# SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
# SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
# SPDX-FileContributor: 2003-24 Bradley M. Bell
# ----------------------------------------------------------------------------
{xrst_begin reverse_one}
{xrst_spell
   dw
}

First Order Reverse Mode
########################

Syntax
******
| *dw* = *f* . ``Reverse`` (1, *w* )

Purpose
*******
We use :math:`F : \B{R}^n \rightarrow \B{R}^m` to denote the
:ref:`glossary@AD Function` corresponding to *f* .
The function :math:`W : \B{R}^n \rightarrow \B{R}` is defined by

.. math::

   W(x) = w_0 * F_0 ( x ) + \cdots + w_{m-1} * F_{m-1} (x)

The result of this operation is the derivative
:math:`dw = W^{(1)} (x)`; i.e.,

.. math::

   dw = w_0 * F_0^{(1)} ( x ) + \cdots + w_{m-1} * F_{m-1}^{(1)} (x)

Note that if :math:`w` is the *i*-th
:ref:`glossary@Elementary Vector` ,
:math:`dw = F_i^{(1)} (x)`.

f
*
The object *f* has prototype

   ``const ADFun`` < *Base* > *f*

Before this call to ``Reverse`` , the value returned by

   *f* . ``size_order`` ()

must be greater than or equal one (see :ref:`size_order-name` ).

x
*
The vector *x* in expression for *dw* above
corresponds to the previous call to :ref:`forward_zero-name`
using this ADFun object *f* ; i.e.,

   *f* . ``Forward`` (0, *x* )

If there is no previous call with the first argument zero,
the value of the :ref:`Independent-name` variables
during the recording of the AD sequence of operations is used
for *x* .

w
*
The argument *w* has prototype

   ``const`` *Vector* & *w*

(see :ref:`reverse_one@Vector` below)
and its size
must be equal to *m* , the dimension of the
:ref:`fun_property@Range` space for *f* .

dw
**
The result *dw* has prototype

   *Vector* *dw*

(see :ref:`reverse_one@Vector` below)
and its value is the derivative :math:`W^{(1)} (x)`.
The size of *dw*
is equal to *n* , the dimension of the
:ref:`fun_property@Domain` space for *f* .

Vector
******
The type *Vector* must be a :ref:`SimpleVector-name` class with
:ref:`elements of type<SimpleVector@Elements of Specified Type>`
*Base* .
The routine :ref:`CheckSimpleVector-name` will generate an error message
if this is not the case.

Example
*******
{xrst_toc_hidden
   example/general/reverse_one.cpp
}
The file
:ref:`reverse_one.cpp-name`
contains an example and test of this operation.

{xrst_end reverse_one}
