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The numerical methods employed in the solution of many scientiic computing problems require the computation of derivatives of a function f : R n ! R m. Both the accuracy and the computationalrequirements of the derivativecomputation are usually of critical importance for the robustness and speed of the numerical solution. ADIFOR (Automatic Diierentiation In… (More)

The formal process of the evaluation of derivatives using some of the various modern methods of computational diierentiation can be recognized as an example for the application of conventional \approximate" numerical techniques on a non-archimedean extension of the real numbers. In many cases, the application of \innnitely small" numbers instead of \small… (More)

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 rst 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… (More)

Equations Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long series. A portable translator program accepts statements of the system of differential equations and produces a portable FORTRAN object code which is then run to solve the system. At each… (More)

and his brother, who will arrive shortly. iii Acknowledgment I would like to thank Dr. Xin Feng for the encouragement, support, and direction he has provided during the past three years. His insightful suggestions, enthusiastic endorsement, and shrewd proverbs have made the completion of this research possible. They provide an example to emulate. I owe a… (More)

Compared to standard numerical methods for initial value problems (IVPs) for ordinary diierential equations (ODEs), validated methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. The authors survey… (More)

We describe algorithms for computing the greatest common divisor GCD of two univariate polynomials with inexactly-known coeecients. Assuming that an estimate for the GCD degree is available e.g., using an SVD-based algorithm, we formulate and solve a nonlinear optimization problem in order to determine the coeecients of the best" GCD. We discuss various… (More)

We apply interval techniques for global optimization to several industrial applications including Swiss Bank (currency trading), BancOne (portfolio management), MacNeal-Schwendler (finite element), GE Medical Systems (Magnetic resonance imaging), Genome Theraputics (gene prediction), inexact greatest common divisor computations from computer algebra, and… (More)

The numerical methods employed in the solution of many scientiic computing problems require the computation of derivatives of a function f : R n ! R m. Both the accuracy and the computationalrequirements of the derivativecomputation are usually of critical importance for the robustness and speed of the numerical solution. ADIFOR (Automatic Diierentiation In… (More)

Structural engineers use design codes formulated to consider uncertainty for both reinforced concrete and structural steel design. For a simple one-bay structural steel frame, we survey typical uncertainties and compute an interval solution for displacements and forces. The naive solutions have large over-estimations, so we explore the Mullen-Muhanna… (More)