Solving for time-varying and static cube roots in real and complex domains via discrete-time ZD models

Abstract

Different from conventional gradient-based neural dynamics, a special type of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and/or static (or termed, time-invariant) problems. The design of Zhang dynamics (ZD) is based on the elimination of an indefinite error function, instead of the elimination of a square-based positive (or at least lower-bounded) energy function usually associated with gradient dynamics (GD). In this paper, we generalize, propose and investigate the continuous-time ZD model and its discrete-time models in two situations (i.e., the time-derivative of the coefficient being known or unknown) for time-varying cube root finding, including the complex-valued continuous-time ZD model for finding cube roots in complex domain. In addition, to find the static scalar-valued cube root, a simplified continuous-time ZD model and its discrete-time model are generated. By focusing on such a static problem solving, Newton-Raphson iteration is found to be a special case of the discrete-time ZD model by utilizing the linear activation function and fixing the step-size value to be 1. Computer-simulation and testing results demonstrate the efficacy of the proposed ZD models (including real-valued ZD models and complex-valued ZD models) for time-varying and static cube root finding, as well as the link and new explanation to Newton-Raphson iteration.

Appendix

In this appendix, the discrete-time iteration model of the proposed complex-valued CTZD model (4) is developed and investigated for static cube root finding in complex domain, where, for example, let us consider the following static complex-valued cube root problem:

$$ f(z)=z^{3}-i=0\in C. $$

(21)

By following the ZD design method and using the Euler forward-difference rule, we obtain the corresponding complex-valued DTZD model as follows:

$$ z_{k+1}=z_{k}-h\frac{\phi(z_{k}^{3}-i)}{3z_{k}^{2}}, $$

(22)

where h > 0 denotes the step-size (or say, step-length), being the same as before. It is worth pointing out that, similar to Newton fractals generated by solving nonlinear complex-valued equations via Newton-Raphson iteration [36], new fractals can be generated by using the complex-valued DTZD model (22) to solve complex-valued static cube root problem (21). In this appendix, the following hyperbolic-sine activation function (with parameter ξ ≥ 1) (which is newly introduced in the ZD research) is exploited, in addition to the aforementioned linear activation function:

$$ \phi(e)=(\exp(\xi e)-\exp(-\xi e))/2. $$

For illustration and verification, the complex-valued DTZD model (22) with step-size h = 0.2 and using different activation functions is applied to the online solution of (21), and the corresponding numerical results are shown in Fig. 10. From Fig. 10a, c, we see that, by using different activation functions, different fractals are generated and visualized through the rectangle-area solution of nonlinear complex-valued equation (21) via the complex-valued DTZD model (22). In addition, the self-similar properties of the fractals are illustrated in Fig. 10b, d via partially enlarged view. Therefore, generating new fractals can be viewed as one of the interesting applications of the proposed ZD method in related scientific and engineering fields.

Fig. 10

Fractals generated by solving (21) via the complex-valued DTZD model (22) with h = 0.2 and using different activation functions, where the horizontal and vertical axes correspond to the real and imaginary parts of the initial states (i.e., z ₀ = x ₀ + iy ₀), respectively, and the color or gray scale corresponds to the iteration number required by DTZD (22) starting with an initial state $z_0\in\{[-2,2]+i[-2,2]\}$ to solve (21)