergence of the NQZ method and the largest eigenvalue for nonnegative tensors, SIAM J. Matrix Anal. Appl., 32 (2011), 806-819. doi: 10.1137/100807120.

[8]

K. Chang, L. Qi and G. Zhou, Singular values of a real rectangular tensor, Journal of Mathematical Analysis and Applications, 370 (2010), 284-294. doi: 10.1016/j.jmaa.2010.04.037.

[9]

L. De Lathauwer, B. De Moor and J. Vandewalle, On the best rank-1 and rank- (R1, ...,Rn) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.

[10]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749. doi: 10.1016/j.laa.2011.02.042.

[11]

S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. math. Soc., 356 (2004), 4931-4950. doi: 10.1090/S0002-9947-04-03470-1.

[12]

J. Gunawardena, From max-plus algebra to nonexpansive maps: a nonlinear theory for discrete event systems, Theoritical Computer Science, 293 (2003), 141-167. doi: 10.1016/S0304-3975(02)00235-9.

[13]

J. Gunawardena (editor), Idempotency, Publications of the Isaac Newton Institute, Cambridge University Press, 1998. doi: 10.1017/CBO9780511662508.

[14]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. doi: 10.1137/07070111X.

[15]

V. N. Kolokoltsov, Nonexpansive maps and option pricing theory, Kybernetika, 34 (1998), 713-724.

[16]

V. N. Kolokoltsov and V. P. Maslov, Idempotency Analysis and Applications, Kluwer Academic, 1997.

[17]

L. H. Lim, Multilinear pagerank: measuring higher order connectivity in linked objects, The Internet: Today and Tomorrow, 2005.

[18]

Y. Liu, G. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor, Journal of Computational and Applied Mathematics, 235 (2010), 286-292. doi: 10.1016/j.cam.2010.06.002.

[19]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099. doi: 10.1137/09074838X.

[20]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107. doi: 10.1109/TAC.2008.923679.

[21]

R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., 75 (1988). doi: 10.1090/memo/0391.

[22]

R. D. Nussbaum, Iterated nonlinear map and Hilbert's projective metric, II, Memoirs of the AMS, 79 (1989). doi: 10.1090/memo/0401.

[23]

M. Morishima, Equilibrium, Stability and Growth, Clarenson, Oxford, England, 2002.

[24]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Mathematical Programming, 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6.

[25]

L. Qi , Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensor, J. Comput. Appl. Math., 221 (2008), 150-157. doi: 10.1016/j.cam.2007.10.012.

[26]

D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel Jurnal of Mathematics, 121 (2001), 221-246. doi: 10.1007/BF02802505.

[27]

R. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1962.

[28]

R. J. Wood and M. J. O'Neill, Finding the spectral radius of a large sparse non-negative matrix, Anziam J., 48 (2007), C330-C345.

[29]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530. doi: 10.1137/090778766.

[30]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors II, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250. doi: 10.1137/100813671.

[31]

G. Zhou, L. Caccetta and L. Qi, Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor, Linear Algebra Appl., 438 (2013), 959-968. doi: 10.1016/j.laa.2011.06.038.

show all references

References:
[1]

M. Akian, S. Gaubert and A. Guterman, Tropical polyhedra are equivalent to mean payoff games, arXiv:0912.2462.

[2]

F. Baccelli, G. Cohen, G. J. Olsder and J. P. Quadrat, Synchronization and Linearity, Wiley Series in Probability and Mathematical Statistics, John Wiley, 1992.

[3]

L. Baratchart, M. Berthod and L. Pottier, Optimization of positive generalized polynomials under lpconstraints Journal of Convex Analysis, 5 (1998), 353-379.

[4]

L. Bloy and R. Verma, On computing the underlying fiber directions from the diffusion orientation distribution function, Medical Image Computing and Computer-Assisted Intervention MICCAI, (2008), 1-8.

