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STUDYING THE BASIN OF CONVERGENCE OF METHODS FOR COMPUTING PERIODIC ORBITS

Computational Intelligence Laboratory, Department of Mathematics, University of Patras Artificial Intelligence Research Center, University of Patras, GR-26110 Patras, Greece

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MICHAEL N. VRAHATIS

Computational Intelligence Laboratory, Department of Mathematics, University of Patras Artificial Intelligence Research Center, University of Patras, GR-26110 Patras, Greece

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https://doi.org/10.1142/S0218127411029653Cited by:2 (Source: Crossref)

Abstract

Starting from the well-known Newton's fractal which is formed by the basin of convergence of Newton's method applied to a cubic equation in one variable in the field ℂ, we were able to find methods for which the corresponding basins of convergence do not exhibit a fractal-like structure. Using this approach we are able to distinguish reliable and robust methods for tackling a specific problem. Also, our approach is illustrated here for methods for computing periodic orbits of nonlinear mappings as well as for fixed points of the Poincaré map on a surface of section.

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References

G. S. Androulakis and M. N. Vrahatis, J. Comput. Appl. Math. 72, 41 (1996), DOI: 10.1016/0377-0427(95)00244-8. Crossref, Web of Science, Google Scholar
G. S. Androulakis, G. D. Magoulas and M. N. Vrahatis, Nonlin. Anal.: Th. Meth. Appl. 30, 4539 (1997). Crossref, Web of Science, Google Scholar
L. Armijo, Pacific J. Math. 16, 1 (1966). Crossref, Web of Science, Google Scholar
S. Banach, Fund. Math. 3, 133 (1922). Crossref, Google Scholar
M. Barnsley , Fractals Everywhere ( Academic Press Professional, Inc. , San Diego, CA, USA , 1988 ) . Google Scholar
H.-G. Beyer, Comput. Meth. Appl. Mech. Engin. 186, 239 (2000), DOI: 10.1016/S0045-7825(99)00386-2. Crossref, Web of Science, Google Scholar
H.-G. Beyer , The Theory of Evolution Strategies ( Springer-Verlag Inc. , NY, USA , 2001 ) . Crossref, Google Scholar
T. C. Bountis and S. Tompaidis, Strong and weak instabilities in a 4D mapping model of FODO cell dynamics, Workshop on Nonlinear Problems in Future Particle Accelerators, eds. G. Turchetti and W. Scandale (World Scientific, Singapore, 1991) pp. 112–127. Google Scholar
T. C. Bountis , The Wondrous World of Fractal ( Leader Books , Athens, Greece , 2004 ) . Google Scholar
G. E. P. Box and M. E. Muller, Ann. Math. Stat. 29, 610 (1958), DOI: 10.1214/aoms/1177706645. Crossref, Web of Science, Google Scholar
J. Bräuninger, Numer. Math. 36, 359 (1981). Crossref, Web of Science, Google Scholar
C. G. Broyden, Math. Comput. 19, 577 (1965). Crossref, Google Scholar
A. Cauchy, Comptes Rendus de l'Académie des Sciences 25, 536 (1847). Google Scholar
A. Cayley, Amer. J. Math. 2, 97 (1879), DOI: 10.2307/2369201. Crossref, Web of Science, Google Scholar
A. R. Conn , K. Scheinberg and L. N. Vicente , Introduction to Derivative-Free Optimization , Society for Industrial and Applied Mathematics ( SIAM , Philadelphia, PA, U.S.A. , 2009 ) . Crossref, Google Scholar
J. E. Dennis and R. B. Schnabel , Numerical Methods for Unconstrained Optimization and Nonlinear Equations , Classics in Applied Mathematics 16 ( SIAM , Philadelphia, PA, USA , 1996 ) . Crossref, Google Scholar
L. Drossoset al., Phys. Rev. E 53, 1206 (1996), DOI: 10.1103/PhysRevE.53.1206. Crossref, Web of Science, Google Scholar
C. Elster and A. Neumaier, Computing 58, 31 (1997), DOI: 10.1007/BF02684470. Crossref, Web of Science, Google Scholar
R. Fletcher and C. M. Reeves, The Computer J. 7, 149 (1964), DOI: 10.1093/comjnl/7.2.149. Crossref, Web of Science, Google Scholar
G. Giorgi , A. Guerraggio and J. Thierfelder , Mathematics of Optimization: Smooth and Nonsmooth Case ( Elsevier Science , Amsterdam, The Netherlands , 2004 ) . Google Scholar
M. C. Gutzwiller , Chaos in Classical and Quantum Mechanics ( Springer-Verlag , NY , 1990 ) . Crossref, Google Scholar
B. Henry, S. Watt and S. Wearne, Austr. New Zealand Industrial and Appl. Math. J. (ANZIAM J.) 42, C735 (2000). Google Scholar
