Solution of nonlinear equations by continuous- and discrete-time Zhang dynamics and more importantly their links to Newton iteration

Different from gradient-based dynamics (GD), a special class of neural dynamics has been found, developed, generalized and investigated by Zhang et al, e.g., for online solution of time-varying and/or static nonlinear equations. The resultant Zhang dynamics (ZD) is designed 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 GD and/or Hopfield-type neural newtorks). In this paper, discrete-time ZD models (different from our previous research on continuous-time ZD models) are developed and investigated. In terms of nonlinear-equations solving, the Newton iteration (also termed, Newton-Raphson iteration) is found to be a special case of the ZD models (by focusing on the static-problem solving, utilizing the linear activation function and fixing the step-size to be 1). Noticing this new relation and explanation, we conduct computer-simulation, testing and comparisons for such discrete-time ZD models (including Newton iteration) for nonlinear equations solving. The numerical results substantiate the theoretical analysis, explanation, unification and efficacy of the discrete-time ZD models on nonlinear equations solving.