Backward DifferenceΒΆ

The basic formula for computing the Backward Difference derivative at the point \(t_i\) is stated as follows:

\[Y(t_i)' = \frac{Y(t_i) - Y(t_i-h_{i-1})}{h_{i-1}}\]

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.

Input Parameters

Parameter Type Constraint Description Remarks
\(Y\) A continuous function or a data vector None None None
\(h_{i-1}\) \(h_{i-1} \in \mathbb{R}^+\) None None None

Output Parameters

Parameter Type Constraint Description Remarks
\(\hat{Y}\) A data vector None First derivative of \(Y\) with respect to \(t\) None

Single Steps using the Algorithm

References

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