Automatic differentiation

In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation,^[1][2] is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.

Automatic differentiation is not:

Figure 1: How automatic differentiation relates to symbolic differentiation

• Symbolic differentiation, nor
• Numerical differentiation (the method of finite differences).

These classical methods run into problems: symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression, while numerical differentiation can introduce round-off errors in the discretization process and cancellation. Both classical methods have problems with calculating higher derivatives, where the complexity and errors increase. Finally, both classical methods are slow at computing the partial derivatives of a function with respect to many inputs, as is needed for gradient-based optimization algorithms. Automatic differentiation solves all of these problems, at the expense of introducing more software dependencies.

The chain rule, forward and reverse accumulation

Fundamental to AD is the decomposition of differentials provided by the chain rule. For the simple composition y = g(h(x)) = g(w) the chain rule gives

$\frac{dy}{dx} = \frac{dy}{dw} \frac{dw}{dx}$

Usually, two distinct modes of AD are presented, forward accumulation (or forward mode) and reverse accumulation (or reverse mode). Forward accumulation specifies that one traverses the chain rule from inside to outside (that is, first one computes dw/dx and then dy/dw, while reverse accumulation has the traversal from outside to inside.

Forward accumulation

Figure 2: Example of forward accumulation with computational graph

In forward accumulation AD, one first fixes the independent variable to which differentiation is performed and computes the derivative of each sub-expression recursively. In a pen-and-paper calculation, one can do so by repeatedly substituting the derivative of the inner functions in the chain rule:

$\frac{\partial y}{\partial x} = \frac{\partial y}{\partial w_1} \frac{\partial w_1}{\partial x} = \frac{\partial y}{\partial w_1} \left(\frac{\partial w_1}{\partial w_2} \frac{\partial w_2}{\partial x}\right) = \frac{\partial y}{\partial w_1} \left(\frac{\partial w_1}{\partial w_2} \left(\frac{\partial w_2}{\partial w_3} \frac{\partial w_3}{\partial x}\right)\right) = \cdots$

This can be generalized to multiple variables as a matrix product of Jacobians.

Compared to reverse accumulation, forward accumulation is very natural and easy to implement as the flow of derivative information coincides with the order of evaluation. One simply augments each variable w with its derivative ẇ (stored as a numerical value, not a symbolic expression),

$\dot w = \frac{\partial w}{\partial x}$

as denoted by the dot. The derivatives are then computed in sync with the evaluation steps and combined with other derivatives via the chain rule.

As an example, consider the function:

$\begin{align} z &= f(x_1, x_2) \\ &= x_1 x_2 + \sin x_1 \\ &= w_1 w_2 + \sin w_1 \\ &= w_3 + w_4 \\ &= w_5 \end{align}$

For clarity, the individual sub-expressions have been labeled with the variables w_i.

The choice of the independent variable to which differentiation is performed affects the seed values ẇ₁ and ẇ₂. Suppose one is interested in the derivative of this function with respect to x₁. In this case, the seed values should be set to:

$\begin{align} \dot w_1 = \frac{\partial x_1}{\partial x_1} = 1 \\ \dot w_2 = \frac{\partial x_2}{\partial x_1} = 0 \end{align}$

With the seed values set, one may then propagate the values using the chain rule as shown in both the table below. Figure 2 shows a pictorial depiction of this process as a computational graph.

$\begin{array}{l|l} \text{Operations to compute value} & \text{Operations to compute derivative} \\ \hline w_1 = x_1 & \dot w_1 = 1 \text{ (seed)} \\ w_2 = x_2 & \dot w_2 = 0 \text{ (seed)} \\ w_3 = w_1 \cdot w_2 & \dot w_3 = w_2 \cdot \dot w_1 + w_1 \cdot \dot w_2 \\ w_4 = \sin w_1 & \dot w_4 = \cos w_1 \cdot \dot w_1 \\ w_5 = w_3 + w_4 & \dot w_5 = \dot w_3 + \dot w_4 \end{array}$

To compute the gradient of this example function, which requires the derivatives of f with respect to not only x₁ but also x₂, one must perform an additional sweep over the computational graph using the seed values $\dot w_1 = 0; \dot w_2 = 1$.

