=================== Backward Difference =================== :doc:`/WorkProcessClassifiers/LocalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index` The basic formula for computing the *Backward Difference* derivative at the point :math:`t_i` is stated as follows: .. math:: Y(t_i)' = \frac{Y(t_i) - Y(t_i-h_{i-1})}{h_{i-1}} where :math:`Y(t_i)` is the function value at time point :math:`t_i`\ , :math:`Y'(t_i)` is the first derivative at time point :math:`t_i`\ , :math:`h_{i-1}` is the step size between the point :math:`t_i` and the preceding time point. .. rubric:: Input Parameters +----------------------------+----------------------------------------------------+------------+-------------+---------+ | Parameter | Type | Constraint | Description | Remarks | +============================+====================================================+============+=============+=========+ | :math:`Y` | A continuous function or a data vector | | | | +----------------------------+----------------------------------------------------+------------+-------------+---------+ | :math:`h_{i-1}` | :math:`h_{i-1} \in \mathbb{R}^+` | | | | +----------------------------+----------------------------------------------------+------------+-------------+---------+ .. rubric:: Output Parameters +----------------------------+---------------+------------+---------------------------------------------------------------------------+---------+ | Parameter | Type | Constraint | Description | Remarks | +============================+===============+============+===========================================================================+=========+ | :math:`\hat{Y}` | A data vector | | First derivative of :math:`Y` with respect to :math:`t` | | +----------------------------+---------------+------------+---------------------------------------------------------------------------+---------+ .. rubric:: Single Steps using the Algorithm * :doc:`/DataPreprocessing/MathematicalComputation/NumericalDifferentiation/ComputingFirstOrderDerivative/ComputingFirstOrderDerivativeWithBackwardDifference/index` .. rubric:: References - R.\ L. Burden and J. D. Faires, Numerical Analysis, Fifth Edition, PWS Publishing Co. Boston, MA, 1993.