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Topic: Automatic differentiation

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	Derivative - Wikipedia, the free encyclopedia
		A function is differentiable at a point x if its derivative exists at that point; a function is differentiable on an interval if it is differentiable at every x within the interval.
		Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
		For complex functions of a complex variable differentiability is a much stronger condition than that the real and imaginary part of the function are differentiable with respect to the real and imaginary part of the argument.
	en.wikipedia.org /wiki/Derivative (2137 words)

	Automatic Differentiation
		The automatic differentiation in contrast with symbolic differentiation propagates numerical values of the derivatives rather symbolic expressions.
		Differentiation of a composite function is performed in the forward mode: first differential tuples for independent variables are generated, then they are propagated through the computational graph of the function.
		The automatic differentiation algorithms are implemented through overloaded elementary functions (exp, pow, sin, cos, etc.), arithmetic operations(+, -, , /), various utility functions including differentiation* operators, and a function to generate differential tuples corresponding to independent variables [6].
	sal-cnc.me.wisc.edu /Research/meshless/AutoDiff/AutoDiff.html (436 words)

	Automatic differentiation - Wikipedia, the free encyclopedia
		In mathematics and computer algebra, automatic differentiation, or AD, sometimes alternatively called algorithmic differentiation, is a method to numerically evaluate the derivative of a function specified by a computer program.
		The drawback with symbolic differentiation is low speed, and the difficulty of converting a computer program into a single expression.
		Automatic differentiation solves all of the mentioned problems.
	en.wikipedia.org /wiki/Automatic_differentiation (422 words)

	[No title]
		Automatic generation of efficient derivative codes thus requires analysis of programs for detection of such properties and systematic methods for their exploitation in composing the derivative codes.
		Automatic differentiation can have a significant impact on what is considered a practical approach and what types of problems can be solved.
		Automatic differentiation commonly performs this evaluation by associating vector storage either with the program variables (in the case of forward-mode automatic differentiation) or with the adjoint variables (in the case of reverse).
	www.ii.uib.no /forskningsgrupper/opt/forskning/AD_bibs/ad1996.bib (3089 words)

	[No title]
		Automatic differentiation enables, in a completely mechanical fashion, algorithmic changes by switching from a quasi-Newton method, where first order derivatives are approximated by finite differences, to a modified Gauss--Newton method using exact first order derivatives.
		The automatic differentiation software tool \textbf{AD}\raisebox{-.25ex}{\textsl{opt}} is used for the evaluation of the first-order derivatives of objective and constraint functions with respect to the control variables.
		Automatic differentiation is used to enable interesting mathematical explorations.", comment = "", keywords = "", referred = "", isbn = "" } @ARTICLE{Fliess1995a, author = "Michel Fliess and Jean L{\'e}vine and Philippe Martin and Pierre Rouchon", year = 1995, title = "Flatness and Defect of Non-Linear Systems: {I}ntroductory Theory and Examples", journal = "Int.
	www.ii.uib.no /forskningsgrupper/opt/forskning/AD_bibs/ad2000.bib (3164 words)

	BibTeX bibliography all_brec.bib (Site not responding. Last check: 2007-10-19)
		Interval techniques for ordinary differential equations are based on using an {\it a priori\/} bound to capture remainder terms.
		They consider the rationale and applications of differentiation arithmetic, outline the general structure of a coder-operators system, and describe the specifications for the coder, the operators, and their interface.", } @Article{Denn81a, author = "John Dennis and David Gay and R. Welsch", title = "Algorithm 573.
		Automatic differentiation is used to calculate the derivatives required.
	www.math.utah.edu /ftp/pub/bibnet/subjects/all_brec.html (1743 words)

	Conference Program Online (Site not responding. Last check: 2007-10-19)
		Automatic differentiation techniques and tools have made significant advances in recent years with major improvements in terms of efficiency, applicability and ease-of-use.
		Without question, incorporating automatic differentiation tools and techniques into optimization is not only useful but, in many cases, essential in order to facilitate the widespread use of state-of-the-art optimization software.
		The speakers will review the latest developments in automatic differentiation and show how automatic differentiation techniques and tools are impacting the solution of nonlinear optimization problems.
	meetings.siam.org /sess/dsp_programsess.cfm?SESSIONCODE=1261 (86 words)

	Automatic Differentiation
		In automatic differentiation, the rules of calculus are applied to these atomic operations and are combined appropriately according to the algorithmic specification of the function.
		In reverse automatic differentiation (RAD), the algorithm first performs a forward sweep for the function evaluation, storing, at the same time, the information required for the derivative computations.
		Automatic differentiation techniques and their application in metrology
	www.npl.co.uk /scientific_software/tutorials/data_analysis/automatic_differentiation (887 words)

	Publications
		It is shown how the techniques of automatic differentiation can be viewed in a broader context as an application of analysis on a nonarchimedean field.
		The rings used in automatic differentiation can be ordered in a natural way and form finite dimensional real algebras which contain infinitesimals.
		A remarkable property of differentiation is that difference quotients with infinitely small differences yield the exact derivative up to an infinitely small error.
	bt.pa.msu.edu /papers-cgi/display.pl?name=adtheory (158 words)

