Finding All Solutions of Systems of Nonlinear Equations Using Spiral Dynamics Inspired Optimization with Clustering
Kuntjoro Adji Sidarto and Adhe Kania
Department of Mathematics, Institut Teknologi Bandung
Bandung 40132, Indonesia
Nowadays the root finding problem for nonlinear system equations is still one of the difficult problems in computational sciences. Many attempts using deterministic and meta-heuristic methods have been done with their advantages and disadvantages, but many of them have fail to converge to all possible roots. In this paper, a novel method of locating and finding all of the real roots from the system of nonlinear equations is proposed mainly using the spiral dynamics inspired optimization by Tamura and Yasuda . The method is improved by the usage of the Sobol sequence of points for generating initial candidates of roots which are uniformly distributed than of pseudo-random generated points. Using clustering technique, the method localizes all potential roots so the optimization is conducted in those points simultaneously. A set of problems as the benchmarks from the literature is given. Having only a single run for each problem, the proposed method has successfully found all possible roots within a bounded domain.
-  K. Tamura and K. Yasuda,“Spiral Dynamics Inspired Optimization,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.15, No.8, pp. 1116-1122, 2011.
-  Y. Z. Luo, G. J. Tang, and L. N. Zhou, “Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method,” Applied Soft Computing, Vol.8, pp. 1068-1073, 2008.
-  R. L. Burden and J. D. Faires, “Numerical Analysis,” 7th ed., Brooks/Cole, 2001.
-  I. G. Tsoulos and A. Stavrakoudis, “On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods,” Nonlinear Analysis: Real World Applications, Vol.11, pp. 2465-2471, 2010.
-  W. F. Sacco and N. Henderson, “Finding all solutions of nonlinear systems using a hybrid meta-heuristic method with Fuzzy Clustering Means,” Applied Soft Computing, Vol.11, pp. 5424-5432, 2011.
-  C. Grosan and A. Abraham, “A new approach for solving Nonlinear equations systems,” IEEE Trans. Systems, Man and Cybernetics, Vol.38, No.3, pp. 698-714, 2008.
-  W. Song, Y. Wang, H. X. Li, and Z. Cai, “Locating multiple optimal solutions of nonlinear equations systems based on multi objective optimization,” IEEE Trans. Evol. Comput., in press.
-  V. Aggarwal, “Solving transcendental equations using Genetic Algorithm,” http://web.mit.edu/varuntextunderscore ag/www/stetextunderscore gas.pdf
-  R. Seydel, “Tools for Computational Finance,” Springer-Verlag, 2002.
-  S. Joe and S. Y. Kuo, “Constructing Sobol sequences with better two dimensional projections,” SIAM J. Sci. Comput., Vol.30, pp. 2635-2654, 2008.
-  K. Chen, P. Giblin, and A. Irving, “Mathematical Explorations with MATLAB,” Cambridge University Press, 1999.
-  S. Krzyworzcka, “Extension of the Lanczos and CGS methods to systems of nonlinear equations,” J. Comput. Appl. Math., Vol.69, No.1, pp. 181-190, 1996.
-  R. B. Kearfott, “Some tests of generalized bisection,” ACM Trans. Math. Softw., Vol.13, No.3, pp. 197-220, 1987.