Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Yang, L., Zhu, Z., Wang, Y.: Exact Solutions of Nonlinear Equations. Physics Letters A 260, 55–59 (1999)
Babolian, E., Biazar, J.: Solution of Nonlinear Equations by Modified Adomian Decomposition Method. Applied Mathematics and Computation 132, 167–172 (2002)
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)
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)
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)
Wang, J.: A Recurrent Neural Network for Real-Time Matrix Inversion. Applied Mathematics and Computation 55, 89–100 (1993)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Cichocki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. Wiley, Chichester (1993)
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)
Mathews, J.H., Fink, K.D.: Numerical Methods Using MATLAB, 4th edn. Pretice Hall, New Jersey (2004)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhang, Y., Yi, C., Ma, W. (2008). Comparison on Gradient-Based Neural Dynamics and Zhang Neural Dynamics for Online Solution of Nonlinear Equations. In: Kang, L., Cai, Z., Yan, X., Liu, Y. (eds) Advances in Computation and Intelligence. ISICA 2008. Lecture Notes in Computer Science, vol 5370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92137-0_30
Download citation
DOI: https://doi.org/10.1007/978-3-540-92137-0_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92136-3
Online ISBN: 978-3-540-92137-0
eBook Packages: Computer ScienceComputer Science (R0)