The computational complexity of one sweep of forward accumulation is proportional to the complexity of the original code.

Forward accumulation is more efficient than reverse accumulation for functions f : ℝⁿ → ℝ^m with m ≫ n as only n sweeps are necessary, compared to m sweeps for reverse accumulation.

Reverse accumulation

Figure 3: Example of reverse accumulation with computational graph

In reverse accumulation AD, one first fixes the dependent variable to be differentiated and computes the derivative with respect to each sub-expression recursively. In a pen-and-paper calculation, one can perform the equivalent by repeatedly substituting the derivative of the outer functions in the chain rule:

$\frac{\partial y}{\partial x} = \frac{\partial y}{\partial w_1} \frac{\partial w_1}{\partial x} = \left(\frac{\partial y}{\partial w_2} \frac{\partial w_2}{\partial w_1}\right) \frac{\partial w_1}{\partial x} = \left(\left(\frac{\partial y}{\partial w_3} \frac{\partial w_3}{\partial w_2}\right) \frac{\partial w_2}{\partial w_1}\right) \frac{\partial w_1}{\partial x} = \cdots$

In reverse accumulation, the quantity of interest is the adjoint, denoted with a bar (w̄); it is a derivative of a chosen dependent variable with respect to a subexpression w:

$\bar w = \frac{\partial y}{\partial w}$

Reverse accumulation traverses the chain rule from outside to inside, or in the case of the computational graph in Figure 3, from top to bottom. The example function is real-valued, and thus there is only one seed for the derivative computation, and only one sweep of the computational graph is needed in order to calculate the (two-component) gradient. This is only half the work when compared to forward accumulation, but reverse accumulation requires the storage of the intermediate variables w_i as well as the instructions that produced them in a data structure known as a Wengert list (or "tape"),^[3] which may represent a significant memory issue if the computational graph is large. This can be mitigated to some extent by storing only a subset of the intermediate variables and then reconstructing the necessary work variables by repeating the evaluations, a technique known as checkpointing.

The operations to compute the derivative using reverse accumulation are shown in the table below (note the reversed order):

$\begin{array}{l} \text{Operations to compute derivative} \\ \hline \bar w_5 = 1 \text{ (seed)} \\ \bar w_4 = \bar w_5 \\ \bar w_3 = \bar w_5 \\ \bar w_2 = \bar w_3 \cdot w_1 \\ \bar w_1 = \bar w_3 \cdot w_2 + \bar w_4 \cdot \cos w_1 \end{array}$

The data flow graph of a computation can be manipulated to calculate the gradient of its original calculation. This is done by adding an adjoint node for each primal node, connected by adjoint edges which parallel the primal edges but flow in the opposite direction. The nodes in the adjoint graph represent multiplication by the derivatives of the functions calculated by the nodes in the primal. For instance, addition in the primal causes fanout in the adjoint; fanout in the primal causes addition in the adjoint; a unary function y = f(x) in the primal causes x̄ = ȳ f′(x) in the adjoint; etc.

Reverse accumulation is more efficient than forward accumulation for functions f : ℝⁿ → ℝ^m with m ≪ n as only m sweeps are necessary, compared to n sweeps for forward accumulation.

Reverse mode AD was first published in 1970 by Seppo Linnainmaa in his master thesis.^[4][5][6]

Backpropagation of errors in multilayer perceptrons, a technique used in machine learning, is a special case of reverse mode AD.

