An Introduction to Affine Arithmetic

Abstract

Abstract. Affine arithmetic (AA) is a model for self-validated computation which, like standard interval arithmetic (IA), produces guaranteed enclosures for computed quantities, taking into account any uncertainties in the input data as well as all internal truncation and roundoff errors. Unlike standard IA, the quantity representations used by AA are first-order approximations, whose error is generally quadratic in the width of input intervals. In many practical applications, the higher asymptotic accuracy of AA more than compensates for the increased cost of its operations.

Load Flow Analysis of a Power System Network in the Presence of Uncertainty using Complex Affine Arithmetic

“…Sources of error during manipulation may be external to the system like less precise and missing input data or a mathematical model that is uncertain by itself or it can be internal to the system. Truncation and round off errors are categorized under internal errors [17]. AA is first formalized and introduced in 1993 by Comba and Stolfi as a basic tool to deal with both internal and external sources of errors and other shortcomings of IA [16,17,20].…”

“…Both IA and AA based analysis are validated by comparing the result with probabilistic Monte Carlo approach. AA is the extension of IA which overcomes the problems associated with IA by considering round-off and truncation errors beside input errors [15]- [17]. Despite their advantage, a stochastic approach like Monte Carlo suffers from underestimation and deterministic approach like IA suffers from critical estimation.…”

Project Duration Evaluated Using Affine Arithmetic

“…The committed error in the affine form approximation depends quadratically on the extent of the input variable ranges, but some authors showed that it is essentially zero, if the function f depends on a single variable (Stolfi and de Figueiredo, 2003), as in Eq. (20).…”

“…However, it is impractical to define a rigorous method to handle this issue and that could be also easily implemented and understood by the operators involved in the operation. For this reason, in this paper the authors propose a different approach to manage the project uncertainties, described by lower and upper limits, using the Affine Arithmetic (AA) technique (Comba and Stolfi, 1993;Stolfi and de Figueiredo, 2003). This technique represents a novel analytical approach derived from the Interval Analysis (IA) (Moore, 2009).…”

Static Analysis of the Accuracy in Control Systems: Principles and Experiments

“…We thus proposed and implemented a relational domain, relying on affine arithmetic [5,22] for the computation of the floating-point value f x . Affine arithmetic uses affine correlation between real variables, and allows to get much tighter results than classical interval arithmetic (the concretisation forms zonotopes : center-symmetric bounded convex polytopes).…”