================== Central Difference ================== :doc:/WorkProcessClassifiers/LocalAlgorithm/index - :doc:/WorkProcessClassifiers/OneDimensionalAlgorithm/index The basic formula for computing the *Central Difference* derivative at the point :math:t_i is stated as follows: .. math:: Y'(t_i) = \frac{Y(t_i+h_i) - Y(t_i-h_{i-1})}{h_{i-1} + h_i} 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, and :math:h_{i} is the step size between the point :math:t_i and the next time point. .. rubric:: Input Parameters +----------------------------+----------------------------------------------------+------------+-------------+---------+ | Parameter | Type | Constraint | Description | Remarks | +============================+====================================================+============+=============+=========+ | :math:Y | A continuous function or a data vector | | | | +----------------------------+----------------------------------------------------+------------+-------------+---------+ | :math:h_i | :math:h_i \in \mathbb{R}^+ | | | | +----------------------------+----------------------------------------------------+------------+-------------+---------+ | :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/ComputingFirstOrderDerivativeWithCentralDifference/index .. rubric:: References - R.\ L. Burden and J. D. Faires, Numerical Analysis, Fifth Edition, PWS Publishing Co. Boston, MA, 1993.