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# ifndef CPPAD_INTRODUCTION_EXP_2_HPP
# define CPPAD_INTRODUCTION_EXP_2_HPP
// SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
// SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
// SPDX-FileContributor: 2003-24 Bradley M. Bell
// ----------------------------------------------------------------------------
/*
{xrst_begin exp_2}
{xrst_spell
apx
yyyymmdd
}
Second Order Exponential Approximation
######################################
Syntax
******
| # ``include`` ``"exp_2.hpp"``
| *y* = ``exp_2`` ( *x* )
Purpose
*******
This is a simple example algorithm that is used to demonstrate
Algorithmic Differentiation
(see :ref:`exp_eps-name` for a more complex example).
Mathematical Form
*****************
The exponential function can be defined by
.. math::
\exp (x) = 1 + x^1 / 1 ! + x^2 / 2 ! + \cdots
The second order approximation for the exponential function is
.. math::
{\rm exp\_2} (x) = 1 + x + x^2 / 2
include
*******
The include command in the syntax is relative to
``cppad-`` *yyyymmdd* / ``introduction/exp_apx``
where ``cppad-`` *yyyymmdd* is the distribution directory
created during the beginning steps of the
:ref:`installation<Install-name>` of CppAD.
x
*
The argument *x* has prototype
``const`` *Type* & *x*
(see *Type* below).
It specifies the point at which to evaluate the
approximation for the second order exponential approximation.
y
*
The result *y* has prototype
*Type* *y*
It is the value of the exponential function
approximation defined above.
Type
****
If *u* and *v* are *Type* objects and *i*
is an ``int`` :
.. csv-table::
:widths: auto
**Operation**,**Result Type**,**Description**
*Type* ( *i* ),*Type*,construct object with value equal to *i*
*Type u* = *v*,*Type*,construct *u* with value equal to *v*
*u* * *v*,*Type*,result is value of :math:`u * v`
*u* / *v*,*Type*,result is value of :math:`u / v`
*u* + *v*,*Type*,result is value of :math:`u + v`
Contents
********
{xrst_toc_table
introduction/exp_2.xrst
introduction/exp_2_cppad.cpp
}
Implementation
**************
The file :ref:`exp_2.hpp-name`
contains a C++ implementation of this function.
Test
****
The file :ref:`exp_2.cpp-name`
contains a test of this implementation.
Exercises
*********
#. Suppose that we make the call
::
double x = .1;
double y = exp_2(x);
What is the value assigned to
``v1`` , ``v2`` , ... ,``v5`` in :ref:`exp_2.hpp-name` ?
#. Extend the routine ``exp_2.hpp`` to
a routine ``exp_3.hpp`` that computes
.. math::
1 + x^2 / 2 ! + x^3 / 3 !
Do this in a way that only assigns one value to each variable
(as ``exp_2`` does).
#. Suppose that we make the call
::
double x = .5;
double y = exp_3(x);
using ``exp_3`` created in the previous problem.
What is the value assigned to the new variables in ``exp_3``
(variables that are in ``exp_3`` and not in ``exp_2`` ) ?
{xrst_end exp_2}
------------------------------------------------------------------------------
*/
// BEGIN C++
template <class Type>
Type exp_2(const Type &x)
{ Type v1 = x; // v1 = x
Type v2 = Type(1) + v1; // v2 = 1 + x
Type v3 = v1 * v1; // v3 = x^2
Type v4 = v3 / Type(2); // v4 = x^2 / 2
Type v5 = v2 + v4; // v5 = 1 + x + x^2 / 2
return v5; // exp_2(x) = 1 + x + x^2 / 2
}
// END C++
# endif
|
