Numerical experiments and verifications of ZFD formula 4NgSFD for first-order derivative discretization and approximation

In order to discretize the continuous-time neural dynamics or control system effectively and to achieve high computational accuracy in approximating the first-order derivative of a target function, a 1-node-ahead differentiation formula within Zhang finite difference (ZFD) framework labeled as 4-node g-square finite difference (4NgSFD) is presented in this paper. The presented formula, which is based on Taylor expansion instead of polynomial-interpolation theory, achieves an error of O(g
²
), where g denotes the sampling period. Moreover, different from the traditional central/backward numerical differentiation formula, the ZFD formula 4NgSFD makes use of an ahead node, a target node and two back nodes in the approximation of the first-order derivative at the target node. Numerical experimental results further substantiate the efficacy for the first-order derivative approximation with the error being O(g
²
) by using the presented ZFD formula 4NgSFD.