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

Different from conventional gradient-based neural dynamics, a special class of neural dynamics have been proposed by Zhang et al. since 12 March 2001 for online solution of time-varying and static (or termed, time-invariant) problems (e.g., nonlinear equations). 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) and/or Hopfield-type neural networks. In this paper, we generalize, develop, investigate and compare the continuous-time ZD (CTZD) and GD models for online solution of time-varying and static square roots. In addition, a simplified continuous-time ZD (S-CTZD) and discrete-time ZD (DTZD) models are generated for static scalar-valued square roots finding. In terms of such scalar square roots finding problem, the Newton iteration (also termed, Newton–Raphson iteration) is found to be a special case of the DTZD models (by focusing on the static-problem solving, utilizing the linear activation function and fixing the step-size to be 1). Computer-simulation results via a power-sigmoid activation function further demonstrate the efficacy of the ZD solvers for online scalar (time-varying and static) square roots finding, in addition to the DTZD's link and new explanation to Newton–Raphson iteration.

Research highlights
► CTZD and GD models are presented for time-varying square root finding.
► S-CTZD model is presented for time-invariant square root finding.
► DTZD model is generalized for time-invariant square root finding.
► The link of the ZD models to Newton–Raphson iteration is shown.
► ZD, GD and Newton–Raphson iteration are simulated for square root finding.