Beyond forward and reverse accumulation

Forward and reverse accumulation are just two (extreme) ways of traversing the chain rule. The problem of computing a full Jacobian of f : ℝⁿ → ℝ^m with a minimum number of arithmetic operations is known as the optimal Jacobian accumulation (OJA) problem, which is NP-complete.^[7] Central to this proof is the idea that there may exist algebraic dependencies between the local partials that label the edges of the graph. In particular, two or more edge labels may be recognized as equal. The complexity of the problem is still open if it is assumed that all edge labels are unique and algebraically independent.

Automatic differentiation using dual numbers

Forward mode automatic differentiation is accomplished by augmenting the algebra of real numbers and obtaining a new arithmetic. An additional component is added to every number which will represent the derivative of a function at the number, and all arithmetic operators are extended for the augmented algebra. The augmented algebra is the algebra of dual numbers.

Replace every number $\,x$ with the number $x + x'\varepsilon$, where $x'$ is a real number, but $\varepsilon$ is an abstract number with the property $\varepsilon^2=0$ (an infinitesimal; see Smooth infinitesimal analysis). Using only this, we get for the regular arithmetic

$\begin{align} (x + x'\varepsilon) + (y + y'\varepsilon) &= x + y + (x' + y')\varepsilon \\ (x + x'\varepsilon) \cdot (y + y'\varepsilon) &= xy + xy'\varepsilon + yx'\varepsilon + x'y'\varepsilon^2 = xy + (x y' + yx')\varepsilon \end{align}$

and likewise for subtraction and division.

Now, we may calculate polynomials in this augmented arithmetic. If $P(x) = p_0 + p_1 x + p_2x^2 + \cdots + p_n x^n$, then

$\begin{align} P(x + x'\varepsilon) &= p_0 + p_1(x + x'\varepsilon) + \cdots + p_n (x + x'\varepsilon)^n \\ &= p_0 + p_1 x + \cdots + p_n x^n + p_1x'\varepsilon + 2p_2xx'\varepsilon + \cdots + np_n x^{n-1} x'\varepsilon \\ &= P(x) + P^{(1)}(x)x'\varepsilon \end{align}$

where $P^{(1)}$ denotes the derivative of $P$ with respect to its first argument, and $x'$, called a seed, can be chosen arbitrarily.

The new arithmetic consists of ordered pairs, elements written $\langle x, x' \rangle$, with ordinary arithmetics on the first component, and first order differentiation arithmetic on the second component, as described above. Extending the above results on polynomials to analytic functions we obtain a list of the basic arithmetic and some standard functions for the new arithmetic:

$\begin{align} \left\langle u,u'\right\rangle + \left\langle v,v'\right\rangle &= \left\langle u + v, u' + v' \right\rangle \\ \left\langle u,u'\right\rangle - \left\langle v,v'\right\rangle &= \left\langle u - v, u' - v' \right\rangle \\ \left\langle u,u'\right\rangle * \left\langle v,v'\right\rangle &= \left\langle u v, u'v + uv' \right\rangle \\ \left\langle u,u'\right\rangle / \left\langle v,v'\right\rangle &= \left\langle \frac{u}{v}, \frac{u'v - uv'}{v^2} \right\rangle \quad ( v\ne 0) \\ \sin\left\langle u,u'\right\rangle &= \left\langle \sin(u) , u' \cos(u) \right\rangle \\ \cos\left\langle u,u'\right\rangle &= \left\langle \cos(u) , -u' \sin(u) \right\rangle \\ \exp\left\langle u,u'\right\rangle &= \left\langle \exp u , u' \exp u \right\rangle \\ \log\left\langle u,u'\right\rangle &= \left\langle \log(u) , u'/u \right\rangle \quad (u>0) \\ \left\langle u,u'\right\rangle^k &= \left\langle u^k , k u^{k - 1} u' \right\rangle \quad (u \ne 0) \\ \left| \left\langle u,u'\right\rangle \right| &= \left\langle \left| u \right| , u' \mbox{sign} u \right\rangle \quad (u \ne 0) \end{align}$

and in general for the primitive function $g$,

$g(\langle u,u' \rangle , \langle v,v' \rangle ) = \langle g(u,v) , g_u(u,v) u' + g_v(u,v) v' \rangle$

where $g_u$ and $g_v$ are the derivatives of $g$ with respect to its first and second arguments, respectively.

