Advertisement

A Fully Complex-Valued Neural Network for Rapid Solution of Complex-Valued Systems of Linear Equations

  • Lin Xiao
  • Weiwei Meng
  • Rongbo Lu
  • Xi Yang
  • Bolin Liao
  • Lei Ding
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9377)

Abstract

In this paper, online solution of complex-valued systems of linear equations is investigated in the complex domain. Different from the conventional real-valued neural network, which is only designed for real-valued linear equations solving, a fully complex-valued gradient neural network (GNN) is developed for online complex-valued systems of linear equations. The advantages of the proposed complex-valued GNN model decrease the unnecessary complexities in theoretical analysis, real-time computation and related applications. In addition, the theoretical analysis of the fully complex-valued GNN model is presented. Finally, simulative results substantiate the effectiveness of the fully complex-valued GNN model for online solution of the complex-valued systems of linear equations in the complex domain.

Keywords

complex domain simulation verification complex-valued linear system neural network 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Day, D., Heroux, M.A.: Solving Complex-Valued Linear Systems via Equivalent Real Formulations. SIAM J. Sci. Comput. 23, 480–498 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Owe, A., Andrey, K.: Real Valued Iterative Methods for Solving Complex Symmetric Linear Systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, X., Song, Q.: Global Stability of Complex-Valued Neural Networks with Both Leakage Time Delay and Discrete Time Delay on Time Scales. Neurocomputing 121, 254–264 (2013)CrossRefGoogle Scholar
  4. 4.
    Subramanian, K., Savitha, R., Suresh, S.: A Complex-Valued Neuro-Fuzzy Inference System and Its Learning Mechanism. Neurocomputing 123, 110–120 (2014)CrossRefGoogle Scholar
  5. 5.
    Venkatesh Babu, R., Suresh, S., Savitha, R.: Human Action Recognition Using a Fast Learning Fully Complex-Valued Classifier. Neurocomputing 89, 202–212 (2012)CrossRefGoogle Scholar
  6. 6.
    Durán-Díaz, I., Cruces, S., Sarmiento-Vega, M.A., Aguilera-Bonet, P.: Cyclic maximization of Non-Gaussianity for Blind Signal Extraction of Complex-Valued Sources. Neurocomputing 74, 2867–2873 (2011)CrossRefGoogle Scholar
  7. 7.
    Zhang, Y., Chen, Z., Chen, K.: Convergence Properties Analysis of Gradient Neural Network for Solving Online Linear Equations. Acta Automatica Sinica 35, 1136–1139 (2009)Google Scholar
  8. 8.
    Yi, C., Zhang, Y.: Analogue Recurrent Neural Network for Linear Algebraic Equation Solving. Electron. Lett. 44, 1078–1079 (2008)CrossRefGoogle Scholar
  9. 9.
    Zhang, Y., Chen, K.: Global Exponential Convergence and Stability of Wang Neural Network for Solving Online Linear Equations. Electron. Lett. 44, 145–146 (2008)CrossRefGoogle Scholar
  10. 10.
    Liao, B., Zhang, Y.: Different Complex ZFs Leading to Different Complex ZNN Models for Time-Varying Complex Generalized Inverse Matrices. IEEE Trans. Neural Netw. Learning Syst. 25, 1621–1631 (2014)CrossRefGoogle Scholar
  11. 11.
    Liao, W., Wang, J., Wang, J.: A Recurrent Neural Network for Solving Complex-Valued Quadratic Programming Problems With Equality Constraints. In: Tan, Y., Shi, Y., Tan, K.C. (eds.) ICSI 2010, Part II. LNCS, vol. 6146, pp. 321–326. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Guo, D., Yi, C., Zhang, Y.: Zhang Neural Network Versus Gradient-based Neural Network for Time-Varying Linear Matrix Equation Solving. Neurocomputing 74, 3708–3712 (2011)CrossRefGoogle Scholar
  14. 14.
    Zhang, Y., Shi, Y., Chen, K., Wang, C.: Global Exponential Convergence and Stability of Gradient-based Neural Network for Online Matrix Inversion. Appl. Math. Comput. 215, 1301–1306 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Zhang, Y., Ge, S.S.: Design and Analysis of a General Recurrent Neural Network Model for Time-Varying Matrix Inversion. IEEE Trans. Neural Netw. 16, 1477–1490 (2005)CrossRefGoogle Scholar
  16. 16.
    Zhang, Y., Ke, Z., Xu, P., Yi, C.: Time-varying Square Roots Finding via Zhang Dynamics Versus Gradient Dynamics and the Former’s Link and New Explanation to Newton-Raphson Iteration. Inform. Process. Lett. 110, 1103–1109 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Xiao, L., Zhang, Y.: Two New Types of Zhang Neural Networks Solving Systems of Time-Varying Nonlinear Inequalities. IEEE Trans. Circuits Syst. I 59, 2363–2373 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Xiao, L., Zhang, Y.: From Different Zhang Functions to Various ZNN Models Accelerated To Finite-Time Convergence for Time-Varying Linear Matrix Equation. Neural Process. Lett. 39, 309–326 (2014)CrossRefGoogle Scholar
  19. 19.
    Xiao, L., Lu, R.: Finite-time Solution to Nonlinear Equation Using Recurrent Neural Dynamics with a Specially-Constructed Activation Function. Neurocomputing 151, 246–251 (2015)CrossRefGoogle Scholar
  20. 20.
    Xiao, L.: A Finite-Time Convergent Neural Dynamics for Online Solution of Time-Varying Linear Complex Matrix Equation. Neurocomputing 167, 254–259 (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

<SimplePara><Emphasis Type="Bold">Open Access</Emphasis> This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. </SimplePara> <SimplePara>The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.</SimplePara>

Authors and Affiliations

  • Lin Xiao
    • 1
  • Weiwei Meng
    • 2
  • Rongbo Lu
    • 1
  • Xi Yang
    • 1
  • Bolin Liao
    • 1
  • Lei Ding
    • 1
  1. 1.College of Information Science and EngineeringJishou UniversityJishouChina
  2. 2.Department of Computer and Information SciencesDelaware State UniversityDoverUSA

Personalised recommendations