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lines 6-71 of file: example/atomic_four/lin_ode/rev_depend.cpp
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{xrst_begin atomic_four_lin_ode_rev_depend.cpp}
{xrst_spell
   cccc
}

Atomic Linear ODE Reverse Dependency Analysis: Example and Test
###############################################################

Purpose
*******
This example demonstrates calculating reverse dependency with
the :ref:`atomic_four_lin_ode-name` class; see
:ref:`atomic_four_lin_ode_rev_depend.hpp-name` .

y(t, x)
*******
We are given a positive integer m and define
:math:`y : \B{R} \times \B{R}^m \rightarrow \B{R}^m` by

.. math::

   y(t, x) = \prod_{i=0}^m x_i t^i / i !

It follows that

.. math::

   \partial_t y_i (t, x) = \left \{ \begin{array}{ll}
      0                   & \R{if} \; i = 0 \\
      x_i y_{i-1} (t, x)  & \R{otherwise}
   \end{array} \right .
   \\
   \partial_t y (t, x) =
   \left( \begin{array}{cccc}
   0        & 0        & \cdots & 0        \\
   x_1      & 0        & \cdots & 0        \\
   0        & x_2      &        & 0        \\
   \vdots   &          & \ddots & \vdots   \\
   0        & 0        & \cdots & x_m
   \end{array} \right)
   y (t, x)
   \W{,}
   y (0, x) =
   \left( \begin{array}{c}
   x_0      \\
   0        \\
   \vdots   \\
   0        \\
   \end{array} \right)

Problem Parameters
******************
The following problem parameters can be changed:
{xrst_literal
   // BEGIN_PROBLEM_PARAMETERS
   // END_PROBLEM_PARAMETERS
}

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

{xrst_end atomic_four_lin_ode_rev_depend.cpp}