When a binary basic arithmetic operation is applied to mixed arguments—the pair $\langle u, u' \rangle$ and the real number $c$—the real number is first lifted to $\langle c, 0 \rangle$. The derivative of a function $f : \mathbb{R}\rightarrow\mathbb{R}$ at the point $x_0$ is now found by calculating $f(\langle x_0, 1 \rangle)$ using the above arithmetic, which gives $\langle f ( x_0 ) , f' ( x_0 ) \rangle$ as the result.

Vector arguments and functions

Multivariate functions can be handled with the same efficiency and mechanisms as univariate functions by adopting a directional derivative operator. That is, if it is sufficient to compute $y' = \nabla f(x)\cdot x'$, the directional derivative $y' \in \mathbb{R}^m$ of $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ at $x \in \mathbb{R}^n$ in the direction $x' \in \mathbb{R}^n$, this may be calculated as $(\langle y_1,y'_1\rangle, \ldots, \langle y_m,y'_m\rangle) = f(\langle x_1,x'_1\rangle, \ldots, \langle x_n,x'_n\rangle)$ using the same arithmetic as above. If all the elements of $\nabla f$ are desired, then $n$ function evaluations are required. Note that in many optimization applications, the directional derivative is indeed sufficient.

High order and many variables

The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate functions. However, the arithmetic rules quickly grow very complicated: complexity will be quadratic in the highest derivative degree. Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, defined on generalized dual numbers, allows to efficiently compute using functions as if they were a new data type. Once the Taylor polynomial of a function is known, the derivatives are easily extracted. Currently there are a few software projects that implement the Taylor polynomial truncated algebra. AuDi, CTaylor and COSY infinity. Only the first two are open source.

Implementation

Forward-mode AD is implemented by a nonstandard interpretation of the program in which real numbers are replaced by dual numbers, constants are lifted to dual numbers with a zero epsilon coefficient, and the numeric primitives are lifted to operate on dual numbers. This nonstandard interpretation is generally implemented using one of two strategies: source code transformation or operator overloading.

Source code transformation (SCT)

Figure 4: Example of how source code transformation could work

The source code for a function is replaced by an automatically generated source code that includes statements for calculating the derivatives interleaved with the original instructions.

Source code transformation can be implemented for all programming languages, and it is also easier for the compiler to do compile time optimizations. However, the implementation of the AD tool itself is more difficult.

Operator overloading (OO)

Figure 5: Example of how operator overloading could work

Operator overloading is a possibility for source code written in a language supporting it. Objects for real numbers and elementary mathematical operations must be overloaded to cater for the augmented arithmetic depicted above. This requires no change in the form or sequence of operations in the original source code for the function to be differentiated, but often requires changes in basic data types for numbers and vectors to support overloading and often also involves the insertion of special flagging operations.

Operator overloading for forward accumulation is easy to implement, and also possible for reverse accumulation. However, current compilers lag behind in optimizing the code when compared to forward accumulation.

