lines 6-66 of file: example/python/numeric/rosen3_step_xam.py

# {xrst_begin numeric_rosen3_step_xam.py}
# {xrst_spell
#     rl
#     rosenbrock
# }
# {xrst_comment_ch #}
#
#
# Example Computing Derivative A Rosenbrock Ode Solution
# ######################################################
#
# y(t, x)
# *******
# The two initial value problems below have the following solution:
#
# .. math::
#
#    y_i (t, x) = ( t^{i+1} / (i+1) ! ) \prod_{j=0}^i x_j
#
# First ODE
# *********
#
# .. math::
#
#    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`
#
# Second ODE
# **********
#
# .. math::
#
#    f(t, y, x)  = ( t^i / i ! ) \prod_{j=0}^i x_j
#
# with the initial condition :math:`y(0) = 0`
#
# 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_rosen3_step_xam.py}