M. R. Hestenes and E. Stiefel, J. Res. Nat. Bureau of Standards 49, 409 (1952). Crossref, Web of Science, Google Scholar
E. T. Jaynes, The Maximum Entropy Formalism, eds. R. D. Levine and M. Tribus (The MIT Press, Cambridge, U.S.A., 1979) pp. 15–118. Google Scholar
V. S. Kalantoniset al., Celest. Mech. Dyn. Astron. 80, 81 (2001), DOI: 10.1023/A:1011970019812. Crossref, Google Scholar
V. S. Kalantoniset al., Astrophys. Space Sci. 288, 489 (2003). Google Scholar
V. S. Kalantoniset al., Astrophys. Space Sci. 288, 479 (2003). Google Scholar
M. Katsanikas and P. A. Patsis, Int. J. Bifurcation and Chaos 21, 467 (2011). Link, Web of Science, Google Scholar
M. Katsanikas, P. A. Patsis and G. Contopoulos, Int. J. Bifurcation and Chaos 21, 2321 (2011). Link, Web of Science, Google Scholar
M. Katsanikas, P. A. Patsis and A. D. Pinotsis, Int. J. Bifurcation and Chaos 21, 2331 (2011). Link, Web of Science, Google Scholar
G. P. Morriss and L. Rondoni, Austr. J. Phys. 49, 51 (1996). Crossref, Web of Science, Google Scholar
J. Nocedal and S. J. Wright , Numerical Optimization ( Springer-Verlag , NY , 2006 ) . Google Scholar
J. M. Ortega and W. C. Rheinboldt , Iterative Solution of Nonlinear Equations in Several Variables , Society for Industrial and Applied Mathematics ( SIAM , Philadelphia, PA, USA , 2000 ) . Crossref, Google Scholar
K. E. Parsopoulos and M. N. Vrahatis, Natural Comput. 1, 235 (2002), DOI: 10.1023/A:1016568309421. Crossref, Google Scholar
K. E. Parsopoulos and M. N. Vrahatis, Computing periodic orbits of nondifferentiable/discontinuous mappings through particle swarm optimization, Proc. IEEE 2003 Swarm Intelligence Symposium (SIS 2003) (IEEE Press, 2003) pp. 34–41. Google Scholar
K. E. Parsopoulos and M. N. Vrahatis, IEEE Trans. Evolut. Comput. 8, 211 (2004), DOI: 10.1109/TEVC.2004.826076. Crossref, Web of Science, Google Scholar
K. E. Parsopoulos, V. C. Georgopoulos and M. N. Vrahatis, A technique for the visualization of population-based algorithms, Proc. IEEE 2008 Congress on Evolutionary Computation (CEC 2008) (IEEE Press, 2008) pp. 1694–1701. Google Scholar
K. E. Parsopoulos and M. N. Vrahatis , Particle Swarm Optimization and Intelligence: Advances and Applications , Information Science Reference ( IGI Global , Hershey PA, U.S.A. , 2010 ) . Crossref, Google Scholar
E. A. Perdios, V. S. Kalantonis and M. N. Vrahatis, Celest. Mech. Dyn. Astron. 84, 231 (2002), DOI: 10.1023/A:1020338609583. Crossref, Web of Science, Google Scholar
A. E. Perdiouet al., Comput. Phys. Commun. 148, 227 (2002), DOI: 10.1016/S0010-4655(02)00560-X. Crossref, Web of Science, Google Scholar
A. Perry, Operat. Res. 26, 1073 (1978), DOI: 10.1287/opre.26.6.1073. Crossref, Web of Science, Google Scholar
E. Polak and G. Ribière, Revue Française d'Informatique et de Recherche Opérationnelle 3, 35 (1969). Web of Science, Google Scholar
W. H. Press et al. , Numerical Recipes 3rd Edition: The Art of Scientific Computing , 3rd edn. ( Cambridge University Press , New York, NY, USA , 2007 ) . Google Scholar
K. A. Sikorski , Optimal Solution of Nonlinear Equations ( Oxford University Press, Inc. , NY, USA , 2001 ) . Crossref, Google Scholar
M. N. Vrahatis, J. Comput. Phys. 119, 105 (1995), DOI: 10.1006/jcph.1995.1119. Crossref, Web of Science, Google Scholar
M. N. Vrahatis, T. C. Bountis and M. Kollmann, Int. J. Bifurcation and Chaos 6, 1425 (1996). Link, Web of Science, Google Scholar
M. N. Vrahatis, H. Isliker and T. C. Bountis, Int. J. Bifurcation and Chaos 7, 2707 (1997). Link, Web of Science, Google Scholar
M. N. Vrahatiset al., J. Comput. Appl. Math. 114, 367 (2000), DOI: 10.1016/S0377-0427(99)00276-9. Crossref, Web of Science, Google Scholar
M. N. Vrahatiset al., Comput. Phys. Commun. 138, 53 (2001), DOI: 10.1016/S0010-4655(01)00190-4. Crossref, Web of Science, Google Scholar
M. N. Vrahatis, O. Ragos and G. S. Androulakis, Nonlin. Anal.: Th. Meth. Appl. 47, 3449 (2001). Crossref, Web of Science, Google Scholar
M. N. Vrahatis, G. A. Tsirogiannis and E. C. Laskari, Math. Comput. Model. 51, 239 (2010), DOI: 10.1016/j.mcm.2009.08.011. Crossref, Web of Science, Google Scholar
P. Wolfe, SIAM Rev. 11, 226 (1969), DOI: 10.1137/1011036. Crossref, Web of Science, Google Scholar
P. Wolfe, SIAM Rev. 13, 185 (1971), DOI: 10.1137/1013035. Crossref, Web of Science, Google Scholar