lines 6-79 of file: example/atomic_four/lin_ode/forward.cpp

{xrst_begin atomic_four_lin_ode_forward.cpp}
{xrst_spell
   cccc
}

Atomic Linear ODE Forward Mode: Example and Test
################################################

Purpose
*******
This example demonstrates using forward mode with
the :ref:`atomic_four_lin_ode-name` class.

f(u)
****
For this example, the function :math:`f(u) = z(r, u)` where
:math:`z(t, u)` solves the following ODE

.. math::

   z_t (t, u) =
   \left( \begin{array}{cccc}
   0   & 0  & 0    & 0   \\
   u_4 & 0  & 0    & 0   \\
   0   & u_5 & 0   & 0   \\
   0   & 0   & u_6 & 0   \\
   \end{array} \right)
   z(t, u)
   \W{,}
   z(0, u) =
   \left( \begin{array}{c}
   u_0 \\
   u_1 \\
   u_2 \\
   u_3 \\
   \end{array} \right)

z(t, u)
*******
The actual solution to this ODE is

.. math::

   z(t, u) =
   \left( \begin{array}{l}
   u_0  \\
   u_1 + u_4 u_0 t \\
   u_2 + u_5 u_1 t + u_5 u_4 u_0 t^2 / 2  \\
   u_3 + u_6 u_2 t + u_6 u_5 u_1 t^2 / 2 + u_6 u_5 u_4 u_0 t^3 / 6
   \end{array} \right)

g(u)
****
We define :math:`g(u) = \partial_{u0} f(u)`.
It follows that

.. math::

   g (u) =
   \left( \begin{array}{l}
   1  \\
   u_4 r \\
   u_5 u_4 r^2 / 2  \\
   u_6 u_5 u_4 r^3 / 6
   \end{array} \right)

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

{xrst_end atomic_four_lin_ode_forward.cpp}