lines 71-160 of file: xrst/algorithm.xrst

{xrst_begin an_ode}
{xrst_spell
   kutta
   llll
   runge
}

An ODE Solution
###############

ODE
***

.. math::

   \begin{array}{llll}
   y_i '(t) & = & x_0                          & \mbox{for} \; i = 0 \\
   y_i '(t) & = & \sum_{j=1}^i x_j y_{i-1} (t) & \mbox{for} \; i > 0  \\
   \end{array}

Parameter Vector
================
We refer to :math:`x` as the parameter vector in the ode.

Initial Value
*************

.. math::

   y_i (0) = 0  \; \mbox{for all} \; i

Solution
********
This initial value problem has the following analytic solution
(which can be used to check function values and derivatives):

.. math::

   \begin{array}{llll}
   y_0 (t) & = & x_0 t \\
   y_1 (t) & = & x_0 x_1 t^2 / 2 ! \\
   y_2 (t) & = & x_0 x_1 x_2 t^3 / 3 !
   \end{array}

.. math ::

   y_i (t) =
      \frac{t^{i+1}}{(i+1)!} \prod_{j=0}^i x_j \; \; \mbox{for all} \; i

Algorithm
*********
The :ref:`rk4_step-name` method is used to approximate
the solution for :math:`y(t)` at :math:`t = 2` .
Note that this approximation has no truncation error for :math:`i < 4` .

option
******
This algorithm uses the ``n_arg`` and ``n_other`` options; see below:

n_arg
=====
This is the size of the vectors *x* and *y* above .
There is an assert checking that *n_arg* > 0.

n_other
=======
This is the number of :ref:`Runge-Kutta steps<rk4_step-name>`
used to approximate the solution of the ODE :math:`y(t)` .
There is an assert checking that *n_other* > 0.

{xrst_toc_hidden
   xrst/rk4_step.xrst
}
Implementation
==============
:ref:`rk4_step-name` , :ref:`cpp_an_ode-name` , :ref:`py_an_ode-name` .

Derivative
**********
The partial of :math:`y_i (t)` with respect to :math:`x_j` is

.. math::

   \frac{ \partial y_i (t) }{  \partial x_j } = \begin{cases}
      y_i (t) / x_j   & \text{if} \; j \leq i  \\
      0               & \text{otherwise}
   \end{cases}


{xrst_end an_ode}