lines 7-61 of file: example/general/mul_level.cpp

{xrst_begin mul_level.cpp}
{xrst_spell
   adouble
   dx
}

Multiple Level of AD: Example and Test
######################################

See Also
********
:ref:`base2ad.cpp-name`

Purpose
*******
In this example, we use ``AD< AD<double> >`` (level two taping),
the compute values of the function :math:`f : \B{R}^n \rightarrow \B{R}` where

.. math::

   f(x) = \frac{1}{2} \left( x_0^2 + \cdots + x_{n-1}^2 \right)

We then use ``AD<double>`` (level one taping) to compute
the directional derivative

.. math::

   f^{(1)} (x) * v  = x_0 v_0 + \cdots + x_{n-1} v_{n-1}

where :math:`v \in \B{R}^n`.
We then use ``double`` (no taping) to compute

.. math::

   \frac{d}{dx} \left[ f^{(1)} (x) * v \right] = v

This is only meant as an example of multiple levels of taping.
The example :ref:`hes_times_dir.cpp-name` computes the same value more
efficiently by using the identity:

.. math::

   \frac{d}{dx} \left[ f^{(1)} (x) * v \right] = f^{(2)} (x) * v

The example :ref:`mul_level_adolc.cpp-name` computes the same values using
Adolc's type ``adouble`` and CppAD's type ``AD<adouble>`` .

Source
******
{xrst_literal
   // BEGIN C++
   // END C++
}

{xrst_end mul_level.cpp}