This paper introduced a smoothing algorithm for calculating the maximal eigenvalue of non-defective positive matrices. Two special matrices were constructed to provide monotonically increasing lower-bound estimates and monotonically decreasing upper-bound estimates of the maximal eigenvalue. The monotonicity and convergence of these estimations was also proven. Finally, the effectiveness of the algorithm was demonstrated with numerical examples.
Citation: Na Li, Qin Zhong. Smoothing algorithm for the maximal eigenvalue of non-defective positive matrices[J]. AIMS Mathematics, 2024, 9(3): 5925-5936. doi: 10.3934/math.2024289
| [1] | Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler . Mathematical methods in systems biology. Mathematical Biosciences and Engineering, 2016, 13(6): i-ii. doi: 10.3934/mbe.201606i |
| [2] | Janine Egert, Clemens Kreutz . Realistic simulation of time-course measurements in systems biology. Mathematical Biosciences and Engineering, 2023, 20(6): 10570-10589. doi: 10.3934/mbe.2023467 |
| [3] | Adam Peddle, William Lee, Tuoi Vo . Modelling chemistry and biology after implantation of a drug-eluting stent. Part Ⅱ: Cell proliferation. Mathematical Biosciences and Engineering, 2018, 15(5): 1117-1135. doi: 10.3934/mbe.2018050 |
| [4] | Anissa Guillemin, Michael P. H. Stumpf . Non-equilibrium statistical physics, transitory epigenetic landscapes, and cell fate decision dynamics. Mathematical Biosciences and Engineering, 2020, 17(6): 7916-7930. doi: 10.3934/mbe.2020402 |
| [5] | Urszula Ledzewicz, Avner Friedman, Jacek Banasiak, Heinz Schättler, Edward M. Lungu . From the guest editors. Mathematical Biosciences and Engineering, 2013, 10(3): i-ii. doi: 10.3934/mbe.2013.10.3i |
| [6] | Avner Friedman, Kang-Ling Liao . The role of the cytokines IL-27 and IL-35 in cancer. Mathematical Biosciences and Engineering, 2015, 12(6): 1203-1217. doi: 10.3934/mbe.2015.12.1203 |
| [7] | Lin Zhang, Yongbin Ge, Zhi Wang . Positivity-preserving high-order compact difference method for the Keller-Segel chemotaxis model. Mathematical Biosciences and Engineering, 2022, 19(7): 6764-6794. doi: 10.3934/mbe.2022319 |
| [8] | Cristina De Ambrosi, Annalisa Barla, Lorenzo Tortolina, Nicoletta Castagnino, Raffaele Pesenti, Alessandro Verri, Alberto Ballestrero, Franco Patrone, Silvio Parodi . Parameter space exploration within dynamic simulations of signaling networks. Mathematical Biosciences and Engineering, 2013, 10(1): 103-120. doi: 10.3934/mbe.2013.10.103 |
| [9] | Fred Brauer, Carlos Castillo-Chavez, Thomas G. Hallam, Jia Li, Jianhong Wu, Yicang Zhou . From the Guest Editors. Mathematical Biosciences and Engineering, 2006, 3(1): i-ix. doi: 10.3934/mbe.2006.3.1i |
| [10] | Maria Vittoria Barbarossa, Christina Kuttler, Jonathan Zinsl . Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells. Mathematical Biosciences and Engineering, 2012, 9(2): 241-257. doi: 10.3934/mbe.2012.9.241 |
This paper introduced a smoothing algorithm for calculating the maximal eigenvalue of non-defective positive matrices. Two special matrices were constructed to provide monotonically increasing lower-bound estimates and monotonically decreasing upper-bound estimates of the maximal eigenvalue. The monotonicity and convergence of these estimations was also proven. Finally, the effectiveness of the algorithm was demonstrated with numerical examples.
