Comparison on Gradient-Based Neural Dynamics and Zhang Neural Dynamics for Online Solution of Nonlinear Equations

  • Yunong Zhang
  • Chenfu Yi
  • Weimu Ma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5370)


For online solution of nonlinear equation f(x) = 0, this paper generalizes a special kind of recurrent neural dynamics by using a recent design method proposed by Zhang et al. Different from gradient-based dynamics (GD), the resultant Zhang dynamics (ZD) is designed based on the elimination of an indefinite error-monitoring function (instead of the elimination of a square-based positive error-function usually associated with GD). For comparative purposes, the gradient-based dynamics is also developed and exploited for solving online such a nonlinear equation f(x) = 0. Computer-simulation results via power-sigmoid activation functions substantiate further the theoretical analysis and efficacy of Zhang neural dynamics on nonlinear equations solving.


Recurrent neural networks neural dynamics nonlinear equations activation functions exponential convergence 


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  1. 1.
    Yang, L., Zhu, Z., Wang, Y.: Exact Solutions of Nonlinear Equations. Physics Letters A 260, 55–59 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babolian, E., Biazar, J.: Solution of Nonlinear Equations by Modified Adomian Decomposition Method. Applied Mathematics and Computation 132, 167–172 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Babolian, E., Biazar, J., Vahidi, A.R.: Solution of a System of Nonlinear Equations by Adomian Decomposition Method. Applied Mathematics and Computation 150, 847–854 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zhang, Y., Leithead, W.E., Leith, D.J.: Time-Series Gaussian Process Regression Based on Toeplitz Computation of O(N 2) Operations and O(N)-level Storage. In: Proceedings of IEEE Conference on Decision and Control, pp. 3711–3716 (2005)Google Scholar
  5. 5.
    Jang, J., Lee, S., Shin, S.: An Optimization Network for Matrix Inversion. In: Anderson, D.Z. (ed.) Neural Information Processing Systems, pp. 397–401. American Institute of Physics, New York (1998)Google Scholar
  6. 6.
    Wang, J.: A Recurrent Neural Network for Real-Time Matrix Inversion. Applied Mathematics and Computation 55, 89–100 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Zhang, Y.: Revisit the Analog Computer and Gradient-Based Neural System for Matrix Inversion. In: Proceedings of IEEE International Symposium on Intelligent Control, pp. 1411–1416 (2005)Google Scholar
  8. 8.
    Zhang, Y.: Towards Piecewise-Linear Primal Neural Networks for Optimization and Redundant Robotics. In: Proceedings of IEEE International Conference on Networking, Sensing and Control, pp. 374–379 (2006)Google Scholar
  9. 9.
    Zhang, Y., Jiang, D., Wang, J.: A Recurrent Neural Network for Solving Sylvester Equation with Time-Varying Coefficients. IEEE Transactions on Neural Networks 13, 1053–1063 (2002)CrossRefGoogle Scholar
  10. 10.
    Zhang, Y., Ge, S.S.: Design and Analysis of a General Recurrent Neural Network Model for Time-Varying Matrix Inversion. IEEE Transactions on Neural Networks 16, 1477–1490 (2005)CrossRefGoogle Scholar
  11. 11.
    Zhang, Y., Wang, J.: Global Exponential Stability of Recurrent Neural Networks for Synthesizing Linear Feedback Control Systems via Pole Assignment. IEEE Transactions on Neural Networks 13, 633–644 (2002)CrossRefGoogle Scholar
  12. 12.
    Zhang, Y., Wang, J.: A Dual Neural Network for Convex Quadratic Programming Subject to Linear Equality and Inequality Constraints. Physics Letters A 298, 271–278 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhang, Y.: A Set of Nonlinear Equations and Inequalities Arising in Robotics and its Online Solution via a Primal Neural Network. Neurocomputing 70, 513–524 (2006)CrossRefGoogle Scholar
  14. 14.
    Steriti, R.J., Fiddy, M.A.: Regularized Image Reconstruction Using SVD and a Neural Network Method for Matrix Inversion. IEEE Transactions on Signal Processing 41, 3074–3077 (1993)CrossRefzbMATHGoogle Scholar
  15. 15.
    Tank, D., Hopfield, J.: Simple Neural Optimization Networks: an A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit. IEEE Transactions on Circuits and Systems 33, 533–541 (1986)CrossRefGoogle Scholar
  16. 16.
    Cichocki, A., Unbehauen, R.: Neural Network for Solving Systems of Linear Equations and Related Problems. IEEE Transactions on Circuits and Systems 39, 124–138 (1992)CrossRefzbMATHGoogle Scholar
  17. 17.
    Cichocki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. Wiley, Chichester (1993)zbMATHGoogle Scholar
  18. 18.
    Manherz, R.K., Jordan, B.W., Hakimi, S.L.: Analog Methods for Computation of the Generalized Inverse. IEEE Transactions on Automatic Control 13, 582–585 (1968)CrossRefGoogle Scholar
  19. 19.
    Mathews, J.H., Fink, K.D.: Numerical Methods Using MATLAB, 4th edn. Pretice Hall, New Jersey (2004)Google Scholar
  20. 20.
    Zhang, Y.: Dual Neural Networks: Design, Analysis, and Application to Redundant Robotics. In: Kang, G.B. (ed.) Progress in Neurocomputing Research, pp. 41–81. Nova Science Publishers, New York (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Yunong Zhang
    • 1
  • Chenfu Yi
    • 1
  • Weimu Ma
    • 1
  1. 1.School of Information Science and TechnologySun Yat-Sen UniversityGuangzhouChina

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