Software

• C/C++

Package	License	Approach	Brief info
ADC Version 4.0	nonfree	OO
Adept	Apache 2.0	OO	First-order forward and reverse modes. Very fast due to its use of expression templates and an efficient tape structure.
ADIC	free for noncommercial	SCT	forward mode
ADMB	BSD	SCT+OO	Uses a template approach. See http://admb-project.org/
ADNumber	dual license	OO	arbitrary order forward/reverse
ADOL-C	CPL 1.0 or GPL 2.0	OO	arbitrary order forward/reverse, part of COIN-OR
AuDi	GPL 2.0	OO	AuDI is an open source, header only, C++ library that allows for AUtomated DIfferentiation implementing the Taylor truncated polynomial algebra (forward mode automated differentiation). It was created with the aim to offer a generic solution to the user in need of an automated differentiation system. AuDi is thread-safe and, when possible, use of Piranha fine-grained parallelization of the truncated polynomial multiplication. The benefits of this fine grained parallelization are well visible for many variables and high orders.
AMPL	free for students	SCT
CasADi	LGPL	SCT	Forward/reverse modes, matrix-valued atomic operations.
ceres-solver	BSD	OO	A portable C++ library that allows for modeling and solving large complicated nonlinear least squares problems
CppAD	EPL 1.0 or GPL 3.0	OO	arbitrary order forward/reverse, AD<Base> for arbitrary Base including AD<Other_Base>, part of COIN-OR; can also be used to produce C source code using the CppADCodeGen library.
CTaylor	free	OO	truncated taylor series, multi variable, high performance, calculating and storing only potentially nonzero derivatives, calculates higher order derivatives, order of derivatives increases when using matching operations until maximum order (parameter) is reached, example source code and executable available for testing performance
Eigen Auto Diff	MPL2	OO
FADBAD++	free for noncommercial	OO	uses operator new
OpenAD	depends on components	SCT
ReverseAD	free	OO	High order reverse mode which evaluates the high order derivative tensor directly instead of a forward/reverse modes hierarchy.
Sacado	GNU GPL	OO	A part of the Trilinos collection, forward/reverse modes.
Stan (software)	BSD	OO	forward- and reverse-mode automatic differentiation with library of special functions, probability functions, matrix operators, and linear algebra solvers; interfaces to MATLAB, R and Python.
TAPENADE	Free for noncommercial	SCT
Tensorflow	Apache 2.0	OO	TensorFlow is a Google-developed Python and C++ library that allows you to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays both on CPU and GPU efficiently.

• Fortran

Package	License	Approach	Brief info
ADF Version 4.0	nonfree	OO
ADIFOR	>>> (free for non-commercial)	SCT
AUTO_DERIV	free for non-commercial	OO
OpenAD	depends on components	SCT
TAF	nonfree	SCT
TAPENADE	Free for noncommercial	SCT
GADfit	Free (GPL 3)	OO	First (forward, reverse) and second (forward) order, principal use is nonlinear curve fitting, includes the differentiation of integrals

• Matlab

Package	License	Approach	Brief info
AD for MATLAB	GNU GPL	OO	Forward (1st & 2nd derivative, Uses MEX files & Windows DLLs)
Adiff	BSD	OO	Forward (1st derivative)
MAD	Proprietary	OO	Forward (1st derivative) full/sparse storage of derivatives
ADiMat	Proprietary	SCT	Forward (1st & 2nd derivative) & Reverse (1st), proprietary server side transform
MADiff	GNU GPL	OO	Reverse

• Python

Package	License	Approach	Brief info
ad	BSD	OO	first and second-order, reverse accumulation, transparent on-the-fly calculations, basic NumPy support, written in pure python
FuncDesigner	BSD	OO	uses NumPy arrays and SciPy sparse matrices, also allows to solve linear/non-linear/ODE systems and to perform numerical optimizations by OpenOpt
ScientificPython	CeCILL	OO	see modules Scientific.Functions.FirstDerivatives and Scientific.Functions.Derivatives
pycppad	BSD	OO	arbitrary order forward/reverse, implemented as wrapper for CppAD including AD<double> and AD< AD<double> >.
pyadolc	BSD	OO	wrapper for ADOL-C, hence arbitrary order derivatives in the (combined) forward/reverse mode of AD, supports sparsity pattern propagation and sparse derivative computations
uncertainties	BSD	OO	first-order derivatives, reverse mode, transparent calculations
algopy	BSD	OO	same approach as pyadolc and thus compatible, support to differentiate through numerical linear algebra functions like the matrix-matrix product, solution of linear systems, QR and Cholesky decomposition, etc.
pyderiv	GNU GPL	OO	automatic differentiation and (co)variance calculation
CasADi	LGPL	SCT	Python front-end to CasADi. Forward/reverse modes, matrix-valued atomic operations.
Tensorflow	Apache 2.0	OO	TensorFlow is a Google-developed Python and C++ library that allows you to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays both on CPU and GPU efficiently.
Theano	BSD	OO	Theano is a Python library that allows you to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays both on CPU and GPU efficiently.
Autograd	MIT	OO	Autograd can reverse-mode differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures. It is closed under its own operation and hence can compute derivatives of any order.