	FADBAD++ Automatic Differentiation Made Easy
		In automatic differentiation the evaluation of a function and its derivatives are calculated simultaneously, using the same code and common temporary values.
		Automatic differentiation is easy to implement in languages with operator overloading such as C++, Ada and PASCAL-XSC.
		Taylor expansion of the solution to an ordinary differential equation using interval arithmetic and using the Forward method to obtain derivatives of the Taylor coefficients with respect to the point of expansion, which are the values of the Taylor coefficients for the solution of the variational problem.
	www2.imm.dtu.dk /~km/FADBAD (2610 words)

	Large, Sparse Jacobian Matrices Computed Accurately and Efficiently with Automatic Differentiation
		The research dispelled the common misconception that automatic differentiation is unable to handle large problems effectively.
		Automatic differentiation relies on the fact that every function, no matter how complicated, is evaluated on a computer as a (potentially very long) sequence of elementary operations such as additions, multiplications, and elementary functions such as the trigonometric and exponential functions.
		Automatic differentiation not only outperformed forward and central difference approximations in terms of accuracy and speed, but also proved to be comparable in speed and just as accurate as hand-coded derivatives.
	www.crpc.rice.edu /newsletters/jul93/wip.jacobian.html (270 words)

	Publications
		This book is a selection of papers from the Second International Workshop on Computational Differentiation held in Sante Fe, New Mexico, February 12-14, 1996, under the sponsorship of the Society for Industrial and Applied Mathematics (SIAM) and the Special Interest Group on Numerical Mathematics of the Association of Computing Machinery.
		Computational differentiation enables practitioners to quickly generate derivative-enhanced versions of their codes, which are required in these "value-added" uses of their simulation models.
		On the other hand, much remains to be done regarding the treatment of potential nonsmoothness, interactions between commonly used numerical techniques and automatic differentiation, and the implementation of automated trade-offs between storing or recomputing intermediates in the context of multidirectional derivative calculations.
	bt.pa.msu.edu /papers-cgi/display.pl?name=sfbook (1088 words)

	Research Focus: The Automatic Differentiation Project
		Automatic differentiation provides a foundation for those efforts by reliably and efficiently computing derivatives of large computer codes, and by providing significant algorithmic speedups, compared to current approaches.
		The Automatic Differentiation (AD) project in the CRPC came into existence in 1991, after Chris Bischof of Argonne National Laboratory visited the Fortran Parallel Programming Project at Rice University with the encouragement of John Dennis at Rice.
		From the beginning, Bischof and Carle believed that even though automatic differentiation was not a new area, its impact on practitioners had been minimal because of the lack of both robust and efficient tools and educational outreach by the AD community.
	www.crpc.rice.edu /newsletters/fal95/resfocus.html (993 words)

	Application of Automatic Differentiation to Reservoir Design Models (Site not responding. Last check: 2007-10-19)
		Automatic differentiation is a technique for computing derivatives accurately and efficiently with minimal human effort.
		The writers report on the use of automatic differentiation and divided difference approaches for computing derivatives for a single-and multiple-reservoir yield model.
		The results show that, for both the single- and the multiple-reservoir model, automatic differentiation computes derivatives exactly and more efficiently than the divided difference implementation.
	www.pubs.asce.org /WWWdisplay.cgi?9801552 (148 words)

	Efficient Derivative Codes through Automatic Differentiation and Interface Contraction: An Application in Biostatistics
		Automatic differentiation has proven capable of producing derivative codes with very little effort on the part of the user.
		Automatic differentiation avoids the truncation errors characteristic of divided difference approximations.
		A case study involving the ADIFOR (Automatic Differentiation of Fortran) tool and a program for maximizing a logistic-normal likelihood function developed from a problem in nutritional epidemiology is examined, and performance figures are presented.
	epubs.siam.org /sam-bin/dbq/article/28180 (290 words)

	SergeFantino.com:Automatic Differentiation
		Windy is a framework and an application for research in Automatic Differentiation.
		Automatic Differentiation (AD) is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as a computer program.
		By applying the chain rule of derivative calculus repeatedly to these operations, derivatives of arbitrary order can be computed automatically, and accurate to working precision.
	perso.wanadoo.fr /serge.fantino/tech/ad.html (126 words)

	Research Experience for Undergraduates
		On the convergence of numerical differentiation for Hermite nodes.
		Automatic differentiation of algorithms (Breckenridge, CO, 1991), 3--16, SIAM, Philadelphia, PA, 1991, MathSciNet.
		Automatic differentiation of algorithms (Breckenridge, CO, 1991), 191--201, SIAM, Philadelphia, PA, 1991, MathSciNet.
	math.fullerton.edu /mathews/n2003/differentiation/NumericalDiffBib/Links/NumericalDiffBib_lnk_3.html (1833 words)

