lines 6-59 of file: example/python/numeric/runge4_step_xam.py

# {xrst_begin numeric_runge4_step_xam.py}
# {xrst_spell
#     kutta
#     rl
# }
# {xrst_comment_ch #}
#
#
# Example Computing Derivative A Runge-Kutta Ode Solution
# #######################################################
#
# ODE
# ***
#
# .. math::
#
#    \partial_t y_i (t, x) =  f(t, y, x) \left\{ \begin{array}{rl}
#         x_0               & {\rm if} \; i = 0 \\
#         x_i y_{i-1} (t)   & {\rm otherwise}
#    \end{array} \right.
#
# with the initial condition :math:`y(0) = 0`
#
# Solution
# ********
# This is a special case for which we know the solution
#
# .. math::
#
#    y_i (t, x) = \left\{ \begin{array}{rl}
#         t  x_0                            & {\rm if} \; i = 0 \\
#         ( t^i / (i+1) ! ) \prod_{j=0}^i x_j   & {\rm otherwise}
#    \end{array} \right.
#
# Derivative of Solution
# **********************
# For this special case, the partial derivative of the solution with respect
# to the j-th component of the vector :math:`x` is
#
# .. math::
#
#    \partial_{x(j)} y_i (t, x) =  \left\{ \begin{array}{rl}
#         y_i (t, x) / x_j      & {\rm if} \; j \leq i \\
#         0                     & {\rm otherwise}
#    \end{array} \right.
#
# Source Code
# ***********
# {xrst_literal
#  # BEGIN_PYTHON
#  # END_PYTHON
# }
#
# {xrst_end numeric_runge4_step_xam.py}