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{xrst_begin exp_eps_for0}

exp_eps: Operation Sequence and Zero Order Forward Sweep
########################################################

Mathematical Form
*****************
Suppose that we use the algorithm :ref:`exp_eps.hpp-name`
to compute ``exp_eps`` ( *x* , *epsilon* )
with *x* is equal to .5
and *epsilon* is equal to .2.
For this case,
the mathematical form for the operation sequence
corresponding to the ``exp_eps`` is

.. math::

   f( x , \varepsilon ) = 1 + x + x^2 / 2

Note that, for these particular values of *x* and *epsilon* ,
this is the same as the mathematical form for
:ref:`exp_2<exp_2_for0@Mathematical Form>` .

Operation Sequence
******************
We consider the
:ref:`operation sequence<glossary@Operation@Sequence>`
corresponding to the algorithm :ref:`exp_eps.hpp-name`
with the argument *x* is equal to .5
and *epsilon* is equal to .2.

Variable
========
We refer to values that depend on the input variables
*x* and *epsilon* as variables.

Parameter
=========
We refer to values that do not depend on the input variables
*x* or *epsilon* as parameters.
Operations where the result is a parameter are not included
in the zero order sweep below.

Index
=====
The Index column contains the index in the operation sequence
of the corresponding atomic operation and variable.
A Forward sweep starts with the first operation
and ends with the last.

Code
====
The Code column contains the C++ source code corresponding
to the corresponding atomic operation in the sequence.

Operation
=========
The Operation column contains the
mathematical function corresponding to each atomic operation in the sequence.

Zero Order
==========
The Zero Order column contains the
:ref:`zero order derivative<exp_2_for0@Zero Order Expansion>`
for the corresponding variable in the operation sequence.
Forward mode refers to the fact that
these coefficients are computed in the same order as the original algorithm;
i.e., in order of increasing index.

Sweep
=====

.. csv-table::
   :widths: auto

   **Index**,,**Code**,,**Operation**,,**Zero Order**
   1,,``abs_x = x;``,,:math:`v_1 = x`,,:math:`v_1^{(0)} = 0.5`
   2,,``temp  = term * abs_x;``,,:math:`v_2 = 1 * v_1`,,:math:`v_2^{(0)} = 0.5`
   3,,``term = temp / Type(k);``,,:math:`v_3 = v_2 / 1`,,:math:`v_3^{(0)} = 0.5`
   4,,``sum  = sum + term;``,,:math:`v_4 = 1 + v_3`,,:math:`v_4^{(0)} = 1.5`
   5,,``temp  = term * abs_x;``,,:math:`v_5 = v_3 * v_1`,,:math:`v_5^{(0)} = 0.25`
   6,,``term = temp / Type(k);``,,:math:`v_6 = v_5 / 2`,,:math:`v_6^{(0)} = 0.125`
   7,,``sum  = sum + term;``,,:math:`v_7 = v_4 + v_6`,,:math:`v_7^{(0)} = 1.625`

Return Value
************
The return value for this case is

.. math::

   1.625 =
   v_7^{(0)} =
   f ( x^{(0)} , \varepsilon^{(0)} )

Comparisons
***********
If *x* were negative,
or if *epsilon* were a much smaller or much larger value,
the results of the following comparisons could be different:
::

   if( Type(0) > x )
   while(term > epsilon)

This in turn would result in a different operation sequence.
Thus the operation sequence above only corresponds to
:ref:`exp_eps.hpp-name`
for values of *x* and *epsilon* within a certain range.
Note that there is a neighborhood
of :math:`x = 0.5` for which the comparisons would have the
same result and hence the operation sequence would be the same.
{xrst_toc_hidden
   introduction/exp_eps_for0.cpp
}

Verification
************
The file :ref:`exp_eps_for0.cpp-name` contains a routine
that verifies the values computed above.

Exercises
*********

#. Suppose that :math:`x^{(0)} = .1`,
   what is the result of a zero order forward sweep for
   the operation sequence above;
   i.e., what are the corresponding values for
   :math:`v_1^{(0)} , v_2^{(0)} , \ldots , v_7^{(0)}`.
#. Create a modified version of
   :ref:`exp_eps_for0.cpp-name` that verifies the values you obtained
   for the previous exercise.
#. Create and run a main program that reports the result of calling
   the modified version
   of :ref:`exp_eps_for0.cpp-name` in the previous exercise.

{xrst_end exp_eps_for0}