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**Complex-valued Zhang neural network for online complex-valued time-varying matrix inversion.**
*(English)*
Zbl 1220.65036

Summary: A new complex-valued recurrent neural network (CVRNN) called complex-valued Zhang neural network (CVZNN) is proposed and simulated to solve the complex-valued time-varying matrix-inversion problems. Such a CVZNN model is designed based on a matrix-valued error function in the complex domain, and utilizes the complex-valued first-order time-derivative information of the complex-valued time-varying matrix for online inversion. Superior to the conventional complex-valued gradient-based neural network (CVGNN) and its related methods, the state matrix of the resultant CVZNN model can globally exponentially converge to the theoretical inverse of the complex-valued time-varying matrix in an error-free manner. Moreover, by exploiting the design parameter \(\gamma >1\), superior convergence can be achieved for the CVZNN model to solve such complex-valued time-varying matrix inversion problems, as compared with the situation without design parameter \(\gamma \) involved (i.e., the situation with \(\gamma =1\)). Computer-simulation results substantiate the theoretical analysis and further demonstrate the efficacy of such a CVZNN model for online complex-valued time-varying matrix inversion.

### MSC:

65F05 | Direct numerical methods for linear systems and matrix inversion |

65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |

### Keywords:

complex-valued recurrent neural network; complex-valued Zhang neural network; complex-valued time-varying matrix inversion; matrix-valued error function; superior convergence; numerical examples
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\textit{Y. Zhang} et al., Appl. Math. Comput. 217, No. 24, 10066--10073 (2011; Zbl 1220.65036)

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### References:

[1] | Zhang, Y.; Wang, J., A dual neural network for convex quadratic programming subject to linear equality and inequality constraints, Phys. lett. A, 298, 271-278, (2002) · Zbl 0995.65063 |

[2] | Jwo, D.; Lai, C., Neural network-based GPS GDOP approximation and classification, GPS solutoins, 11, 51-60, (2007) |

[3] | Ge, S.S.; Lee, T.H.; Harris, C.J., Adaptive neural network control of robotic manipulators, (1998), World Scientific London, UK |

[4] | Sturges, R.H., Analog matrix inversion (robot kinematics), IEEE J. robot. autmat., 4, 157-162, (1988) |

[5] | Steriti, R.J.; Fiddy, M.A., Regularized image reconstruction using SVD and a neural network method for matrix inversion, IEEE trans. sign. proc., 41, 3074-3077, (1993) · Zbl 0825.68654 |

[6] | Zhang, Y.; Leithead, W.E., Exploiting hessian matrix and trust-region algorithm in hyperparameters estimation of Gaussian process, Appl. math. comput., 171, 1264-1281, (2005) · Zbl 1097.65019 |

[7] | Sarkar, T.; Siarkiewicz, K.; Stratton, R., Survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems, IEEE trans. antenn. propag., 29, 847-856, (1981) · Zbl 0947.65501 |

[8] | Wang, J., A recurrent neural network for real-time matrix inversion, Appl. math. comput., 55, 89-100, (1993) · Zbl 0772.65015 |

[9] | Song, J.; Yam, Y., Complex recurrent neural network for computing the inverse and psedo-inverse of the complex matrix, Appl. math. comput., 93, 195-205, (1998) · Zbl 0943.65040 |

[10] | 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) · Zbl 1194.65056 |

[11] | Zhang, Y.; Ma, W.; Cai, B., From Zhang neural network to Newton iteration for matrix inversion, IEEE trans. circuit syst. I, 56, 7, 1405-1415, (2009) |

[12] | 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, 6, 1477-1490, (2005) |

[13] | Zhang, Y.; Chen, K.; Tan, H.Z., Performance analysis of gradient neural network exploited for online time-varying matrix inversion, IEEE trans. automat. contr., 54, 8, 1940-1945, (2009) · Zbl 1367.92010 |

[14] | Zhang, Y.; Jiang, D.; Wang, J., A recurrent neural network for solving Sylvester equation with time-varying coefficients, IEEE trans. neural netw., 13, 5, 1053-1063, (2002) |

[15] | Zhang, Y.; Li, Z., Zhang neural network for online solution of time-varying convex quadratic program subject to time-varying linear-equality constraints, Phys. lett. A, 373, 1639-1643, (2009) · Zbl 1229.92008 |

[16] | Zhang, Y.; Yi, C.; Ma, W., Simulation and verification of Zhang neural network for online time-varying matrix inversion, Simul. model. pract. theory, 17, 10, 1603-1617, (2009) |

[17] | Mead, C., Analog VLSI and neural systems, (1989), Addison-Wesley Reading, MA · Zbl 0715.68002 |

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