==================
Forward Difference
==================
:doc:`/WorkProcessClassifiers/LocalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

The basic formula  for computing the *Forward 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}

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` 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}^+`                   |            |             |         |
+------------------------+------------------------------------------------+------------+-------------+---------+

.. rubric:: Output Parameters

+----------------------------+---------------+------------+--------------------------------------------------------------------------------+---------+
| Parameter                  | Type          | Constraint | Description                                                                    | Remarks |
+============================+===============+============+================================================================================+=========+
| :math:`\hat{Y}`            | A data vector |            | First derivative of :math:`Y`          with respect to time :math:`t`          |         |
+----------------------------+---------------+------------+--------------------------------------------------------------------------------+---------+

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/MathematicalComputation/NumericalDifferentiation/ComputingFirstOrderDerivative/ComputingFirstOrderDerivativeWithForwardDifference/index`

.. rubric:: References

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