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

Abstract

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 $(\mathrm {i}.\mathrm {e}.$, 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 (ie, 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.