lines 7-100 of file: example/general/mul_level_ode.cpp

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Taylor's Ode Solver: A Multi-Level AD Example and Test
######################################################

See Also
********
:ref:`taylor_ode.cpp-name` , :ref:`base2ad.cpp-name` , :ref:`mul_level_adolc_ode.cpp-name`

Purpose
*******
This is a realistic example using
two levels of AD; see :ref:`mul_level-name` .
The first level uses ``AD<double>`` to tape the solution of an
ordinary differential equation.
This solution is then differentiated with respect to a parameter vector.
The second level uses ``AD< AD<double> >``
to take derivatives during the solution of the differential equation.
These derivatives are used in the application
of Taylor's method to the solution of the ODE.

ODE
***
For this example the function
:math:`y : \B{R} \times \B{R}^n \rightarrow \B{R}^n` is defined by
:math:`y(0, x) = 0` and
:math:`\partial_t y(t, x) = g(y, x)` where
:math:`g : \B{R}^n \times \B{R}^n \rightarrow \B{R}^n` is defined by

.. math::

   g(y, x) =
   \left( \begin{array}{c}
         x_0     \\
         x_1 y_0 \\
         \vdots  \\
         x_{n-1} y_{n-2}
   \end{array} \right)

ODE Solution
************
The solution for this example can be calculated by
starting with the first row and then using the solution
for the first row to solve the second and so on.
Doing this we obtain

.. math::

   y(t, x ) =
   \left( \begin{array}{c}
      x_0 t                  \\
      x_1 x_0 t^2 / 2        \\
      \vdots                 \\
      x_{n-1} x_{n-2} \ldots x_0 t^n / n !
   \end{array} \right)

Derivative of ODE Solution
**************************
Differentiating the solution above,
with respect to the parameter vector :math:`x`,
we notice that

.. math::

   \partial_x y(t, x ) =
   \left( \begin{array}{cccc}
   y_0 (t,x) / x_0      & 0                   & \cdots & 0      \\
   y_1 (t,x) / x_0      & y_1 (t,x) / x_1     & 0      & \vdots \\
   \vdots               & \vdots              & \ddots & 0      \\
   y_{n-1} (t,x) / x_0  & y_{n-1} (t,x) / x_1 & \cdots & y_{n-1} (t,x) / x_{n-1}
   \end{array} \right)

Taylor's Method Using AD
************************
We define the function :math:`z(t, x)` by the equation

.. math::

   z ( t , x ) = g[ y ( t , x ) ] = h [ x , y( t , x ) ]

see :ref:`taylor_ode-name` for the method used to compute the
Taylor coefficients w.r.t :math:`t` of :math:`y(t, x)`.

Source
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   // BEGIN C++
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{xrst_end mul_level_ode.cpp}