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CppAD User's Manual
###################

.. image:: {xrst_dir coin.png}

License
*******
CppAD is distributed by
`COIN-OR <https://www.coin-or.org/>`_
with the Eclipse Public License
`EPL-2.0 <https://spdx.org/licenses/EPL-2.0.html>`_
or the GNU General Public License
`GPL-2.0-or-later <https://spdx.org/licenses/EPL-2.0.html>`_

Git Repository
**************
`<https://github.com/coin-or/CppAD>`_

Versions
********

.. list-table::
   :widths: auto

   * - This version
     - cppad-20250209
   * - Documentation for latest version
     - `latest <https://cppad.readthedocs.io/latest>`_
   * - Documentation for most recent stable version
     - `stable-2025 <https://cppad.readthedocs.io/stable-2025>`_
   * - Most recent release of this stable version
     - `release-2025 <https://codeload.github.com/coin-or/CppAD/tar.gz/refs/tags/20250000.2>`_

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Other Links
***********

.. list-table::
   :widths: auto

   * - :ref:`install-name`
     - :ref:`get_started<get_started.cpp-name>`
     - :ref:`whats_new-name`
     - :ref:`addon-name`
     - `project manager <https://www.seanet.com/~bradbell/>`_


Algorithmic Differentiation
***************************
We refer to the automatic creation of an algorithm that
computes derivative values from an algorithm that computes function values
as *Algorithmic Differentiation* ,
also referred to as *Automatic Differentiation* or just AD.
A brief introduction to AD can be found in
`wikipedia <https://en.wikipedia.org/wiki/Automatic_differentiation>`_.
The web site
`autodiff.org <https://www.autodiff.org/>`_
is dedicated to research about, and promoting the use of, AD.

Features
********

Operator Overloading
====================
CppAD uses operator overloading of the C++ template class :ref:`AD-name`
to compute derivatives of algorithms defined using AD objects; see
the :ref:`introduction-name` for a description of how this is accomplished.

Base Type
=========
The operator overloading uses a templated base type that
can be user defined; see :ref:`base_require-name` .
The required definitions for ``AD<float>`` and
``AD<double>`` are included as part of CppAD.

Recording Operations
====================
A sequence of ``AD`` < *Base* >
:ref:`operations<glossary@Operation@Sequence>`
can be recorded and stored in an
:ref:`AD function object<ADFun-name>` .
This object can then be used to evaluate
function values and arbitrary order derivatives,
and sparsity patterns of derivative values using the *Base* type.

Multi-Threading
===============
CppAD supports an arbitrary :ref:`multi threading<multi_thread-name>` environment.
Examples are provided using Boost threads, Posix threads, and Openmp threads.

Optimizing Operations Sequences
===============================
During the recording of a function,
the :ref:`Independent-name` variables are know and a forward dependency
analysis is used to determine which operations should be recorded.
Once the :ref:`Dependent-name` variables are determined,
a reverse dependency analysis can be preformed.
This :ref:`optimize-name` routine uses a reverse dependency analysis,
and other techniques,
to remove unnecessary operations.

Dynamic Parameters
==================
CppAD enables one to specify a vector of
:ref:`glossary@Parameter@Dynamic` parameters.
The value of the function and derivatives can depend on these parameters,
but no derivatives are taken with respect to these parameters.
This enables CppAD to reduce the derivative calculations; e.g.,
the derivative of variable times a variable has two terms
while a variable times a parameter only has one.

Derivative Calculations
=======================
Arbitrary order
:ref:`Forward-name` and :ref:`Reverse-name`
mode derivative calculations
can be preformed using an ``ADFun`` object.
Easy to user drivers that compute the
entire :ref:`Jacobian-name` and a specific :ref:`Hessian-name` are included.

Sparsity
========
Both forward and reverse mode can be used to calculation the sparsity
pattern for Jacobians and Hessians; see :ref:`sparsity_pattern-name` .
Where a Jacobian or Hessian is sparse,
both forward and reverse mode can be combined with the sparsity pattern
to speed up the calculation of
:ref:`sparse derivatives<sparse_derivative-name>` .
In addition, a :ref:`subgraph<subgraph_jac_rev-name>` method,
that does not require a sparsity pattern, can be used
to speed up these derivative calculations.

Recording Derivative Operations
===============================
A ``ADFun`` object can be converted into an object that
evaluates derivatives using the type ``AD`` < *Base* > ; see :ref:`base2ad-name` .
This enables one to record new functions that
are expressed using derivatives of other functions.

Atomic Functions
================
User defined derivative calculations for arbitrary functions
can also be included in
a recording using :ref:`atomic functions<atomic_two-name>` .
A special :ref:`checkpoint<chkpoint_one-name>` class is included
which allows one to record a function and reuse it as
an atomic operation in other functions.
There also is a special interface for user defined unary
:ref:`discrete functions<Discrete-name>` ; i.e.,
functions that depend on the
:ref:`independent variables<glossary@Tape@Independent Variable>`
but which have identically zero derivatives (e.g., a step function).

Logical Comparisons
===================
Logical comparisons can be included in an operation sequence
using AD :ref:`conditional expressions<CondExp-name>` .

Vectors
=======
The CppAD API allows one to use any
:ref:`SimpleVector-name` class.
The preprocessor symbol :ref:`CPPAD_TESTVECTOR<testvector-name>`
is template vector class which is used for correctness testing.
Many of the correctness tests use this template vector class
which you can choose during the :ref:`cmake-name` configuration command.

Software Engineering
********************
CppAD is developed using the software engineering procedures described
on the project manager's
`software <https://www.seanet.com/~bradbell/software.html>`_
web page.

Testing
*******

Correctness
===========
There is an extensive set of correctness tests; see :ref:`cmake_check-name` .

Speed
=====
A set of programs for doing :ref:`speed-name` comparisons between
`Adolc <https://github.com/coin-or/ADOL-C>`_,
CppAD,
`Fadbad <http://uning.dk//>`_,
and
`Sacado <https://trilinos.github.io/sacado.html>`_
are included.

Utilities
*********
CppAD includes a set of C++ :ref:`utilities<utility-name>` that are useful
for general operator overloaded numerical methods.

Release Notes
*************
This :ref:`whats_new-name` for a list of recent extensions and bug fixes.

Example
*******
The file :ref:`get_started.cpp-name`
contains an example and test of using CppAD to compute
the derivative of a polynomial.
There are many other
:ref:`examples<Example-name>` .
All of the examples are also correctness tests,
which ensures that they work properly.

Contents
********
{xrst_toc_table
   xrst/install/install.xrst
   xrst/theory/theory.xrst
   include/cppad/core/user_ad.hpp
   include/cppad/core/ad_fun.hpp
   xrst/preprocessor.xrst
   xrst/multi_thread.xrst
   include/cppad/utility/xrst/utility.xrst
   include/cppad/ipopt/solve.hpp
   xrst/example.xrst
   speed/speed.xrst
}


{xrst_end user_guide}