一点超前数值差分公式的提出、研究与实践

根据数值微分理论,若给定未知目标函数在指定区间上的离散采样点数据,可使用数值差分公式求目标点处的一阶导数近似值。但对于靠近边界的目标点而言,多点中心差分公式可能因单边数据点不足而无法使用。另外,目标函数的一阶导数在目标点处可能发生加速变化,而前(后)向差分公式只考虑了单边数据点,可能无法适应该变化,使导数值误差较大。实际上,针对靠近右边界的目标点,可将后向差分公式在形式上"前移"一点来计算一阶导数,因此,一点超前数值差分公式被提出与研究。计算机数值实验表明:一点超前数值差分公式可使所求目标点一阶导数值具有较高的计算精度。

Proposing,Investigation and Practice on One-Node-Ahead Numerical Differentiation Formulas

Based on the numerical differential theory,it is available to calculate the approximate first derivative of the target-node by using numerical differentiation formulas when the discrete sampling points of the unknown target function on specified interval are given.But for the target-nodes close to the boundary,it may be unable to use the center differentiation formulas involving multiple nodes because of the lack of sampling points on one side of the target-node.Besides,an accelerating change of the first derivative of the target-node may occur in some target functions.However,the use of forward/backward differentiation formulas simply takes the nodes on one side of the target-node into consideration,which probably makes the formulas difficult to adapt to such a change,and thus leads to less accuracy in estimating the first derivative of the target-node.Actually,for the target-nodes close to the right boundary,it is available to move the backward differentiation formulas one node ahead to calculate the first derivatives.Therefore,one-node-ahead numerical differentiation formulas are proposed and investigated.Experimental results verify and show that the first derivatives of the target-nodes with high computational precision can be obtained by using the one-node-ahead numerical differentiation formulas.