Central Difference

Local Algorithm - One-Dimensional Algorithm

The basic formula for computing the Central Difference derivative at the point $t_i$ is stated as follows:

$$Y'(t_i) = \frac{Y(t_i+h_i) - Y(t_i-h_{i-1})}{h_{i-1} + h_i}$$

where $Y(t_i)$ is the function value at time point $t_i$, $Y'(t_i)$ is the first derivative at time point $t_i$, $h_{i-1}$ is the step size between the point $t_i$ and the preceding time point, and $h_{i}$ is the step size between the point $t_i$ and the next time point.

Input Parameters

Parameter	Type	Constraint	Description	Remarks
$Y$	A continuous function or a data vector
$h_i$	$h_i \in \mathbb{R}^+$
$h_{i-1}$	$h_{i-1} \in \mathbb{R}^+$

Output Parameters

Parameter	Type	Constraint	Description	Remarks
$\hat{Y}$	A data vector		first derivative of $Y$ with respect to $t$

Single Steps using the Algorithm

• Computing the First Order Derivative with Central Difference

References

• R. L. Burden and J. D. Faires, Numerical Analysis, Fifth Edition, PWS Publishing Co. Boston, MA, 1993.