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.