• Lua

Package	License	Approach	Brief info
Torch	BSD	OO	Torch is a LuaJIT library used for Deep Learning. Its nn package is divided into modular objects that share a common Module interface. Modules have a forward and backward function that allow them to feedforward and backpropagate (first-order derivatives). Modules can be joined together using module composites to create complex task-tailored graphs.
SciLua	MIT	OO	SciLua, a framework for general purpose scientific computing in LuaJIT, features complete and transparent support for forward-mode automatic differentiation.

• .NET

Package	License	Approach	Brief info
AutoDiff	GNU LGPL	OO	Automatic differentiation with C# operator overloading.
FuncLib	MIT	OO	Automatic differentiation and numerical optimization, operator overloading, unlimited order of differentiation, compilation to IL code for very fast evaluation.
DiffSharp	GNU LGPL	OO	An automatic differentiation library implemented in the F# language. It supports C# and the other CLI languages. The library provides gradients, Hessians, Jacobians, directional derivatives, and matrix-free Hessian- and Jacobian-vector products, which can be incorporated with minimal change into existing algorithms. Operations can be nested to any level, meaning that you can compute exact higher-order derivatives and differentiate functions that are internally making use of differentiation.

• Haskell

Package	License	Approach	Brief info
ad	BSD	OO	Forward Mode (1st derivative or arbitrary order derivatives via lazy lists and sparse tries) Reverse Mode Combined forward-on-reverse Hessians. Uses Quantification to allow the implementation automatically choose appropriate modes. Quantification prevents perturbation/sensitivity confusion at compile time.
fad	BSD	OO	Forward Mode (lazy list). Quantification prevents perturbation confusion at compile time.
rad	BSD	OO	Reverse Mode. (Subsumed by 'ad'). Quantification prevents sensitivity confusion at compile time.

• Java

Package	License	Approach	Brief info
JAutoDiff	MIT	OO	Provides a framework to compute derivatives of functions on arbitrary types of field using generics. Coded in 100% pure Java.
Apache Commons Math	Apache License v2	OO	This class is an implementation of the extension to Rall's numbers described in Dan Kalman's paper[8]
Deriva	Eclipse Public License v1.0	DSL+Code Generation	Deriva automates algorithmic differentiation in Java and Clojure projects. It defines DSL for building extended arithmetic expressions (the extension being support for conditionals, allowing to express non analytic functions). The DSL is used to generate flat byte-code at runtime, providing implementation without overhead of function calls.
Jap	Public	OO/SCT	Jap is a tools using Virtual Operator Overloading for java class. Jap was developed in the thesis of Phuong PHAM-QUANG 2008-2011.

• Julia

Package	License	Approach	Brief info
ForwardDiff.jl	MIT	OO	A unified package for forward-mode automatic differentiation, combining both DualNumbers and vector-based gradient accumulations.
DualNumbers.jl	MIT	OO	Implements a Dual number type which can be used for forward-mode automatic differentiation of first derivatives via operator overloading.
HyperDualNumers.jl	MIT	OO	Implements a Hyper number type which can be used for forward-mode automatic differentiation of first and second derivatives via operator overloading.
ReverseDiffSource.jl	MIT	SCT	Implements reverse-mode automatic differentiation for gradients and high-order derivatives given user-supplied expressions or generic functions. Accepts a subset of valid Julia syntax, including intermediate assignments.
TaylorSeries.jl	MIT	OO	Implements truncated multivariate power series for high-order integration of ODEs and forward-mode automatic differentiation of arbitrary order derivatives via operator overloading.

