Zhang neural network without using time-derivative information for constant and time-varying matrix inversion

Time-varying square roots finding via Zhang dynamics versus gradient dynamics and the former's link and new explanation to Newton–Raphson iteration

“…By following Zhang et al's neural-dynamic design method [13][14][15][16]21,22,[24][25][26], to monitor and control the time-varying square roots finding process, we could firstly define an indefinite error function (of which the word "indefinite" means that such an error function can be positive, zero, negative or even lower-unbounded):…”

“…Generally speaking, any monotonically-increasing odd activation function φ(·) can be used for the construction of the neural dynamics. Since March 2001 [21], we have introduced and used five types of activation functions (i.e., linear activation function, power activation function, power-sum activation function, sigmoid activation function and power-sigmoid activation function) for the proposed ZD models (for more details, see [13][14][15][16][17][18]22,[24][25][26]). Moreover, similar to usual neural-dynamic approaches, design parameter γ in ZD (2) [and hereafter in GD (3)], being the reciprocal of a capacitance parameter in the hardware implementation, should be set as large as hardware permits (e.g., in analog circuits or VLSI [11,12]) or selected appropriately (e.g., between 10 3 and 10 8 ) for experimental and/or simulative purposes.…”

Continuous and discrete time Zhang dynamics for time-varying 4th root finding

“…Following Zhang et al's neural-dynamic design method [11,12,16,17,[19][20][21], to monitor and control the process of time-varying 4th root finding, we could firstly define an indefinite error function (of which the word "indefinite" means that such an error function can be positive, zero, negative or even lowerunbounded):…”

“…Different from the gradient-based neural-dynamic approach, a special kind of neural dynamics has been formally proposed by Zhang et al [11-14, 16, 17] for time-varying and/or static problems solving, such as Sylvester equation solving [9], time-varying convex quadratic program [12], nonlinear equations [18], matrix inversion [19][20][21]. Besides, the proposed Zhang dynamics (ZD) is designed based on an indefinite error-function instead of a square-based positive (or at least lower-bounded) energy-function usually associated with gradient-based models or Lyapunov design-analysis methods [13].…”

Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation

“…However, it may not be efficient enough for most numerical algorithms because of their serial-processing nature performed on digital computers [4]. In recent years, due to the in-depth research in neural networks, the dynamic system approach using recurrent neural networks is one of the important parallel-processing methods for solving optimization and equation problems [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. For example, many studies have been reported on the real-time solution of algebraic equations, including matrix inversion and Sylvester equation [5-8, 11, 14, 18, 19].…”