[5]

L. Collatz, Einschliessungssatz für die charakteristischen Zahlen von Matrizen, Math. Zeit., 48 (1942), 221-226.

[6]

K. C. Chang, K. J. Pearson and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), 507-520.

[7]

K. C. Chang, K. J. Pearson and T. Zhang, Primitivity, the convergence of the NQZ method and the largest eigenvalue for nonnegative tensors, SIAM J. Matrix Anal. Appl., 32 (2011), 806-819. doi: 10.1137/100807120.

[8]

K. Chang, L. Qi and G. Zhou, Singular values of a real rectangular tensor, Journal of Mathematical Analysis and Applications, 370 (2010), 284-294. doi: 10.1016/j.jmaa.2010.04.037.

[9]

L. De Lathauwer, B. De Moor and J. Vandewalle, On the best rank-1 and rank- (R1, ...,Rn) approximation of higher-order tensors, SIAM J. Matrix Anal. Appl., 21 (2000), 1324-1342.

[10]

S. Friedland, S. Gaubert and L. Han, Perron-Frobenius theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl., 438 (2013), 738-749. doi: 10.1016/j.laa.2011.02.042.

[11]

S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. math. Soc., 356 (2004), 4931-4950. doi: 10.1090/S0002-9947-04-03470-1.

[12]

J. Gunawardena, From max-plus algebra to nonexpansive maps: a nonlinear theory for discrete event systems, Theoritical Computer Science, 293 (2003), 141-167. doi: 10.1016/S0304-3975(02)00235-9.

[13]

J. Gunawardena (editor), Idempotency, Publications of the Isaac Newton Institute, Cambridge University Press, 1998. doi: 10.1017/CBO9780511662508.

[14]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. doi: 10.1137/07070111X.

[15]

V. N. Kolokoltsov, Nonexpansive maps and option pricing theory, Kybernetika, 34 (1998), 713-724.

[16]

V. N. Kolokoltsov and V. P. Maslov, Idempotency Analysis and Applications, Kluwer Academic, 1997.

[17]

L. H. Lim, Multilinear pagerank: measuring higher order connectivity in linked objects, The Internet: Today and Tomorrow, 2005.

[18]

Y. Liu, G. Zhou and N. F. Ibrahim, An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor, Journal of Computational and Applied Mathematics, 235 (2010), 286-292. doi: 10.1016/j.cam.2010.06.002.

[19]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099. doi: 10.1137/09074838X.

[20]

Q. Ni, L. Qi and F. Wang, An eigenvalue method for testing the positive definiteness of a multivariate form, IEEE Transactions on Automatic Control, 53 (2008), 1096-1107. doi: 10.1109/TAC.2008.923679.

[21]

R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc., 75 (1988). doi: 10.1090/memo/0391.

[22]

R. D. Nussbaum, Iterated nonlinear map and Hilbert's projective metric, II, Memoirs of the AMS, 79 (1989). doi: 10.1090/memo/0401.

[23]

M. Morishima, Equilibrium, Stability and Growth, Clarenson, Oxford, England, 2002.

[24]

L. Qi, F. Wang and Y. Wang, Z-eigenvalue methods for a global polynomial optimization problem, Mathematical Programming, 118 (2009), 301-316. doi: 10.1007/s10107-007-0193-6.

[25]

L. Qi , Y. Wang and E. X. Wu, D-eigenvalues of diffusion kurtosis tensor, J. Comput. Appl. Math., 221 (2008), 150-157. doi: 10.1016/j.cam.2007.10.012.

[26]

D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel Jurnal of Mathematics, 121 (2001), 221-246. doi: 10.1007/BF02802505.

[27]

R. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1962.

[28]

R. J. Wood and M. J. O'Neill, Finding the spectral radius of a large sparse non-negative matrix, Anziam J., 48 (2007), C330-C345.

[29]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 31 (2010), 2517-2530. doi: 10.1137/090778766.

[30]

Q. Yang and Y. Yang, Further results for Perron-Frobenius theorem for nonnegative tensors II, SIAM J. Matrix Anal. Appl., 32 (2011), 1236-1250. doi: 10.1137/100813671.

[31]

G. Zhou, L. Caccetta and L. Qi, Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor, Linear Algebra Appl., 438 (2013), 959-968. doi: 10.1016/j.laa.2011.06.038.

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