Any ZeaD Formula of Six Instants Having No Quartic or Higher Precision with Proof

In recent years, Zhang et al discretization (ZeaD) as a new class of time-discretization methods has been proposed, named and applied by Zhang et al. Note that ZeaD formulas can accurately discretize Zhang neural networks (i.e., ZNN, or say, Zhang dynamics) models as well as ordinary differential equation systems. In previous work, various ZeaD formulas have been presented and unified, including Euler forward formula as 2-instant ZeaD formula that is convergent with a truncation error being proportional to the first power of sampling period and Taylor-type discretization formula as 4-instant ZeaD formula that is convergent with a truncation error being proportional to the second power of sampling period. During our pursuit of ZeaD formulas that are convergent with a higher precision, we discover that there exists no 6-instant ZeaD formula that is convergent with a quartic (i.e., biquadratic, of degree 4) or higher precision. The truncation error of any 6-instant ZeaD formula is proportional to the third power of sampling period or bigger. The contributions are theoretically proved in this paper as well.