	[No title]
		They consider the rationale and applications of differentiation arithmetic, outline the general structure of a coder-operators system, and describe the specifications for the coder, the operators, and their interface." } @ARTICLE { Denn81a, AUTHOR = "Dennis, John and Gay, David and Welsch, R. TITLE = "Algorithm 573.
		Various difficulties such as integration across discontinuities and implicit differential equations are discussed and accurate and efficient remedies are provided.
		It is shown that the algebraic complexity of computing a function of several variables and its partial derivatives with respect to all of the variables is at most a constant (four, five, six or seven which is independent of the number of variables) times as large as that of computing the function alone.
	www.eng.mu.edu /corlissg/FtpStuff/ValidODE/Bibliog/autodiff.bib (1657 words)

	CEF 1997: Computing Implied Volatilities Using Automatic Differentiation
		Another alternative, divided differences, does not directly produce a derivative code but rather approximates the derivatives by evaluating f at multiple input points, While it requires little effort to produce these derivatives, the main drawback is that it is not early to determine the accuracy of the approximation due to inherent errors.
		AD works by automatically applying the chain rule of differential calculus at the elementary operator or intrinsic function level (e.g., add, multiply, sine and cosine).
		We have developed an extensible AD tool called ADIC (Automatic Differentiation of C) that is based on source-to-source transformation.
	bucky.stanford.edu /cef97/abstracts/roh.html (832 words)

	DAEPACK (Site not responding. Last check: 2007-10-19)
		automatically generate new code which determines the sparsity pattern of the model for a given set of inputs (sparsity pattern generation),
		automatic generation of a discontinuity-locked model and extraction of hidden discontinuities, allowing state-of-the art state event location algorithms to be applied to general FORTRAN models so that hybrid discrete/continuous dynamic simulation can be performed efficiently and robustly and parametric sensitivity calculations can be performed correctly (discontinuity locking), and
		ATTENTION: Description of a short course on automatic differentiation and other code generation techniques taught in January 2002 at MIT can be found here.
	yoric.mit.edu /daepack/daepack.html (354 words)

	ADIC: An Extensible Automatic Differentiation Tool for ANSI-C - Bischof, Roh, M-Oats (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		Automatic differentiation (AD) techniques augment the program with derivative computation by applying the chain rule of calculus to elementary operations in an automated fashion.
		This article introduces ADIC (Automatic Differentiation of C), a new...
		24 Automatic differentiation in Odyssee (context) - Rostaing, Dalmas et al.
	citeseer.ist.psu.edu /31986.html (916 words)

	A Collection of Automatic Differentiation Tools (Site not responding. Last check: 2007-10-19)
		Automatic differentiation (AD) is a technique for augmenting computer programs with derivative computations.
		In contrast, the reverse mode of automatic differentiation propagates derivatives of the final result with respect to an intermediate quantity, adjoint quantity.
		The following list of automatic differentiation tools provides a short introduction into the capabilities of the listed AD tool, as provided by their developers and provides pointers to developers and additional information.
	www.mcs.anl.gov /Projects/autodiff/AD_Tools (514 words)

	The Efficient Computation of Structured Gradients using Automatic Differentiation (Site not responding. Last check: 2007-10-19)
		The advent of robust automatic differentiation tools is an exciting and important development in scientific computing.
		It is particularly noteworthy that the gradient of a scalar-valued function of many variables can be computed with essentially the same time complexity as required to evaluate the function itself.
		This is true, in theory, when the "reverse mode" of automatic differentiation is used (whereas the "forward mode" introduces an additional factor corresponding to the problem dimension).
	epubs.siam.org /sam-bin/dbq/article/32079 (185 words)

	Application of automatic differentiation in TOUGH2
		Automatic differentiation (AD) is a way to accurately and efficiently compute derivatives of a function written in computer codes.
		Our result with the test problem set indicates that the AD-generated derivative code could improve the convergence behavior in the linear solution step, taking less computational time to compute one linear matrix system.
		Jong G. Kim and Stefan Finsterle, "Application of automatic differentiation in TOUGH2" (May 12, 2003).
	repositories.cdlib.org /lbnl/LBNL-52503 (187 words)

	IMA Automatic Differentiation Resource (Site not responding. Last check: 2007-10-19)
		This homepage was established as a result of the IMA Special Workshop, "Template-Driven Automatic Differentiation for Large-Scale Scientific and Engineering Applications", held at the IMA from June 29 to July 3, 1997.
		By Automatic Differentiation (AD), we mean procedures that take computer programs that evaluate a function and produces programs that calculate the derivatives of the function.
		This page is meant to serve as a resource for scientists and engineers who may wish to use automatic differentiation in their work.
	www.math.umn.edu /~santosa/adhome.html (430 words)

	Automatic Differentiation software available (FORTRAN) (Site not responding. Last check: 2007-10-19)
		A description/announcement follows: AUTOMATIC DIFFERENTIATION FOR FORTRAN Given a function coded in Fortran, GRAD produces Fortran code to compute the derivatives with respect to specified variables (i.e.
		Sometimes computer algebra packages can help, but these are generally inadequate when the functions to be differentiated are defined by computer programs containing intermediate variables, loops, and conditionals.
		GRAD is described in detail in: Garcia, O. A system for the differentiation of Fortran code and an application to parameter estimation in forest growth models.
	www.metla.fi /archive/forest/1993/03/msg00000.html (291 words)

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