Abstract
After the theorem which is used to determine whether all roots of a polynomial are in unit circle is given, and two particle swarm optimizations for finding the maximum module of the roots of a polynomial based on the theorem are proposed. Finally, several computer simulation results show that using these algorithms to find the maximum module of roots of a polynomial are more efficient and feasible, the convergent speed is much faster and the accuracy of results is much higher.
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References
Horn, P.A., Johnson, C.H.: Matrix Analysis. Cambridge University Press, Cambridge (1990)
Farmer, M.R., Loizou, G.: Locating Multiple Zeros Interactively. Comput. Math. Appl. 11(6), 595–603 (1985)
Kittaneh, F.: Singular Values of Companion Matrices and Bounds on Zeros of Polynomials. Society for Industrial and Applied Mathematics (1995)
Guan, Y.: An iterative arithmetic for the largest module of roots of a polynomial. Journal of Hangzhou Teachers College (Natural Science Edition) 4(1), 22–24 (2005) (in Chinese)
Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, pp. 1942–1948 (1995)
Shen, H.Y., Peng, X.Q., Wang, J.N., Hu, Z.K.: A mountain clustering based on improved PSO algorithm. In: Wang, L., Chen, K., S. Ong, Y. (eds.) ICNC 2005. LNCS, vol. 3612, pp. 477–481. Springer, Heidelberg (2005)
Eberhart, R.C., Shi, Y.: Extracting rules from fuzzy neural network by particle swarm optimization. In: Proceedings of IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, USA (1998)
Ralston, A., Will, H.S.: Mathematical Methods for Digital Computer. John Wiley & Sons Inc., Chichester (1960)
Cheng, J.S.: A Parallel Algorithm for Finding Roots of Complex Polynomial. J. of Comput. Sci. & Technol. 5(1), 71–81 (1990)
Shi, Y., Eberhart, R.C.: A modified particle swarm optimizer. In: Proceedings of the IEEE International Conference on Evolutionary Computation, Anchorage, Alaska, USA, pp. 69–73 (1998)
Song, Y.Z.: The existence the region of polynomial zeros. ACTA Mathematic Sinica 36(2), 254–258 (1993) (in Chinese)
Zhao, W.J.: Bounds for the zeros of polynomials. Journal of Qingdao University 13(2), 14–19 (2000) (in Chinese)
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Qu, L., He, D. (2010). Finding the Maximum Module of the Roots of a Polynomial by Particle Swarm Optimization. In: Tan, Y., Shi, Y., Tan, K.C. (eds) Advances in Swarm Intelligence. ICSI 2010. Lecture Notes in Computer Science, vol 6145. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13495-1_27
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DOI: https://doi.org/10.1007/978-3-642-13495-1_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13494-4
Online ISBN: 978-3-642-13495-1
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