. The expansion of complicated functions of many variables in Taylor polynomials is an important problem for many applications, and in practice can be performed rather conveniently (even to high orders) using polynomial algebras. An important application of these methods is the field of beam physics, where often expansions in about six variables to orders between five and ten are used. However, often it is necessary to also know bounds for the remainder term of the Taylor formula if the arguments lie within certain intervals. In principle such bounds can be obtained by interval bounding of the (n+1)-st derivative, which in turn can be obtained with polynomial algebra; but in practice the method is rather inefficient and susceptible to blow-up because of the need of repeated interval evaluations of the derivative. Here we present a new method that allows the computation of sharp remainder intervals in parallel with the accumulation derivatives up to order n. The method is useful for a...

Elements and techniques of state-of-the-art automatically verified constrained global optimization algorithms are reviewed, including a description of ways of rigorously verifying feasibility for equality constraints and a careful consideration of the role of active inequality constraints. Previously developed algorithms and general work on the subject are also listed. Limitations of present knowledge are mentioned, and advice is given on which techniques to use in various contexts. Applications are discussed. 1 INTRODUCTION, BASIC IDEAS AND LITERATURE We consider the constrained global optimization problem minimize OE(X) subject to c i (X) = 0; i = 1; : : : ; m (1.1) a i j x i j b i j ; j = 1; : : : ; q; where X = (x 1 ; : : : ; xn ) T . A general constrained optimization problem, including inequality constraints g(X) 0 can be put into this form by introducing slack variables s, replacing by s + g(X) = 0, and appending the bound constraint 0 s ! 1; see x2.2. 2 Chapter 1 W...

Most interval branch and bound methods for nonlinear algebraic systems have to date been based on implicit underlying assumptions of continuity of derivatives. In particular, much of the theory of interval Newton methods is based on this assumption. However, derivative continuity is not necessary to obtain effective bounds on the range of such functions. Furthermore, if the first derivatives just have jump discontinuities, then interval extensions can be obtained that are appropriate for interval Newton methods. Thus, problems such as minimax or l 1 approximations can be solved simply, formulated as unconstrained nonlinear optimization problems. In this paper, interval extensions and computation rules are given for the unary operation jxj, the binary operation maxfx; yg and a more general "jump" function Ø(s; x; y). These functions are incorporated into an automatic differentiation and code list interpretation environment. Experimental results are given for nonlinear systems involvin...

After several publications about our new global optimization method, we decided to provide the public with a simple implementation of the new method. This implementation has been written using PROFIL/BIAS together with the PROFIL extensions. Some example programs which show some simple applications of the method are also contained in the package. On the one hand, the sample programs give a good example of how to use automatic differentiation. On the other hand, a little bit more sophisticated solutions where speed is of most interest are also presented. In the latter case, the user is free to write special subroutines for computing the derivatives. This is a brief description of the implementation of the global minimization method and its usage. It is neither intended to explain the specific implementation details of the package nor to explain the mathematical background. 2 CONTENTS Contents 1 Introduction 3 2 Installation 3 2.1 Installing the Library and Include Files : : : : : : :...

In many practical problems in which derivatives are calculated, their basic purpose is to be used in the modeling of a functional dependence, often based on a Taylor expansion to first or higher orders. While the practical computation of such derivatives is greatly facilitated and in many cases is possible only through the use of forward or reverse computational differentiation, there is usually no direct information regarding the accuracy of the functional model based on the Taylor expansion. We show how, in parallel to the accumulation of derivatives, error bounds of all functional dependencies can be carried along the computation. The additional effort is minor, and the resulting bounds are usually rather sharp, in particular at higher orders. This Remainder Differential Algebraic Method is more straightforward and can yield tighter bounds than the mere interval bounding of the Taylor remainder's (n + 1)st order derivative obtained via forward differentiation. The method can be appl...

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