ABSTRACT
Different from the conventional gradient-based neural dynamics, a special class of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and constant problems (e.g., nonlinear equations). The design of zeroing dynamics (ZD) is based on the elimination of an indefinite error function, instead of the elimination of a square-based nonnegative, or at least lowerbounded energy function usually associated with the design of gradient dynamics (GD). In this chapter, we propose, generalize, develop, investigate, and compare the continuous-time ZD (CTZD) and GD models for online solution of time-varying and constant square roots. In addition, a simplified continuous-time ZD (S-CTZD) model and a discrete-time ZD (DTZD) model are generated for constant scalar-valued square root finding. In terms of constant square root finding, Newton iteration is found to be a special case of the S-DTZD model (by focusing on the constant problem solving, utilizing the linear activation function, and fixing the step size to be 1). Computer-simulation and numerical-experiment results via a power-sigmoid activation function further illustrate the efficacy of the ZD models for online time-varying and constant square roots finding, in addition to the link and new explanation to Newton iteration.
