lines 6-101 of file: lib/python/cppad_py/fun_reverse.py

# {xrst_begin py_fun_reverse}
# {xrst_spell
#     xq
#     yq
# }
# {xrst_comment_ch #}
#
#
# Reverse Mode AD
# ###############
#
# Syntax
# ******
# *xq* = *f*\ ``.reverse`` ( *q* , *yq* )
#
# f
# *
# This is either a
# :ref:`d_fun<py_fun_ctor@Syntax@d_fun>` or
# :ref:`a_fun<py_fun_ctor@Syntax@a_fun>` function object
# and is effectively constant; i.e., not changed.
#
# Notation
# ********
#
# f(x)
# ====
# We use the notation :math:`f: \B{R}^n \rightarrow \B{R}^m`
# for the function corresponding to *f* .
# Note that *n* is the size of :ref:`ax<py_fun_ctor@ax>`
# and *m* is the size of :ref:`ay<py_fun_ctor@ay>`
# in to the constructor for *f* .
#
# X(t), S
# =======
# This is the same function as
# :ref:`x(t)<py_fun_forward@X(t)>` in the previous call to
# *f*\ ``.forward`` .
# We use :math:`S \in \B{R}^{n \times q}` to denote the Taylor coefficients
# of :math:`X(t)`.
#
# Y(t), T
# =======
# This is the same function as
# :ref:`y(t)<py_fun_forward@Y(t)>` in the previous call to
# *f*\ ``.forward`` .
# We use :math:`T \in \B{R}^{m \times q}` to denote the Taylor coefficients
# of :math:`Y(t)`.
# We also use the notation :math:`T(S)` to express the fact that
# the Taylor coefficients for :math:`Y(t)` are a function of the
# Taylor coefficients of :math:`X(t)`.
#
# G(T)
# ====
# We use the notation :math:`G : \B{R}^{m \times p} \rightarrow \B{R}`
# for a function that the calling routine chooses.
#
# q
# *
# This argument has type ``int`` and is positive.
# It is the number of the Taylor coefficient (for each variable)
# that we are computing the derivative with respect to.
# It must be greater than zero, and less than or equal
# the number of Taylor coefficient stored in *f* ; i.e.,
# :ref:`f_size_order()<py_fun_property@size_order>`.
#
# yq
# **
# If *f* is a ``d_fun`` ( ``a_fun`` ) object, *yq*
# is a numpy vector with ``float`` ( ``a_double`` ) elements,
# *m* rows and *q* columns.
# For 0 <= *i* < *m* and 0 <= *k* < *q* ,
# *yq* [ *i* , *k* ] is the partial derivative of
# :math:`G(T)` with respect to the *k*-th order Taylor coefficient
# for the *i*-th component function; i.e.,
# the partial derivative of :math:`G(T)` w.r.t. :math:`Y_i^{(k)} (t) / k !`.
#
# xq
# **
# The result is a numpy vector with ``float`` ( ``a_double`` ) elements,
# *n* rows and *q* columns.
# For 0 <= *j* < *n* and 0 <= *k* < *q* ,
# *xq* [ *j* , *k* ] is the partial derivative of
# :math:`G(T(S))` with respect to the *k*-th order Taylor coefficient
# for the *j*-th component function; i.e.,
# the partial derivative of
# :math:`G(T(S))` w.r.t. :math:`S_j^{(k)} (t) / k !`.
#
# {xrst_toc_hidden
#  example/python/core/fun_reverse_xam.py
# }
# Example
# *******
# :ref:`fun_reverse_xam.py-name`
#
# {xrst_end py_fun_reverse}