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Automatic differentiation uses recurrence relations based on the rules of calculus. Consequently, the results are guaranteed to be correct only if the relevant mathematical assumptions are satisfied at least in a neighborhood of the current argument. Computer programs may violate these conditions by branching or by calling intrinsic functions such as abs, max, sqrt, and asin at points where their derivative is undefined or infinite. The resulting dependence between the program's input and output variables may still be differentiable, because branch values fit together smoothly or nondifferentiabilities cancel each other out. We have two objectives. First, we would like to assure the user that the function being evaluated is indeed locally differentiable because all intrinsics are evaluated at smooth arguments and none of the branching tests are critical. Second, the derivative program should run even when the assumptions of the chain rule are not strictly satisfied. In this case, the n...

. Automatic differentiation (AD) is a methodology for developing sensitivity-enhanced versions of arbitrary computer programs. In this paper, we provide some background information on AD and address some frequently asked questions. We introduce the ADIFOR and ADIC tools for the automatic differentiation of Fortran 77 and ANSI-C programs, respectively, and give an example of applying ADIFOR in the context of the optimization of multibody systems. 1. Introduction Assume that we have a code for the computation of a function f and f : x 2 R n 7! y 2 R m , and we wish to compute the derivatives of y with respect to x. We call x the independent variable and y the dependent variable. In computing derivatives, we should keep the following issues in mind: Reliability: The computed derivatives should ideally be accurate to machine precision. Computational Cost: In many applications, the computation of derivatives is the dominant computational burden. Hence, the amount of memory and runt...

1 1 Introduction 1 2 A Simple Example 1 3 Restrictions 8 4 Mathematical Pitfalls 15 5 Workarounds and Shortcuts 16 5.1 Input and Output : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 5.2 ADPRE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 5.3 A Generic Makefile : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 6 Advanced Topics 24 6.1 Efficient Common Block Organization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 24 6.2 ADMakefile : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 7 An Advanced Example 25 7.1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 7.2 Creating the ADIFOR script file : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 7.3 ...

Automatic differentiation is a technique of computing the derivative of a function or a subroutine written in a higher level language such as FORTRAN or C. Significant progress has been made in this field in the last few years. Here, we give a short exposition to automatic differentiation and demonstrate its applicability to several fields of engineering analysis. 1 Introduction Engineering analysis often requires the calculation of gradients of functions written in the form of a program or a subroutine in a higher level language such as FORTRAN or C. Often, such functions are not available in an explicit form. A common practice is to approximate the gradients by finite differences. However, finite difference gradients are known to be expensive and inaccurate in several circumstances. Symbolic differentiation can be used if the function being computed is available in the form of a set of expressions. Analytical methods of differentiation have been developed in several areas, but they ...

Automatic differentiation is a technique used for generating computer programs which compute the value of the derivative of a function given by an original computer program. This article shows that automatic differentiation may "fail" if it is applied to an iterative equation solver based on bisection, in particular, it fails to reproduce the user's expectations. A careful investigation of this behavior gives insight into the technique of automatic differentiation. Maple is of great help to explore this problem. Introduction Let f : IR n ! IR be a differentiable function and P a program which for given arguments computes the function value. By automatic differentiation P is transformed into a program P 0 which computes both the partial derivatives and the function value of f . The basic idea behind automatic differentiation is rather simple. The computation of the derivatives is obtained by applying the elementary rules of differentiation to each statement in the computation sche...

Automatic differentiation uses recurrence relations based on the rules of calculus. Consequently, the results are guaranteed to be correct only if the relevant mathematical assumptions are satisfied at least in a neighborhood of the current argument. Computer programs may violate these conditions by branching or by calling intrinsic functions such as abs, max, sqrt, and asin at points where their derivative is undefined or infinite. The resulting dependence between the program's input and output variables may still be differentiable, because branch values fit together smoothly or nondifferentiabilities cancel each other out. We have two objectives. First, we would like to assure the user that the function being evaluated is indeed locally differentiable because all intrinsics are evaluated at smooth arguments and none of the branching tests are critical. Second, the derivative program should run even when the assumptions of the chain rule are not strictly satisfied. In this case, the n...

The Modelica language has been enhanced with a notation for partial derivatives of Modelica functions. This paper presents how Dymola [4] enables the use of partial derivatives in certain modeling applications in the Modelica language. It is shown that using partial derivatives is natural and supported in Dymola, and solves several advanced modeling problems. 1