• Clojure

Package	License	Approach	Brief info
Deriva	Eclipse Public License v1.0	DSL+Code Generation	Deriva automates algorithmic differentiation in Java and Clojure projects. It defines DSL for building extended arithmetic expressions (the extension being support for conditionals, allowing to express non analytic functions). The DSL is used to generate flat byte-code at runtime, providing implementation without overhead of function calls.

• Golang

Package	License	Approach	Brief info
ad	Apache 2.0	DSL+Code Generation	Creates source code corresponding to algebraic expressions. See https://autodiff.info for a demo.

• R

Package	License	Approach	Brief info
PBSadmb	GNU GPL	SCT+OO	Uses a template approach. See http://admb-project.org/ .

References

1. Neidinger, Richard D. (2010). "Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming" (PDF). SIAM Review 52 (3): 545–563. doi:10.1137/080743627.
2. http://www.ec-securehost.com/SIAM/SE24.html
3. Bartholomew-Biggs, Michael; Brown, Steven; Christianson, Bruce; Dixon, Laurence (2000). "Automatic differentiation of algorithms" (PDF). Journal of Computational and Applied Mathematics 124 (1-2): 171–190. doi:10.1016/S0377-0427(00)00422-2.
4. Linnainmaa, Seppo (1970). The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors. Master's Thesis (in Finnish), Univ. Helsinki, 6-7.
5. Linnainmaa, Seppo (1976). Taylor expansion of the accumulated rounding error. BIT Numerical Mathematics, 16(2), 146-160.
6. Griewank, Andreas (2012). Who Invented the Reverse Mode of Differentiation?. Optimization Stories, Documenta Matematica, Extra Volume ISMP (2012), 389-400.
7. Naumann, Uwe (April 2008). "Optimal Jacobian accumulation is NP-complete". Mathematical Programming 112 (2): 427–441. doi:10.1007/s10107-006-0042-z. |contribution= ignored (help)
8. Kalman, Dan (June 2002). "Doubly Recursive Multivariate Automatic Differentiation" (PDF). Mathematics Magazine 75 (3): 187–202. doi:10.2307/3219241.

Literature

• Rall, Louis B. (1981). Automatic Differentiation: Techniques and Applications. Lecture Notes in Computer Science 120. Springer. ISBN 3-540-10861-0.
• Griewank, Andreas; Walther, Andrea (2008). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. Other Titles in Applied Mathematics 105 (2nd ed.). SIAM. ISBN 978-0-89871-659-7.
• Neidinger, Richard (2010). "Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming" (PDF). SIAM Review 52 (3): 545–563. doi:10.1137/080743627. Retrieved 2013-03-15.

External links

• www.autodiff.org, An "entry site to everything you want to know about automatic differentiation"
• Automatic Differentiation of Parallel OpenMP Programs
• Automatic Differentiation, C++ Templates and Photogrammetry
• Automatic Differentiation, Operator Overloading Approach
• Compute analytic derivatives of any Fortran77, Fortran95, or C program through a web-based interface Automatic Differentiation of Fortran programs
• Description and example code for forward Automatic Differentiation in Scala
• Adjoint Algorithmic Differentiation: Calibration and Implicit Function Theorem
• , Exact First- and Second-Order Greeks by Algorithmic Differentiation
• , Adjoint Algorithmic Differentiation of a GPU Accelerated Application
• , Adjoint Methods in Computational Finance Software Tool Support for Algorithmic Differentiation