| [1] | A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Philadelphia: Society for Industrial and Applied Mathematics, 1994. https://doi.org/10.1137/1.9781611971262 |
| [2] |
W. Ledermann, Bounds for the greatest latent root of a positive matrix, J. Lond. Math. Soc., s1-25 (1950), 265–268. https://doi.org/10.1112/jlms/s1-25.4.265 doi: 10.1112/jlms/s1-25.4.265
|
| [3] |
A. Ostrowski, Bounds for the greatest latent root of a positive matrix, J. Lond. Math. Soc., s1-27 (1952), 253–256. https://doi.org/10.1112/jlms/s1-27.2.253 doi: 10.1112/jlms/s1-27.2.253
|
| [4] |
A. Brauer, The theorems of Ledermann and Ostrowski on positive matrices, Duke Math. J., 24 (1957), 265–274. https://doi.org/10.1215/S0012-7094-57-02434-1 doi: 10.1215/S0012-7094-57-02434-1
|
| [5] | H. Minc, Nonnegative Matrices, New York: Wiley, 1988. |
| [6] |
S. L. Liu, Bounds for the greatest characteristic root of a nonnegative matrix, Linear Algebra Appl., 239 (1996), 151–160. https://doi.org/10.1016/S0024-3795(96)90008-7 doi: 10.1016/S0024-3795(96)90008-7
|
| [7] | L. M. Liu, T. Z. Huang, X. Q. Liu, New bounds for the greatest eigenvalue of a nonnegative matrix, J. Univ. Electron. Sci. Techn. China, 2007,343–345. |
| [8] |
P. Liao, Bounds for the Perron root of nonnegative matrices and spectral radius of iteration matrices, Linear Algebra Appl., 530 (2017), 253–265. https://doi.org/10.1016/j.laa.2017.05.021 doi: 10.1016/j.laa.2017.05.021
|
| [9] |
H. Cheriyath, N. Agarwal, On the Perron root and eigenvectors associated with a subshift of finite type, Linear Algebra Appl., 633 (2022), 42–70. https://doi.org/10.1016/j.laa.2021.10.003 doi: 10.1016/j.laa.2021.10.003
|
| [10] |
P. Liao, Bounds for the Perron root of positive matrices, Linear Multilinear A., 71 (2023), 1849–1857. https://doi.org/10.1080/03081087.2022.2081310 doi: 10.1080/03081087.2022.2081310
|
| [11] |
Y. Y. Yan, D. Z. Cheng, J. E. Feng, H. T. Li, J. M. Yue, Survey on applications of algebraic state space theory of logical systems to finite state machines, Sci. China Inform. Sci., 66 (2023), 111201. https://doi.org/10.1007/s11432-022-3538-4 doi: 10.1007/s11432-022-3538-4
|
| [12] | D. Z. Cheng, H. S. Qi, Y. Zhao, An introduction to semi-tensor product of matrices and its applications, World Scientific, 2012. https://doi.org/10.1142/8323 |
| [13] |
J. M. Yue, Y. Y. Yan, H. Deng, Matrix approach to formulate and search k-ESS of graphs using the STP theory, J. Math., 2021 (2021), 1–12. https://doi.org/10.1155/2021/7230661 doi: 10.1155/2021/7230661
|
| 1. | Charles L. Wiseman, Jarrod P. Holmes, Carmen Calfa, Shaker R. Dakhil, Saveri Bhattacharya, George E. Peoples, Markus D. Lacher, Miguel Lopez-Lago, Alex Kharazi, Giuseppe Del Priore, Mingjin Chang, Daniel L. Adams, William V. Williams, Results of a phase I/IIa trial of SV-BR-1-GM inoculation with low-dose cyclophosphamide and interferon alpha (Bria-IMT) in metastatic breast cancer, 2024, 20, 2164-5515, 10.1080/21645515.2024.2379864 |
Na Li, Qin Zhong. Smoothing algorithm for the maximal eigenvalue of non-defective positive matrices[J]. AIMS Mathematics, 2024, 9(3): 5925-5936. doi: 10.3934/math.2024289
DownLoad: